Termination w.r.t. Q proof of /tmp/tmpglkMcE/LengthOfFiniteLists_complete

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U61(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U22(X)) → A__U22(mark(X))
A__U31(tt, V) → A__U32(a__isNatList(V))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U61(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U22(X)) → A__U22(mark(X))
A__U31(tt, V) → A__U32(a__isNatList(V))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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 11 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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.

MARK(length(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATILISTKIND(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = 0
POL(A__U41(x₁, x₂, x₃)) = 0
POL(A__U42(x₁, x₂)) = 0
POL(A__U51(x₁, x₂, x₃)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(A__U61(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁
POL(a__U12(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₃
POL(a__U42(x₁, x₂)) = x₁ + x₂
POL(a__U43(x₁)) = x₁
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U52(x₁, x₂)) = x₁
POL(a__U53(x₁)) = x₁
POL(a__U61(x₁, x₂)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1 + x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__isNatKind(X) → isNatKind(X)
a__zeros → cons(0, zeros)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIList(zeros) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U43(tt) → tt
a__U53(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U32(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatKind(0) → tt
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(length(X)) → a__length(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0) → 0
mark(tt) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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.

MARK(cons(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U53(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(and(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATILISTKIND(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__U11(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = 0
POL(A__U41(x₁, x₂, x₃)) = x₃
POL(A__U42(x₁, x₂)) = x₂
POL(A__U51(x₁, x₂, x₃)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = 1 + x₁
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₁
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = 1 + x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂
POL(U52(x₁, x₂)) = 1 + x₁ + x₂
POL(U53(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂ + max(x₁, x₂)
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__AND(tt, X) → MARK(X) we obtained the following new rules:

A__AND(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
A__U21(tt, V1) → A__ISNAT(V1)
A__AND(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 15 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                    ↳ QDP

                                    ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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.

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))

The remaining pairs can at least be oriented weakly.

A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1 + x₂
POL(A__ISNATILISTKIND(x₁)) = 1 + x₁
POL(A__ISNATKIND(x₁)) = 1 + x₁
POL(MARK(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)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                    ↳ QDP

                                    ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ QDPSizeChangeProof

                                    ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U51(tt, V1, V2) → A__ISNAT(V1)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

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:

A__U52(tt, V2) → A__ISNATLIST(V2)
The graph contains the following edges 2 >= 1
A__U11(tt, V1) → A__ISNATLIST(V1)
The graph contains the following edges 2 >= 1
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
A__U51(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U21(tt, V1) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
The graph contains the following edges 1 > 2
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
The graph contains the following edges 1 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ UsableRulesProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ UsableRulesProof

                                        ↳ QDP

                                          ↳ QDPSizeChangeProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(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:

MARK(U11(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U51(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U42(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U61(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(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.

A__U61(tt, L) → A__LENGTH(mark(L))

The remaining pairs can at least be oriented weakly.

A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( a__U53(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( a__U42(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( a__zeros ) =

/	0	\
\	0	/

M( a__U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U41(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( mark(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( a__isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( a__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	1	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( a__U21(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U12(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__U43(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

M( nil ) =

/	1	\
\	1	/

M( a__U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( a__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( U42(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__U12(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U43(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

M( a__U32(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( a__U41(x₁, ..., x₃) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__isNatKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( a__U31(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U53(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U32(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( a__isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a__U22(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( A__LENGTH(x₁) ) =

[

]

x₁

M( A__U61(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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:

a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__isNatKind(X) → isNatKind(X)
a__zeros → cons(0, zeros)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIList(zeros) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U43(tt) → tt
a__U53(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U32(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatKind(0) → tt
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(length(X)) → a__length(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0) → 0
mark(tt) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.