Termination w.r.t. Q proof of /tmp/tmpYyDPZF/PEANO_complete-noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 AND

  ↳9 QDP

    ↳10 QDPOrderProof (⇔)

    ↳11 QDP

    ↳12 DependencyGraphProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 PisEmptyProof (⇔)

    ↳23 TRUE

  ↳24 QDP

    ↳25 QDPOrderProof (⇔)

    ↳26 QDP

    ↳27 DependencyGraphProof (⇔)

    ↳28 TRUE

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → U16¹(isNat(activate(V2)))
U15¹(tt, V2) → ISNAT(activate(V2))
U15¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U22¹(tt, V1) → ISNAT(activate(V1))
U22¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V2) → U32¹(isNatKind(activate(V2)))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U51¹(tt, N) → ISNATKIND(activate(N))
U51¹(tt, N) → ACTIVATE(N)
U52¹(tt, N) → ACTIVATE(N)
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U61¹(tt, M, N) → ISNATKIND(activate(M))
U61¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U62¹(tt, M, N) → ISNAT(activate(N))
U62¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ACTIVATE(N)
U63¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → S(plus(activate(N), activate(M)))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U64¹(tt, M, N) → ACTIVATE(N)
U64¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → U41¹(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U51¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(4) Obligation:

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

U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U51¹(isNat(N), N)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U52¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U51¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U64¹(tt, M, N) → ACTIVATE(N)
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ACTIVATE(N)
U63¹(tt, M, N) → ACTIVATE(M)
U62¹(tt, M, N) → ISNAT(activate(N))
U62¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ISNATKIND(activate(M))
U61¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ACTIVATE(N)
U31¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → ACTIVATE(V2)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U51¹(isNat(N), N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U64¹(tt, M, N) → ACTIVATE(N)
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ACTIVATE(N)
U63¹(tt, M, N) → ACTIVATE(M)
U62¹(tt, M, N) → ISNAT(activate(N))
U62¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ISNATKIND(activate(M))
U61¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ACTIVATE(N)
U31¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → ACTIVATE(V2)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3) = U12¹(x2, x3)
tt = tt
U13¹(x1, x2, x3) = U13¹(x2, x3)
isNatKind(x1) = isNatKind
activate(x1) = x1
U14¹(x1, x2, x3) = U14¹(x2, x3)
U15¹(x1, x2) = U15¹(x1, x2)
isNat(x1) = x1
ISNAT(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
U11¹(x1, x2, x3) = U11¹(x2, x3)
ISNATKIND(x1) = x1
U31¹(x1, x2) = U31¹(x2)
ACTIVATE(x1) = x1
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
U51¹(x1, x2) = x2
U52¹(x1, x2) = x2
n__s(x1) = x1
U21¹(x1, x2) = x2
U22¹(x1, x2) = x2
s(x1) = x1
U61¹(x1, x2, x3) = U61¹(x2, x3)
U62¹(x1, x2, x3) = U62¹(x2, x3)
U63¹(x1, x2, x3) = U63¹(x2, x3)
U64¹(x1, x2, x3) = U64¹(x2, x3)
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
U61(x1, x2, x3) = U61(x2, x3)
U41(x1) = x1
U32(x1) = x1
U31(x1, x2) = x1
U23(x1) = x1
U62(x1, x2, x3) = U62(x2, x3)
U63(x1, x2, x3) = U63(x2, x3)
U51(x1, x2) = U51(x2)
U52(x1, x2) = U52(x2)
U11(x1, x2, x3) = U11(x3)
U64(x1, x2, x3) = U64(x2, x3)
U21(x1, x2) = x2
U12(x1, x2, x3) = U12(x3)
U13(x1, x2, x3) = U13(x3)
U14(x1, x2, x3) = U14(x3)
U15(x1, x2) = x2
U16(x1) = x1
U22(x1, x2) = x2

Lexicographic path order with status [LPO].
Quasi-Precedence:

[tt, isNatKind, 0, n_₀] > [n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U11^1₂ > U12^1₂ > U13^1₂ > [U14^1₂, U15^1₂]
[tt, isNatKind, 0, n_₀] > [n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U31^1₁ > [U14^1₂, U15^1₂]
[tt, isNatKind, 0, n_₀] > [n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > [PLUS₂, U61^1₂, U62^1₂, U63^1₂, U64^1₂] > [U14^1₂, U15^1₂]
[tt, isNatKind, 0, n_₀] > [n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U51₁ > U52₁ > [U14^1₂, U15^1₂]
[tt, isNatKind, 0, n_₀] > [n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U11₁ > [U12₁, U13₁] > U14₁ > [U14^1₂, U15^1₂]

Status:

n_{_plus₂}: [1,2]
U62₂: [2,1]
U12^1₂: [2,1]
U31^1₁: [1]
U61^1₂: [1,2]
PLUS₂: [2,1]
tt: []
U63^1₂: [1,2]
U11^1₂: [1,2]
U52₁: [1]
U12₁: [1]
U14^1₂: [1,2]
U11₁: [1]
isNatKind: []
plus₂: [1,2]
U64^1₂: [1,2]
U14₁: [1]
U62^1₂: [1,2]
0: []
U61₂: [2,1]
n_₀: []
U15^1₂: [1,2]
U13₁: [1]
U64₂: [2,1]
U13^1₂: [1,2]
U51₁: [1]
U63₂: [2,1]

The following usable rules [FROCOS05] were oriented:

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
plus(N, s(M)) → U61(isNat(M), M, N)
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
activate(n__0) → 0
activate(X) → X
U41(tt) → tt
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U23(tt) → tt
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__0) → tt
U64(tt, M, N) → s(plus(activate(N), activate(M)))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))

(6) Obligation:

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

U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U52¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U51¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, N) → ACTIVATE(N)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U61¹(isNat(M), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U61¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = x2
tt = tt
U62¹(x1, x2, x3) = x2
isNatKind(x1) = isNatKind
activate(x1) = x1
U63¹(x1, x2, x3) = x2
isNat(x1) = isNat
U64¹(x1, x2, x3) = x2
PLUS(x1, x2) = x2
s(x1) = s(x1)
0 = 0
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
n__plus(x1, x2) = n__plus(x1, x2)
U61(x1, x2, x3) = U61(x1, x2, x3)
n__s(x1) = n__s(x1)
U41(x1) = U41
U32(x1) = U32
U31(x1, x2) = U31
U23(x1) = x1
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U51(x1, x2) = U51(x1, x2)
U52(x1, x2) = x2
U11(x1, x2, x3) = x1
U64(x1, x2, x3) = U64(x1, x2, x3)
U21(x1, x2) = U21
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = U13
U14(x1, x2, x3) = U14
U15(x1, x2) = U15
U16(x1) = U16
U22(x1, x2) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, n_₀] > U51₂ > [tt, isNatKind, isNat, s₁, n_{_s₁}, U41, U32, U31, U21, U13, U14, U15, U16]
[plus₂, n_{_plus₂}, U61₃, U62₃, U63₃, U64₃] > U51₂ > [tt, isNatKind, isNat, s₁, n_{_s₁}, U41, U32, U31, U21, U13, U14, U15, U16]

Status:

n_{_plus₂}: [2,1]
U13: []
plus₂: [2,1]
U21: []
U32: []
U61₃: [2,3,1]
U31: []
U15: []
U64₃: [2,3,1]
n_{_s₁}: [1]
0: []
U63₃: [2,3,1]
U51₂: [1,2]
isNat: []
U16: []
tt: []
n_₀: []
U41: []
U14: []
isNatKind: []
s₁: [1]
U62₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
plus(N, s(M)) → U61(isNat(M), M, N)
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
activate(n__0) → 0
activate(X) → X
U41(tt) → tt
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U23(tt) → tt
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__0) → tt
U64(tt, M, N) → s(plus(activate(N), activate(M)))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))

(11) Obligation:

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

U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(13) TRUE

(14) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__s(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = x1
n__s(x1) = n__s(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

n_{_s₁}: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1) = ISNATKIND(x1)
n__s(x1) = n__s(x1)
activate(x1) = x1
0 = 0
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
n__plus(x1, x2) = n__plus(x1, x2)
s(x1) = s(x1)
U61(x1, x2, x3) = U61(x1, x2, x3)
isNat(x1) = isNat
U41(x1) = x1
tt = tt
U32(x1) = x1
U31(x1, x2) = U31
isNatKind(x1) = isNatKind
U23(x1) = x1
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U51(x1, x2) = x2
U52(x1, x2) = x2
U11(x1, x2, x3) = U11
U64(x1, x2, x3) = U64(x1, x2, x3)
U21(x1, x2) = U21
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = U13
U14(x1, x2, x3) = x1
U15(x1, x2) = U15
U16(x1) = U16
U22(x1, x2) = U22

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, n_₀] > [isNat, tt, U31, isNatKind, U11, U21, U13, U15, U16, U22]
[plus₂, n_{_plus₂}, U61₃, U62₃, U63₃, U64₃] > [ISNATKIND₁, n_{_s₁}, s₁] > [isNat, tt, U31, isNatKind, U11, U21, U13, U15, U16, U22]

Status:

n_{_plus₂}: [1,2]
U13: []
plus₂: [1,2]
U21: []
U22: []
ISNATKIND₁: [1]
U61₃: [3,2,1]
U31: []
U11: []
U15: []
U64₃: [3,2,1]
n_{_s₁}: [1]
0: []
U63₃: [3,2,1]
isNat: []
U16: []
tt: []
n_₀: []
s₁: [1]
isNatKind: []
U62₃: [3,2,1]

The following usable rules [FROCOS05] were oriented:

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
plus(N, s(M)) → U61(isNat(M), M, N)
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
activate(n__0) → 0
activate(X) → X
U41(tt) → tt
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U23(tt) → tt
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__0) → tt
U64(tt, M, N) → s(plus(activate(N), activate(M)))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))

(21) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(23) TRUE

(24) Obligation:

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

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1
n__s(x1) = n__s(x1)
U21¹(x1, x2) = x2
isNatKind(x1) = isNatKind
activate(x1) = x1
tt = tt
U22¹(x1, x2) = x2
0 = 0
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
n__plus(x1, x2) = n__plus(x1, x2)
s(x1) = s(x1)
U61(x1, x2, x3) = U61(x1, x2, x3)
isNat(x1) = isNat
U41(x1) = x1
U32(x1) = x1
U31(x1, x2) = x1
U23(x1) = U23
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U51(x1, x2) = x2
U52(x1, x2) = x2
U11(x1, x2, x3) = U11
U64(x1, x2, x3) = U64(x1, x2, x3)
U21(x1, x2) = U21
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = U13
U14(x1, x2, x3) = U14
U15(x1, x2) = x1
U16(x1) = x1
U22(x1, x2) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, n_₀]
[plus₂, n_{_plus₂}, U61₃, U62₃, U63₃, U64₃] > [isNatKind, tt, isNat, U23, U11, U21, U13, U14] > [n_{_s₁}, s₁]

Status:

n_{_plus₂}: [2,1]
U13: []
plus₂: [2,1]
U21: []
U61₃: [2,3,1]
U11: []
U64₃: [2,3,1]
n_{_s₁}: [1]
0: []
U63₃: [2,3,1]
isNat: []
tt: []
n_₀: []
U14: []
isNatKind: []
s₁: [1]
U23: []
U62₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
plus(N, s(M)) → U61(isNat(M), M, N)
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
activate(n__0) → 0
activate(X) → X
U41(tt) → tt
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U23(tt) → tt
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__0) → tt
U64(tt, M, N) → s(plus(activate(N), activate(M)))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))

(26) Obligation:

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

U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) DependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) PisEmptyProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) PisEmptyProof (EQUIVALENT transformation)

(23) TRUE

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) TRUE