Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
U711(mark(X1), X2) → U711(X1, X2)
ACTIVE(U32(X)) → ACTIVE(X)
PROPER(U52(X1, X2)) → U521(proper(X1), proper(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U52(tt, V2)) → U531(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → U711(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
ACTIVE(U61(tt, V1, V2)) → U621(isNat(V1), V2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U21(X1, X2)) → U211(active(X1), X2)
U321(ok(X)) → U321(X)
U311(mark(X1), X2) → U311(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
U621(ok(X1), ok(X2)) → U621(X1, X2)
ACTIVE(isNatList(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(take(s(M), cons(N, IL))) → U911(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
ACTIVE(isNatIListKind(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(U53(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(U63(X)) → ACTIVE(X)
U711(ok(X1), ok(X2)) → U711(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U71(X1, X2)) → U711(active(X1), X2)
ACTIVE(isNatList(take(V1, V2))) → U611(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
U121(ok(X)) → U121(X)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U431(ok(X)) → U431(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(U51(tt, V1, V2)) → U521(isNat(V1), V2)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
U631(ok(X)) → U631(X)
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILISTKIND(IL)
U311(ok(X1), ok(X2)) → U311(X1, X2)
ACTIVE(U11(tt, V1)) → U121(isNatList(V1))
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(U43(X)) → U431(active(X))
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PROPER(U42(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U121(mark(X)) → U121(X)
PROPER(U91(X1, X2, X3, X4)) → U911(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U61(X1, X2, X3)) → U611(proper(X1), proper(X2), proper(X3))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
U431(mark(X)) → U431(X)
ACTIVE(U91(tt, IL, M, N)) → TAKE(M, IL)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
U531(ok(X)) → U531(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U63(X)) → PROPER(X)
ACTIVE(U63(X)) → U631(active(X))
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNatKind(M))
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U81(X)) → ACTIVE(X)
PROPER(U81(X)) → U811(proper(X))
PROPER(U21(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatKind(N))
U111(ok(X1), ok(X2)) → U111(X1, X2)
ACTIVE(U62(tt, V2)) → U631(isNatIList(V2))
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U61(tt, V1, V2)) → ISNAT(V1)
PROPER(isNatIListKind(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNat(length(V1))) → U111(isNatIListKind(V1), V1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U62(X1, X2)) → U621(active(X1), X2)
ISNATKIND(ok(X)) → ISNATKIND(X)
U221(mark(X)) → U221(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(U41(tt, V1, V2)) → U421(isNat(V1), V2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U621(mark(X1), X2) → U621(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
U811(mark(X)) → U811(X)
PROPER(U42(X1, X2)) → U421(proper(X1), proper(X2))
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U321(active(X))
U521(ok(X1), ok(X2)) → U521(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(U61(X1, X2, X3)) → U611(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
U811(ok(X)) → U811(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → U621(proper(X1), proper(X2))
ACTIVE(take(0, IL)) → ISNATILISTKIND(IL)
U221(ok(X)) → U221(X)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(U53(X)) → ACTIVE(X)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2)) → U521(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
PROPER(isNatList(X)) → PROPER(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
U631(mark(X)) → U631(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
U211(mark(X1), X2) → U211(X1, X2)
ACTIVE(U53(X)) → U531(active(X))
ACTIVE(take(0, IL)) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U211(ok(X1), ok(X2)) → U211(X1, X2)
ACTIVE(U91(X1, X2, X3, X4)) → U911(active(X1), X2, X3, X4)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U81(X)) → U811(active(X))
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(length(X)) → PROPER(X)
PROPER(U63(X)) → U631(proper(X))
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(U71(tt, L)) → LENGTH(L)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
PROPER(U32(X)) → U321(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(take(0, IL)) → U811(and(isNatIList(IL), isNatIListKind(IL)))
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ACTIVE(s(X)) → ACTIVE(X)
U321(mark(X)) → U321(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
TOP(mark(X)) → PROPER(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
PROPER(U22(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(N)
PROPER(isNatKind(X)) → PROPER(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(U43(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), isNatIListKind(IL))
PROPER(U61(X1, X2, X3)) → PROPER(X2)
AND(mark(X1), X2) → AND(X1, X2)
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
PROPER(isNatIListKind(X)) → ISNATILISTKIND(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
ACTIVE(U71(tt, L)) → S(length(L))
PROPER(U21(X1, X2)) → U211(proper(X1), proper(X2))
U531(mark(X)) → U531(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(M)
ACTIVE(isNatIList(cons(V1, V2))) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U121(proper(X))
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2) → U421(X1, X2)
U421(ok(X1), ok(X2)) → U421(X1, X2)
ACTIVE(U22(X)) → U221(active(X))
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → U531(proper(X))
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
ACTIVE(U42(X1, X2)) → U421(active(X1), X2)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
ACTIVE(U91(tt, IL, M, N)) → CONS(N, take(M, IL))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(tt, V2)) → U431(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
PROPER(U71(X1, X2)) → U711(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X2)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
PROPER(U52(X1, X2)) → PROPER(X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
ACTIVE(U62(tt, V2)) → ISNATILIST(V2)
U111(mark(X1), X2) → U111(X1, X2)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(U52(X1, X2)) → PROPER(X1)
ACTIVE(U31(tt, V)) → U321(isNatList(V))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
PROPER(s(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → U311(isNatIListKind(V), V)
PROPER(take(X1, X2)) → PROPER(X2)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → U431(proper(X))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
PROPER(U22(X)) → U221(proper(X))
TOP(mark(X)) → TOP(proper(X))
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
U711(mark(X1), X2) → U711(X1, X2)
ACTIVE(U32(X)) → ACTIVE(X)
PROPER(U52(X1, X2)) → U521(proper(X1), proper(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U52(tt, V2)) → U531(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → U711(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
ACTIVE(U61(tt, V1, V2)) → U621(isNat(V1), V2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U21(X1, X2)) → U211(active(X1), X2)
U321(ok(X)) → U321(X)
U311(mark(X1), X2) → U311(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
U621(ok(X1), ok(X2)) → U621(X1, X2)
ACTIVE(isNatList(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(take(s(M), cons(N, IL))) → U911(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
ACTIVE(isNatIListKind(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(U53(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(U63(X)) → ACTIVE(X)
U711(ok(X1), ok(X2)) → U711(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U71(X1, X2)) → U711(active(X1), X2)
ACTIVE(isNatList(take(V1, V2))) → U611(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
U121(ok(X)) → U121(X)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U431(ok(X)) → U431(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(U51(tt, V1, V2)) → U521(isNat(V1), V2)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
U631(ok(X)) → U631(X)
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILISTKIND(IL)
U311(ok(X1), ok(X2)) → U311(X1, X2)
ACTIVE(U11(tt, V1)) → U121(isNatList(V1))
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(U43(X)) → U431(active(X))
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PROPER(U42(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U121(mark(X)) → U121(X)
PROPER(U91(X1, X2, X3, X4)) → U911(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U61(X1, X2, X3)) → U611(proper(X1), proper(X2), proper(X3))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
U431(mark(X)) → U431(X)
ACTIVE(U91(tt, IL, M, N)) → TAKE(M, IL)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
U531(ok(X)) → U531(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U63(X)) → PROPER(X)
ACTIVE(U63(X)) → U631(active(X))
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNatKind(M))
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U81(X)) → ACTIVE(X)
PROPER(U81(X)) → U811(proper(X))
PROPER(U21(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatKind(N))
U111(ok(X1), ok(X2)) → U111(X1, X2)
ACTIVE(U62(tt, V2)) → U631(isNatIList(V2))
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U61(tt, V1, V2)) → ISNAT(V1)
PROPER(isNatIListKind(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNat(length(V1))) → U111(isNatIListKind(V1), V1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U62(X1, X2)) → U621(active(X1), X2)
ISNATKIND(ok(X)) → ISNATKIND(X)
U221(mark(X)) → U221(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(U41(tt, V1, V2)) → U421(isNat(V1), V2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U621(mark(X1), X2) → U621(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
U811(mark(X)) → U811(X)
PROPER(U42(X1, X2)) → U421(proper(X1), proper(X2))
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U321(active(X))
U521(ok(X1), ok(X2)) → U521(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(U61(X1, X2, X3)) → U611(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
U811(ok(X)) → U811(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → U621(proper(X1), proper(X2))
ACTIVE(take(0, IL)) → ISNATILISTKIND(IL)
U221(ok(X)) → U221(X)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(U53(X)) → ACTIVE(X)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2)) → U521(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
PROPER(isNatList(X)) → PROPER(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
U631(mark(X)) → U631(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
U211(mark(X1), X2) → U211(X1, X2)
ACTIVE(U53(X)) → U531(active(X))
ACTIVE(take(0, IL)) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U211(ok(X1), ok(X2)) → U211(X1, X2)
ACTIVE(U91(X1, X2, X3, X4)) → U911(active(X1), X2, X3, X4)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U81(X)) → U811(active(X))
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(length(X)) → PROPER(X)
PROPER(U63(X)) → U631(proper(X))
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(U71(tt, L)) → LENGTH(L)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
PROPER(U32(X)) → U321(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(take(0, IL)) → U811(and(isNatIList(IL), isNatIListKind(IL)))
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ACTIVE(s(X)) → ACTIVE(X)
U321(mark(X)) → U321(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
TOP(mark(X)) → PROPER(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
PROPER(U22(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(N)
PROPER(isNatKind(X)) → PROPER(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(U43(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), isNatIListKind(IL))
PROPER(U61(X1, X2, X3)) → PROPER(X2)
AND(mark(X1), X2) → AND(X1, X2)
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
PROPER(isNatIListKind(X)) → ISNATILISTKIND(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
ACTIVE(U71(tt, L)) → S(length(L))
PROPER(U21(X1, X2)) → U211(proper(X1), proper(X2))
U531(mark(X)) → U531(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(M)
ACTIVE(isNatIList(cons(V1, V2))) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U121(proper(X))
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2) → U421(X1, X2)
U421(ok(X1), ok(X2)) → U421(X1, X2)
ACTIVE(U22(X)) → U221(active(X))
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → U531(proper(X))
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
ACTIVE(U42(X1, X2)) → U421(active(X1), X2)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
ACTIVE(U91(tt, IL, M, N)) → CONS(N, take(M, IL))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(tt, V2)) → U431(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
PROPER(U71(X1, X2)) → U711(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X2)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
PROPER(U52(X1, X2)) → PROPER(X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
ACTIVE(U62(tt, V2)) → ISNATILIST(V2)
U111(mark(X1), X2) → U111(X1, X2)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(U52(X1, X2)) → PROPER(X1)
ACTIVE(U31(tt, V)) → U321(isNatList(V))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
PROPER(s(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → U311(isNatIListKind(V), V)
PROPER(take(X1, X2)) → PROPER(X2)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → U431(proper(X))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
PROPER(U22(X)) → U221(proper(X))
TOP(mark(X)) → TOP(proper(X))
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 31 SCCs with 127 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(ok(X)) → ISNATKIND(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(ok(X)) → ISNATKIND(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ISNATKIND(ok(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(ok(X)) → ISNATILIST(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(ok(X)) → ISNATILIST(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ISNATILIST(ok(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(ok(X)) → ISNAT(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(ok(X)) → ISNAT(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ISNAT(ok(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(ok(X)) → ISNATLIST(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(ok(X)) → ISNATLIST(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ISNATLIST(ok(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
AND(mark(X1), X2) → AND(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
AND(mark(X1), X2) → AND(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- AND(ok(X1), ok(X2)) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
- TAKE(mark(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- TAKE(X1, mark(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 4 > 4
- U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U811(mark(X)) → U811(X)
U811(ok(X)) → U811(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U811(ok(X)) → U811(X)
U811(mark(X)) → U811(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U811(mark(X)) → U811(X)
The graph contains the following edges 1 > 1
- U811(ok(X)) → U811(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- LENGTH(ok(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
- LENGTH(mark(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(ok(X)) → S(X)
S(mark(X)) → S(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(ok(X)) → S(X)
S(mark(X)) → S(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- S(ok(X)) → S(X)
The graph contains the following edges 1 > 1
- S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U711(mark(X1), X2) → U711(X1, X2)
U711(ok(X1), ok(X2)) → U711(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U711(mark(X1), X2) → U711(X1, X2)
U711(ok(X1), ok(X2)) → U711(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U711(mark(X1), X2) → U711(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U711(ok(X1), ok(X2)) → U711(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U631(ok(X)) → U631(X)
U631(mark(X)) → U631(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U631(ok(X)) → U631(X)
U631(mark(X)) → U631(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U631(ok(X)) → U631(X)
The graph contains the following edges 1 > 1
- U631(mark(X)) → U631(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U621(mark(X1), X2) → U621(X1, X2)
U621(ok(X1), ok(X2)) → U621(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U621(mark(X1), X2) → U621(X1, X2)
U621(ok(X1), ok(X2)) → U621(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U621(mark(X1), X2) → U621(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U621(ok(X1), ok(X2)) → U621(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U611(mark(X1), X2, X3) → U611(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
- U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U531(mark(X)) → U531(X)
U531(ok(X)) → U531(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U531(mark(X)) → U531(X)
U531(ok(X)) → U531(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U531(mark(X)) → U531(X)
The graph contains the following edges 1 > 1
- U531(ok(X)) → U531(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U521(mark(X1), X2) → U521(X1, X2)
U521(ok(X1), ok(X2)) → U521(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U521(mark(X1), X2) → U521(X1, X2)
U521(ok(X1), ok(X2)) → U521(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U521(mark(X1), X2) → U521(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U521(ok(X1), ok(X2)) → U521(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U511(mark(X1), X2, X3) → U511(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
- U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U431(mark(X)) → U431(X)
U431(ok(X)) → U431(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U431(ok(X)) → U431(X)
U431(mark(X)) → U431(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U431(mark(X)) → U431(X)
The graph contains the following edges 1 > 1
- U431(ok(X)) → U431(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U421(mark(X1), X2) → U421(X1, X2)
U421(ok(X1), ok(X2)) → U421(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U421(mark(X1), X2) → U421(X1, X2)
U421(ok(X1), ok(X2)) → U421(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U421(mark(X1), X2) → U421(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U421(ok(X1), ok(X2)) → U421(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
- U411(mark(X1), X2, X3) → U411(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U321(ok(X)) → U321(X)
U321(mark(X)) → U321(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U321(ok(X)) → U321(X)
U321(mark(X)) → U321(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U321(ok(X)) → U321(X)
The graph contains the following edges 1 > 1
- U321(mark(X)) → U321(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U311(mark(X1), X2) → U311(X1, X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U311(mark(X1), X2) → U311(X1, X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U311(mark(X1), X2) → U311(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U311(ok(X1), ok(X2)) → U311(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U221(mark(X)) → U221(X)
U221(ok(X)) → U221(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U221(mark(X)) → U221(X)
U221(ok(X)) → U221(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U221(mark(X)) → U221(X)
The graph contains the following edges 1 > 1
- U221(ok(X)) → U221(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(ok(X1), ok(X2)) → U211(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(ok(X1), ok(X2)) → U211(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U211(ok(X1), ok(X2)) → U211(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
- U211(mark(X1), X2) → U211(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U121(mark(X)) → U121(X)
U121(ok(X)) → U121(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U121(mark(X)) → U121(X)
U121(ok(X)) → U121(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U121(mark(X)) → U121(X)
The graph contains the following edges 1 > 1
- U121(ok(X)) → U121(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U111(mark(X1), X2) → U111(X1, X2)
U111(ok(X1), ok(X2)) → U111(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U111(ok(X1), ok(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- U111(mark(X1), X2) → U111(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U111(ok(X1), ok(X2)) → U111(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(isNat(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U32(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U53(X)) → PROPER(X)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U81(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PROPER(isNat(X)) → PROPER(X)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U12(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U32(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U81(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- PROPER(U71(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U52(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U51(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U42(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U12(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(isNatIList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U62(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(isNatList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U31(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U22(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(take(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U63(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(isNatKind(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U61(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U41(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U32(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U21(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(and(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
The graph contains the following edges 1 > 1
- PROPER(U11(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(length(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U31(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U41(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
- PROPER(U43(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
The graph contains the following edges 1 > 1
- PROPER(U61(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
- PROPER(U71(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U61(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(and(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U51(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
- PROPER(isNatIListKind(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(take(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U53(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U52(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U81(X)) → PROPER(X)
The graph contains the following edges 1 > 1
- PROPER(U41(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U21(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
- PROPER(U62(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U42(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U11(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
- PROPER(U51(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- ACTIVE(U31(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U81(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(take(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
- ACTIVE(U12(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U43(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U22(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U32(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U71(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U21(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(and(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U53(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(length(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U11(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U42(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U52(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(take(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U63(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
- ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
- ACTIVE(U62(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
Q DP problem:
The TRS P consists of the following rules:
TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = 2·x1
POL(U11(x1, x2)) = x1 + 2·x2
POL(U12(x1)) = 2·x1
POL(U21(x1, x2)) = x1 + x2
POL(U22(x1)) = 2·x1
POL(U31(x1, x2)) = x1 + 2·x2
POL(U32(x1)) = x1
POL(U41(x1, x2, x3)) = 2·x1 + x2 + x3
POL(U42(x1, x2)) = x1 + x2
POL(U43(x1)) = x1
POL(U51(x1, x2, x3)) = x1 + x2 + x3
POL(U52(x1, x2)) = x1 + 2·x2
POL(U53(x1)) = x1
POL(U61(x1, x2, x3)) = x1 + x2 + x3
POL(U62(x1, x2)) = x1 + x2
POL(U63(x1)) = x1
POL(U71(x1, x2)) = 2·x1 + 2·x2
POL(U81(x1)) = x1
POL(U91(x1, x2, x3, x4)) = 2·x1 + 2·x2 + 2·x3 + x4
POL(active(x1)) = 2·x1
POL(and(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(isNat(x1)) = x1
POL(isNatIList(x1)) = 2·x1
POL(isNatIListKind(x1)) = x1
POL(isNatKind(x1)) = 2·x1
POL(isNatList(x1)) = x1
POL(length(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = 2·x1
POL(proper(x1)) = x1
POL(s(x1)) = 2·x1
POL(take(x1, x2)) = 2·x1 + 2·x2
POL(tt) = 0
POL(zeros) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(X)) → TOP(active(X))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U43(tt))) → TOP(mark(tt))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(isNatIListKind(zeros))) → TOP(mark(tt))
TOP(ok(U53(tt))) → TOP(mark(tt))
TOP(ok(isNatIListKind(nil))) → TOP(mark(tt))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(U63(tt))) → TOP(mark(tt))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(U22(tt))) → TOP(mark(tt))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(U81(tt))) → TOP(mark(nil))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U43(tt))) → TOP(mark(tt))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(U53(tt))) → TOP(mark(tt))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(U63(tt))) → TOP(mark(tt))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(U22(tt))) → TOP(mark(tt))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(isNatIListKind(zeros))) → TOP(mark(tt))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNatIListKind(nil))) → TOP(mark(tt))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U81(tt))) → TOP(mark(nil))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 17 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 1
POL(U12(x1)) = 1
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 1
POL(U31(x1, x2)) = 1
POL(U32(x1)) = 1
POL(U41(x1, x2, x3)) = 1
POL(U42(x1, x2)) = 1
POL(U43(x1)) = 1
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 1
POL(U53(x1)) = 1
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 1
POL(U63(x1)) = 1
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = 1
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 1
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = 1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 1
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 1
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 1 + x3
POL(U42(x1, x2)) = x2
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 1
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1 + x1
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 1
POL(U12(x1)) = 1
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 1
POL(U31(x1, x2)) = 1
POL(U32(x1)) = 1
POL(U41(x1, x2, x3)) = 1
POL(U42(x1, x2)) = 1
POL(U43(x1)) = 1
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 1
POL(U53(x1)) = 1
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 1
POL(U63(x1)) = 1
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = 1
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 1
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = 1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 1
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = x1
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 1
POL(U12(x1)) = 1
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 1
POL(U31(x1, x2)) = 1
POL(U32(x1)) = 1
POL(U41(x1, x2, x3)) = 1
POL(U42(x1, x2)) = 1
POL(U43(x1)) = 1
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 1
POL(U53(x1)) = 1
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 1
POL(U63(x1)) = 1
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = 1
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 1
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = 1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 1
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = x1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = x2
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = x3
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = x3
POL(U62(x1, x2)) = x2
POL(U63(x1)) = 0
POL(U71(x1, x2)) = x2
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1 + x2
POL(active(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x2
POL(isNat(x1)) = x1
POL(isNatIList(x1)) = 1 + x1
POL(isNatIListKind(x1)) = x1
POL(isNatKind(x1)) = x1
POL(isNatList(x1)) = x1
POL(length(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 1
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 1
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 1
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = x1
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 1
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 1
POL(U12(x1)) = 1
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 1
POL(U31(x1, x2)) = 1
POL(U32(x1)) = 1
POL(U41(x1, x2, x3)) = 1
POL(U42(x1, x2)) = 1
POL(U43(x1)) = 1
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 1
POL(U53(x1)) = 1
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 1
POL(U63(x1)) = 1
POL(U71(x1, x2)) = 1
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = 1
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = 1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 1
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(and(tt, x0))) → TOP(mark(x0))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = 1 + x2
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(and(x1, x2)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
The remaining pairs can at least be oriented weakly.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 1
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 0
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
The remaining pairs can at least be oriented weakly.
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(TOP(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2, x3)) = 1
POL(U62(x1, x2)) = 0
POL(U63(x1)) = 0
POL(U71(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(and(x1, x2)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
The remaining pairs can at least be oriented weakly.
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U52(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U62(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U41(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatIListKind(x1) ) = | | + | | · | x1 |
| M( U71(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U21(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( U51(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U91(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( U31(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U42(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U11(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatKind(x1) ) = | | + | | · | x1 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.