Termination w.r.t. Q proof of /tmp/tmpegbcc-/MYNAT_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:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
ACTIVE(U22(tt)) → MARK(tt)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U22(X)) → U22¹(mark(X))
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(U32(tt, V2)) → U33¹(isNat(V2))
U12¹(active(X1), X2) → U12¹(X1, X2)
MARK(U31(X1, X2, X3)) → MARK(X1)
U12¹(X1, mark(X2)) → U12¹(X1, X2)
U33¹(active(X)) → U33¹(X)
ACTIVE(U41(tt, N)) → MARK(N)
X(active(X1), X2) → X(X1, X2)
MARK(U41(X1, X2)) → MARK(X1)
ACTIVE(plus(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(U12(tt, V2)) → ISNAT(V2)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
ACTIVE(plus(N, s(M))) → AND(isNat(N), isNatKind(N))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
ACTIVE(plus(N, 0)) → ISNATKIND(N)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U13(X)) → MARK(X)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V2)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
PLUS(X1, active(X2)) → PLUS(X1, X2)
S(mark(X)) → S(X)
U13¹(mark(X)) → U13¹(X)
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
ACTIVE(U13(tt)) → MARK(tt)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
U41¹(mark(X1), X2) → U41¹(X1, X2)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(x(N, 0)) → U61¹(and(isNat(N), isNatKind(N)))
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V2)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V1, V2)) → ISNAT(V1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
S(active(X)) → S(X)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
ACTIVE(plus(N, 0)) → U41¹(and(isNat(N), isNatKind(N)), N)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
AND(X1, active(X2)) → AND(X1, X2)
MARK(U51(X1, X2, X3)) → U51¹(mark(X1), X2, X3)
AND(active(X1), X2) → AND(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
U22¹(mark(X)) → U22¹(X)
ACTIVE(U21(tt, V1)) → U22¹(isNat(V1))
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
ACTIVE(x(N, 0)) → AND(isNat(N), isNatKind(N))
PLUS(active(X1), X2) → PLUS(X1, X2)
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V2)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
ACTIVE(x(N, s(M))) → U71¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U32¹(mark(X1), X2) → U32¹(X1, X2)
MARK(U61(X)) → U61¹(mark(X))
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
MARK(U32(X1, X2)) → MARK(X1)
U32¹(active(X1), X2) → U32¹(X1, X2)
MARK(and(X1, X2)) → AND(mark(X1), X2)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
X(X1, active(X2)) → X(X1, X2)
ACTIVE(plus(N, s(M))) → ISNATKIND(N)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(plus(N, s(M))) → ISNATKIND(M)
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(plus(N, s(M))) → ISNAT(N)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(x(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(U51(tt, M, N)) → S(plus(N, M))
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
ACTIVE(U12(tt, V2)) → U13¹(isNat(V2))
ISNATKIND(active(X)) → ISNATKIND(X)
ACTIVE(isNat(plus(V1, V2))) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ACTIVE(U51(tt, M, N)) → PLUS(N, M)
ACTIVE(x(N, s(M))) → ISNAT(N)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(isNat(x(V1, V2))) → U31¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
ACTIVE(plus(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
X(X1, mark(X2)) → X(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
U22¹(active(X)) → U22¹(X)
MARK(x(X1, X2)) → MARK(X2)
ACTIVE(and(tt, X)) → MARK(X)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)
MARK(s(X)) → MARK(X)
ACTIVE(plus(N, 0)) → AND(isNat(N), isNatKind(N))
MARK(U41(X1, X2)) → U41¹(mark(X1), X2)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(U33(tt)) → MARK(tt)
X(mark(X1), X2) → X(X1, X2)
ACTIVE(isNat(0)) → MARK(tt)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U71¹(mark(X1), X2, X3)
ACTIVE(U71(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(isNat(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
U32¹(X1, mark(X2)) → U32¹(X1, X2)
ACTIVE(x(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U13¹(active(X)) → U13¹(X)
MARK(plus(X1, X2)) → MARK(X1)
MARK(U13(X)) → U13¹(mark(X))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U61(X)) → MARK(X)
AND(mark(X1), X2) → AND(X1, X2)
MARK(U33(X)) → MARK(X)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U61¹(mark(X)) → U61¹(X)
ACTIVE(U61(tt)) → MARK(0)
MARK(tt) → ACTIVE(tt)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(x(N, s(M))) → ISNATKIND(N)
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(U31(tt, V1, V2)) → U32¹(isNat(V1), V2)
ACTIVE(x(N, 0)) → ISNATKIND(N)
MARK(x(X1, X2)) → MARK(X1)
ACTIVE(U32(tt, V2)) → ISNAT(V2)
U41¹(active(X1), X2) → U41¹(X1, X2)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isNat(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
U33¹(mark(X)) → U33¹(X)
ACTIVE(plus(N, s(M))) → U51¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ACTIVE(x(N, s(M))) → AND(isNat(N), isNatKind(N))
U41¹(X1, active(X2)) → U41¹(X1, X2)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V2)
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
ACTIVE(x(N, s(M))) → ISNATKIND(M)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
ACTIVE(U31(tt, V1, V2)) → ISNAT(V1)
U21¹(X1, active(X2)) → U21¹(X1, X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
ACTIVE(U11(tt, V1, V2)) → U12¹(isNat(V1), V2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U31(X1, X2, X3)) → U31¹(mark(X1), X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
ACTIVE(x(N, 0)) → ISNAT(N)
ACTIVE(U71(tt, M, N)) → X(N, M)
ACTIVE(isNatKind(0)) → MARK(tt)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U61¹(active(X)) → U61¹(X)
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V1)
MARK(0) → ACTIVE(0)
MARK(U33(X)) → U33¹(mark(X))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
ACTIVE(isNatKind(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
ACTIVE(U22(tt)) → MARK(tt)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(U22(X)) → U22¹(mark(X))
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(U32(tt, V2)) → U33¹(isNat(V2))
U12¹(active(X1), X2) → U12¹(X1, X2)
MARK(U31(X1, X2, X3)) → MARK(X1)
U12¹(X1, mark(X2)) → U12¹(X1, X2)
U33¹(active(X)) → U33¹(X)
ACTIVE(U41(tt, N)) → MARK(N)
X(active(X1), X2) → X(X1, X2)
MARK(U41(X1, X2)) → MARK(X1)
ACTIVE(plus(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(U12(tt, V2)) → ISNAT(V2)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
ACTIVE(plus(N, s(M))) → AND(isNat(N), isNatKind(N))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
ACTIVE(plus(N, 0)) → ISNATKIND(N)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U13(X)) → MARK(X)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V2)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
PLUS(X1, active(X2)) → PLUS(X1, X2)
S(mark(X)) → S(X)
U13¹(mark(X)) → U13¹(X)
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
ACTIVE(U13(tt)) → MARK(tt)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
U41¹(mark(X1), X2) → U41¹(X1, X2)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(x(N, 0)) → U61¹(and(isNat(N), isNatKind(N)))
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V2)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V1, V2)) → ISNAT(V1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
S(active(X)) → S(X)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
ACTIVE(plus(N, 0)) → U41¹(and(isNat(N), isNatKind(N)), N)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
AND(X1, active(X2)) → AND(X1, X2)
MARK(U51(X1, X2, X3)) → U51¹(mark(X1), X2, X3)
AND(active(X1), X2) → AND(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
U22¹(mark(X)) → U22¹(X)
ACTIVE(U21(tt, V1)) → U22¹(isNat(V1))
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
ACTIVE(x(N, 0)) → AND(isNat(N), isNatKind(N))
PLUS(active(X1), X2) → PLUS(X1, X2)
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V2)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
ACTIVE(x(N, s(M))) → U71¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U32¹(mark(X1), X2) → U32¹(X1, X2)
MARK(U61(X)) → U61¹(mark(X))
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
MARK(U32(X1, X2)) → MARK(X1)
U32¹(active(X1), X2) → U32¹(X1, X2)
MARK(and(X1, X2)) → AND(mark(X1), X2)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
X(X1, active(X2)) → X(X1, X2)
ACTIVE(plus(N, s(M))) → ISNATKIND(N)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(plus(N, s(M))) → ISNATKIND(M)
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(plus(N, s(M))) → ISNAT(N)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(x(N, s(M))) → AND(isNat(M), isNatKind(M))
ACTIVE(U51(tt, M, N)) → S(plus(N, M))
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
ACTIVE(U12(tt, V2)) → U13¹(isNat(V2))
ISNATKIND(active(X)) → ISNATKIND(X)
ACTIVE(isNat(plus(V1, V2))) → U11¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ACTIVE(U51(tt, M, N)) → PLUS(N, M)
ACTIVE(x(N, s(M))) → ISNAT(N)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(isNat(x(V1, V2))) → U31¹(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
ACTIVE(plus(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
X(X1, mark(X2)) → X(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
U22¹(active(X)) → U22¹(X)
MARK(x(X1, X2)) → MARK(X2)
ACTIVE(and(tt, X)) → MARK(X)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)
MARK(s(X)) → MARK(X)
ACTIVE(plus(N, 0)) → AND(isNat(N), isNatKind(N))
MARK(U41(X1, X2)) → U41¹(mark(X1), X2)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(U33(tt)) → MARK(tt)
X(mark(X1), X2) → X(X1, X2)
ACTIVE(isNat(0)) → MARK(tt)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U71¹(mark(X1), X2, X3)
ACTIVE(U71(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(isNat(x(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
U32¹(X1, mark(X2)) → U32¹(X1, X2)
ACTIVE(x(N, s(M))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U13¹(active(X)) → U13¹(X)
MARK(plus(X1, X2)) → MARK(X1)
MARK(U13(X)) → U13¹(mark(X))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U61(X)) → MARK(X)
AND(mark(X1), X2) → AND(X1, X2)
MARK(U33(X)) → MARK(X)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U61¹(mark(X)) → U61¹(X)
ACTIVE(U61(tt)) → MARK(0)
MARK(tt) → ACTIVE(tt)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(x(N, s(M))) → ISNATKIND(N)
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(U31(tt, V1, V2)) → U32¹(isNat(V1), V2)
ACTIVE(x(N, 0)) → ISNATKIND(N)
MARK(x(X1, X2)) → MARK(X1)
ACTIVE(U32(tt, V2)) → ISNAT(V2)
U41¹(active(X1), X2) → U41¹(X1, X2)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isNat(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))
U33¹(mark(X)) → U33¹(X)
ACTIVE(plus(N, s(M))) → U51¹(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ACTIVE(x(N, s(M))) → AND(isNat(N), isNatKind(N))
U41¹(X1, active(X2)) → U41¹(X1, X2)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V2)
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
ACTIVE(x(N, s(M))) → ISNATKIND(M)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
ACTIVE(U31(tt, V1, V2)) → ISNAT(V1)
U21¹(X1, active(X2)) → U21¹(X1, X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
ACTIVE(U11(tt, V1, V2)) → U12¹(isNat(V1), V2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(U31(X1, X2, X3)) → U31¹(mark(X1), X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
ACTIVE(x(N, 0)) → ISNAT(N)
ACTIVE(U71(tt, M, N)) → X(N, M)
ACTIVE(isNatKind(0)) → MARK(tt)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U61¹(active(X)) → U61¹(X)
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V1)
MARK(0) → ACTIVE(0)
MARK(U33(X)) → U33¹(mark(X))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
ACTIVE(isNatKind(plus(V1, V2))) → AND(isNatKind(V1), isNatKind(V2))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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 19 SCCs with 79 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ISNATKIND(active(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1
ISNATKIND(mark(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

From the DPs we obtained the following set of size-change graphs:

AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(active(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(X1, mark(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AND(X1, active(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

X(X1, active(X2)) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

X(X1, active(X2)) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)

From the DPs we obtained the following set of size-change graphs:

X(X1, active(X2)) → X(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
X(X1, mark(X2)) → X(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
X(mark(X1), X2) → X(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
X(active(X1), X2) → X(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U61¹(mark(X)) → U61¹(X)
U61¹(active(X)) → U61¹(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U61¹(mark(X)) → U61¹(X)
U61¹(active(X)) → U61¹(X)

From the DPs we obtained the following set of size-change graphs:

U61¹(mark(X)) → U61¹(X)
The graph contains the following edges 1 > 1
U61¹(active(X)) → U61¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

PLUS(active(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(mark(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(X1, mark(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
PLUS(X1, active(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U41¹(X1, active(X2)) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U41¹(X1, active(X2)) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U41¹(X1, active(X2)) → U41¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U41¹(active(X1), X2) → U41¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U41¹(X1, mark(X2)) → U41¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U41¹(mark(X1), X2) → U41¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U33¹(active(X)) → U33¹(X)
U33¹(mark(X)) → U33¹(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U33¹(active(X)) → U33¹(X)
U33¹(mark(X)) → U33¹(X)

From the DPs we obtained the following set of size-change graphs:

U33¹(active(X)) → U33¹(X)
The graph contains the following edges 1 > 1
U33¹(mark(X)) → U33¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U32¹(active(X1), X2) → U32¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U32¹(X1, mark(X2)) → U32¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U32¹(mark(X1), X2) → U32¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U32¹(X1, active(X2)) → U32¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U22¹(active(X)) → U22¹(X)
U22¹(mark(X)) → U22¹(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U22¹(active(X)) → U22¹(X)
U22¹(mark(X)) → U22¹(X)

From the DPs we obtained the following set of size-change graphs:

U22¹(active(X)) → U22¹(X)
The graph contains the following edges 1 > 1
U22¹(mark(X)) → U22¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U21¹(X1, active(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U21¹(X1, active(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U21¹(X1, active(X2)) → U21¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U21¹(active(X1), X2) → U21¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U21¹(X1, mark(X2)) → U21¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U21¹(mark(X1), X2) → U21¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U13¹(active(X)) → U13¹(X)
U13¹(mark(X)) → U13¹(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U13¹(active(X)) → U13¹(X)
U13¹(mark(X)) → U13¹(X)

From the DPs we obtained the following set of size-change graphs:

U13¹(active(X)) → U13¹(X)
The graph contains the following edges 1 > 1
U13¹(mark(X)) → U13¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

From the DPs we obtained the following set of size-change graphs:

ISNAT(active(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
ISNAT(mark(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U12¹(X1, mark(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(mark(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U12¹(X1, active(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(active(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(x(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → ACTIVE(U13(mark(X)))
MARK(s(X)) → MARK(X)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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(U33(X)) → ACTIVE(U33(mark(X)))
MARK(U61(X)) → ACTIVE(U61(mark(X)))
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.

ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(x(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 1
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 1
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(tt) = 0
POL(x(x₁, x₂)) = 1

The following usable rules [17] were oriented:

U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U32(X1, mark(X2)) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U33(active(X)) → U33(X)
U33(mark(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U13(active(X)) → U13(X)
U13(mark(X)) → U13(X)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
x(X1, active(X2)) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U61(active(X)) → U61(X)
U61(mark(X)) → U61(X)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(x(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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.

ACTIVE(plus(N, s(M))) → MARK(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(isNatKind(x(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNat(V1), V2))
MARK(x(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(and(isNatKind(V1), isNatKind(V2)))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(U71(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U41(tt, N)) → MARK(N)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(U32(tt, V2)) → MARK(U33(isNat(V2)))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNat(V1), V2))
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(U51(tt, M, N)) → MARK(s(plus(N, M)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
ACTIVE(x(N, s(M))) → MARK(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
MARK(U32(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(plus(N, 0)) → MARK(U41(and(isNat(N), isNatKind(N)), N))
ACTIVE(U12(tt, V2)) → MARK(U13(isNat(V2)))
ACTIVE(x(N, 0)) → MARK(U61(and(isNat(N), isNatKind(N))))

The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))

Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
plus(x1, x2) = plus(x1, x2)
s(x1) = s(x1)
MARK(x1) = MARK(x1)
U51(x1, x2, x3) = U51(x1, x2, x3)
and(x1, x2) = and(x1, x2)
isNat(x1) = x1
isNatKind(x1) = x1
x(x1, x2) = x(x1, x2)
U31(x1, x2, x3) = U31(x1, x2, x3)
mark(x1) = x1
U41(x1, x2) = U41(x1, x2)
U12(x1, x2) = U12(x1, x2)
U11(x1, x2, x3) = U11(x1, x2, x3)
tt = tt
U21(x1, x2) = U21(x1, x2)
U71(x1, x2, x3) = U71(x1, x2, x3)
U22(x1) = x1
U32(x1, x2) = U32(x1, x2)
U33(x1) = x1
U13(x1) = x1
0 = 0
U61(x1) = x1
active(x1) = x1

Recursive path order with status [2].
Quasi-Precedence:

[tt, 0] > [ACTIVE₁, MARK₁, x₂, U71₃] > [plus₂, U51₃] > s₁ > U21₂ > U12₂
[tt, 0] > [ACTIVE₁, MARK₁, x₂, U71₃] > [plus₂, U51₃] > and₂ > U12₂
[tt, 0] > [ACTIVE₁, MARK₁, x₂, U71₃] > [plus₂, U51₃] > U41₂ > U12₂
[tt, 0] > [ACTIVE₁, MARK₁, x₂, U71₃] > [plus₂, U51₃] > U11₃ > U12₂
[tt, 0] > [ACTIVE₁, MARK₁, x₂, U71₃] > U31₃ > U32₂ > U12₂

Status:

plus₂: [2,1]
U32₂: [1,2]
U11₃: multiset
U12₂: multiset
x₂: [1,2]
and₂: [2,1]
0: multiset
U21₂: [2,1]
ACTIVE₁: [1]
U31₃: multiset
MARK₁: [1]
tt: multiset
U41₂: [2,1]
s₁: [1]
U51₃: [2,3,1]
U71₃: [3,2,1]

The following usable rules [17] were oriented:

x(X1, active(X2)) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(mark(X1), X2) → x(X1, X2)
mark(0) → active(0)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatKind(0)) → mark(tt)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(U41(tt, N)) → mark(N)
mark(U33(X)) → active(U33(mark(X)))
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
mark(U13(X)) → active(U13(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U61(X)) → active(U61(mark(X)))
mark(isNat(X)) → active(isNat(X))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(and(tt, X)) → mark(X)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
mark(isNatKind(X)) → active(isNatKind(X))
mark(s(X)) → active(s(mark(X)))
mark(U22(X)) → active(U22(mark(X)))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
U61(active(X)) → U61(X)
U61(mark(X)) → U61(X)
mark(tt) → active(tt)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U32(X1, mark(X2)) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U33(active(X)) → U33(X)
U33(mark(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
active(isNat(0)) → mark(tt)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
active(U61(tt)) → mark(0)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(U33(tt)) → mark(tt)
U13(active(X)) → U13(X)
U13(mark(X)) → U13(X)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
active(U22(tt)) → mark(tt)
active(U13(tt)) → mark(tt)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U33(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U13(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(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 13 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ UsableRulesProof

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

MARK(U22(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U13(X)) → active(U13(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U61(X)) → active(U61(mark(X)))
mark(0) → active(0)
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

MARK(U22(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

From the DPs we obtained the following set of size-change graphs:

MARK(U22(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U33(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U13(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U61(X)) → MARK(X)
The graph contains the following edges 1 > 1