Termination w.r.t. Q proof of /tmp/tmpcripRs/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 DependencyGraphProof (⇔)

        ↳23 TRUE

    ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 PisEmptyProof (⇔)

        ↳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.

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)
U51¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
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)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
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: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x0, x1, x2, x3) = U12¹(x0, x3)
U13¹(x0, x1, x2, x3) = U13¹(x0, x1, x2, x3)
U14¹(x0, x1, x2, x3) = U14¹(x0, x2, x3)
U15¹(x0, x1, x2) = U15¹(x0, x2)
ISNAT(x0, x1) = ISNAT(x1)
U11¹(x0, x1, x2, x3) = U11¹(x0, x3)
ISNATKIND(x0, x1) = ISNATKIND(x1)
U31¹(x0, x1, x2) = U31¹(x0, x2)
ACTIVATE(x0, x1) = ACTIVATE(x1)
PLUS(x0, x1, x2) = PLUS(x0, x2)
U51¹(x0, x1, x2) = U51¹(x0, x2)
U52¹(x0, x1, x2) = U52¹(x2)
U21¹(x0, x1, x2) = U21¹(x2)
U22¹(x0, x1, x2) = U22¹(x2)
U61¹(x0, x1, x2, x3) = U61¹(x0, x3)
U62¹(x0, x1, x2, x3) = U62¹(x0, x2, x3)
U63¹(x0, x1, x2, x3) = U63¹(x0, x3)
U64¹(x0, x1, x2, x3) = U64¹(x0, x2, x3)

Tags:
U12¹ has argument tags [31,0,2,32] and root tag 24
U13¹ has argument tags [4,8,31,4] and root tag 24
U14¹ has argument tags [8,4,31,4] and root tag 24
U15¹ has argument tags [4,63,3] and root tag 17
ISNAT has argument tags [43,4] and root tag 1
U11¹ has argument tags [60,24,15,32] and root tag 29
ISNATKIND has argument tags [0,4] and root tag 1
U31¹ has argument tags [16,0,4] and root tag 2
ACTIVATE has argument tags [17,1] and root tag 16
PLUS has argument tags [4,6,8] and root tag 8
U51¹ has argument tags [8,12,4] and root tag 7
U52¹ has argument tags [10,36,3] and root tag 15
U21¹ has argument tags [1,5,4] and root tag 1
U22¹ has argument tags [6,31,4] and root tag 1
U61¹ has argument tags [8,57,3,4] and root tag 8
U62¹ has argument tags [8,16,8,4] and root tag 8
U63¹ has argument tags [8,0,0,4] and root tag 8
U64¹ has argument tags [6,52,8,4] and root tag 8

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U12¹(x1, x2, x3) = x2
tt = tt
U13¹(x1, x2, x3) = x2
isNatKind(x1) = x1
activate(x1) = x1
U14¹(x1, x2, x3) = U14¹
U15¹(x1, x2) = x2
isNat(x1) = isNat
ISNAT(x1) = ISNAT
n__plus(x1, x2) = n__plus(x1, x2)
U11¹(x1, x2, x3) = x2
ISNATKIND(x1) = ISNATKIND(x1)
U31¹(x1, x2) = x1
ACTIVATE(x1) = ACTIVATE
PLUS(x1, x2) = x1
0 = 0
U51¹(x1, x2) = U51¹
U52¹(x1, x2) = U52¹
n__s(x1) = x1
U21¹(x1, x2) = U21¹
U22¹(x1, x2) = x1
s(x1) = x1
U61¹(x1, x2, x3) = x2
U62¹(x1, x2, x3) = x1
U63¹(x1, x2, x3) = x2
U64¹(x1, x2, x3) = U64¹
n__0 = n__0
U52(x1, x2) = U52(x2)
plus(x1, x2) = plus(x1, x2)
U51(x1, x2) = U51(x1, x2)
U31(x1, x2) = U31(x1, x2)
U41(x1) = x1
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U61(x1, x2, x3) = U61(x2, x3)
U12(x1, x2, x3) = U12
U32(x1) = U32(x1)
U22(x1, x2) = U22
U62(x1, x2, x3) = U62(x2, x3)
U13(x1, x2, x3) = U13
U23(x1) = U23
U63(x1, x2, x3) = U63(x2, x3)
U14(x1, x2, x3) = U14
U64(x1, x2, x3) = U64(x2, x3)
U15(x1, x2) = x1
U16(x1) = U16

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

[n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > [isNat, U11, U21, U12, U13, U14] > U22 > [tt, U14^1, U23, U16] > [U64^1, U52₁]
[n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U51₂ > [U64^1, U52₁]
[n_{_plus₂}, plus₂, U61₂, U62₂, U63₂, U64₂] > U31₂ > U32₁ > [tt, U14^1, U23, U16] > [U64^1, U52₁]
[0, U51^1, n_₀] > [isNat, U11, U21, U12, U13, U14] > U22 > [tt, U14^1, U23, U16] > [U64^1, U52₁]
[0, U51^1, n_₀] > ISNAT > U21^1 > [U64^1, U52₁]
[0, U51^1, n_₀] > ISNATKIND₁ > [U64^1, U52₁]
[0, U51^1, n_₀] > ACTIVATE > [U64^1, U52₁]
[0, U51^1, n_₀] > U51₂ > [U64^1, U52₁]
U52^1 > ACTIVATE > [U64^1, U52₁]

Status:

tt: multiset
U14^1: []
isNat: multiset
ISNAT: multiset
n_{_plus₂}: multiset
ISNATKIND₁: multiset
ACTIVATE: multiset
0: multiset
U51^1: multiset
U52^1: multiset
U21^1: multiset
U64^1: multiset
n_₀: multiset
U52₁: multiset
plus₂: multiset
U51₂: multiset
U31₂: multiset
U11: multiset
U21: multiset
U61₂: multiset
U12: multiset
U32₁: multiset
U22: multiset
U62₂: multiset
U13: multiset
U23: []
U63₂: multiset
U14: multiset
U64₂: multiset
U16: []

The following usable rules [FROCOS05] were oriented:

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

(6) 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))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNATKIND(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 4 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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U61¹(x0, x1, x2, x3) = U61¹(x0)
U62¹(x0, x1, x2, x3) = U62¹(x0, x1, x2)
U63¹(x0, x1, x2, x3) = U63¹(x0, x2)
U64¹(x0, x1, x2, x3) = U64¹(x0, x1)
PLUS(x0, x1, x2) = PLUS(x0, x2)

Tags:
U61¹ has argument tags [5,12,14,3] and root tag 0
U62¹ has argument tags [19,17,19,0] and root tag 2
U63¹ has argument tags [19,8,8,1] and root tag 2
U64¹ has argument tags [8,17,9,16] and root tag 2
PLUS has argument tags [17,0,5] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U61¹(x1, x2, x3) = U61¹(x2)
tt = tt
U62¹(x1, x2, x3) = U62¹
isNatKind(x1) = isNatKind
activate(x1) = x1
U63¹(x1, x2, x3) = U63¹
isNat(x1) = isNat
U64¹(x1, x2, x3) = x2
PLUS(x1, x2) = PLUS
s(x1) = s(x1)
n__0 = n__0
0 = 0
U52(x1, x2) = x2
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U51(x1, x2) = x2
n__s(x1) = n__s(x1)
U31(x1, x2) = U31
U41(x1) = x1
U11(x1, x2, x3) = x1
U21(x1, x2) = U21
U61(x1, x2, x3) = U61(x1, x2, x3)
U12(x1, x2, x3) = U12
U32(x1) = x1
U22(x1, x2) = U22
U62(x1, x2, x3) = U62(x1, x2, x3)
U13(x1, x2, x3) = x1
U23(x1) = U23
U63(x1, x2, x3) = U63(x1, x2, x3)
U14(x1, x2, x3) = x1
U64(x1, x2, x3) = U64(x1, x2, x3)
U15(x1, x2) = x1
U16(x1) = x1

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

[n_₀, 0]
[n_{_plus₂}, plus₂, U61₃, U62₃, U63₃, U64₃] > [U61^1₁, s₁, n_{_s₁}] > [U62^1, U63^1] > [tt, isNatKind, isNat, PLUS, U31, U21, U12, U22, U23]

Status:

U61^1₁: multiset
tt: multiset
U62^1: multiset
isNatKind: []
U63^1: multiset
isNat: []
PLUS: []
s₁: multiset
n_₀: multiset
0: multiset
n_{_plus₂}: [2,1]
plus₂: [2,1]
n_{_s₁}: multiset
U31: []
U21: []
U61₃: [2,3,1]
U12: []
U22: []
U62₃: [2,3,1]
U23: []
U63₃: [2,3,1]
U64₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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

(11) Obligation:

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

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

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(13) TRUE

(14) 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.

(15) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNATKIND(x0, x1) = ISNATKIND(x0)

Tags:
ISNATKIND has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNATKIND(x1) = x1
n__s(x1) = n__s(x1)
activate(x1) = x1
n__0 = n__0
0 = 0
U52(x1, x2) = U52(x2)
tt = tt
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U51(x1, x2) = U51(x1, x2)
isNat(x1) = isNat
isNatKind(x1) = isNatKind
s(x1) = s(x1)
U61(x1, x2, x3) = U61(x1, x2, x3)
U11(x1, x2, x3) = x1
U21(x1, x2) = U21
U31(x1, x2) = x1
U41(x1) = x1
U62(x1, x2, x3) = U62(x1, x2, x3)
U12(x1, x2, x3) = x1
U22(x1, x2) = x1
U32(x1) = x1
U63(x1, x2, x3) = U63(x1, x2, x3)
U13(x1, x2, x3) = U13
U23(x1) = x1
U64(x1, x2, x3) = U64(x1, x2, x3)
U14(x1, x2, x3) = x1
U15(x1, x2) = U15
U16(x1) = U16

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

[n_₀, 0]
[n_{_plus₂}, plus₂, U61₃, U62₃, U63₃, U64₃] > [n_{_s₁}, s₁] > [tt, isNat, isNatKind, U21, U13, U15, U16] > U52₁
[n_{_plus₂}, plus₂, U61₃, U62₃, U63₃, U64₃] > U51₂ > U52₁

Status:

n_{_s₁}: [1]
n_₀: multiset
0: multiset
U52₁: [1]
tt: multiset
n_{_plus₂}: multiset
plus₂: multiset
U51₂: [1,2]
isNat: []
isNatKind: []
s₁: [1]
U61₃: multiset
U21: []
U62₃: multiset
U63₃: multiset
U13: []
U64₃: multiset
U15: []
U16: []

The following usable rules [FROCOS05] were oriented:

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

(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:

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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNAT(x0, x1) = ISNAT(x1)
U21¹(x0, x1, x2) = U21¹(x0)
U22¹(x0, x1, x2) = U22¹(x0, x2)

Tags:
ISNAT has argument tags [5,2] and root tag 0
U21¹ has argument tags [2,0,4] and root tag 0
U22¹ has argument tags [1,2,2] and root tag 2

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNAT(x1) = ISNAT
n__s(x1) = n__s(x1)
U21¹(x1, x2) = U21¹(x2)
isNatKind(x1) = isNatKind
activate(x1) = x1
tt = tt
U22¹(x1, x2) = U22¹
n__0 = n__0
0 = 0
U52(x1, x2) = x2
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U51(x1, x2) = x2
isNat(x1) = isNat
s(x1) = s(x1)
U31(x1, x2) = x1
U41(x1) = x1
U61(x1, x2, x3) = U61(x1, x2, x3)
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U32(x1) = U32
U62(x1, x2, x3) = U62(x1, x2, x3)
U12(x1, x2, x3) = x1
U22(x1, x2) = U22
U63(x1, x2, x3) = U63(x1, x2, x3)
U13(x1, x2, x3) = x1
U23(x1) = x1
U64(x1, x2, x3) = U64(x1, x2, x3)
U14(x1, x2, x3) = x1
U15(x1, x2) = x1
U16(x1) = x1

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

[isNatKind, tt, n_₀, 0, n_{_plus₂}, plus₂, isNat, U61₃, U11, U21, U32, U62₃, U22, U63₃, U64₃] > [ISNAT, n_{_s₁}, U21^1₁, U22^1, s₁]

Status:

ISNAT: multiset
n_{_s₁}: [1]
U21^1₁: [1]
isNatKind: []
tt: multiset
U22^1: []
n_₀: multiset
0: multiset
n_{_plus₂}: [1,2]
plus₂: [1,2]
isNat: []
s₁: [1]
U61₃: [3,2,1]
U11: []
U21: []
U32: []
U62₃: [3,2,1]
U22: []
U63₃: [3,2,1]
U64₃: [3,2,1]

The following usable rules [FROCOS05] were oriented:

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

(21) Obligation:

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

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

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(23) TRUE

(24) 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.

(25) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x0, x1) = ACTIVATE(x1)

Tags:
ACTIVATE has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE
n__s(x1) = n__s(x1)

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

[ACTIVATE, n_{_s₁}]

Status:

ACTIVATE: multiset
n_{_s₁}: multiset

The following usable rules [FROCOS05] were oriented: none

(26) 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.

(27) PisEmptyProof (EQUIVALENT transformation)

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

(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) DependencyGraphProof (EQUIVALENT transformation)

(23) TRUE

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) PisEmptyProof (EQUIVALENT transformation)

(28) TRUE