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.


U151(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(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))
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)
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)  =  ISNAT(x1)
n__plus(x1, x2)  =  n__plus(x1, x2)
U111(x1, x2, x3)  =  U111(x2, x3)
ISNATKIND(x1)  =  ISNATKIND(x1)
U311(x1, x2)  =  U311(x2)
ACTIVATE(x1)  =  ACTIVATE(x1)
PLUS(x1, x2)  =  PLUS(x1, x2)
0  =  0
U511(x1, x2)  =  U511(x2)
U521(x1, x2)  =  U521(x2)
n__s(x1)  =  x1
U211(x1, x2)  =  U211(x2)
U221(x1, x2)  =  U221(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)
U12(x1, x2, x3)  =  U12
U13(x1, x2, x3)  =  U13
U11(x1, x2, x3)  =  U11
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15
U16(x1)  =  U16
plus(x1, x2)  =  plus(x1, x2)
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U32(x1)  =  x1
U41(x1)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
n__0  =  n__0
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)

Recursive Path Order [RPO].
Precedence:
[U12^12, tt, U13^12, isNatKind, U14^12, U15^12, ISNAT1, nplus2, U11^12, ISNATKIND1, U31^11, ACTIVATE1, PLUS2, 0, U51^11, U52^11, U21^11, U22^11, U61^12, U62^12, U63^12, U64^12, U12, U13, U11, U15, U16, plus2, n0, U632, U642, U612, U622]


The following usable rules [FROCOS05] were oriented:

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

(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))
U141(tt, V1, V2) → U151(isNat(activate(V1)), activate(V2))
U111(tt, V1, V2) → U121(isNatKind(activate(V1)), activate(V1), activate(V2))
U311(tt, V2) → ISNATKIND(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))
U311(tt, V2) → ACTIVATE(V2)

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 16 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))
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)  =  U611(x1, x2, x3)
tt  =  tt
U621(x1, x2, x3)  =  U621(x2, x3)
isNatKind(x1)  =  isNatKind
activate(x1)  =  x1
U631(x1, x2, x3)  =  U631(x2, x3)
isNat(x1)  =  isNat
U641(x1, x2, x3)  =  U641(x2, x3)
PLUS(x1, x2)  =  PLUS(x1, x2)
s(x1)  =  s(x1)
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U32(x1)  =  U32
U41(x1)  =  U41
U23(x1)  =  x1
U31(x1, x2)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
n__s(x1)  =  n__s(x1)
n__0  =  n__0
U11(x1, x2, x3)  =  U11
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  U15
U16(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[nplus2, plus2, U633, U643, U613, U623] > [tt, isNatKind, isNat, s1, U32, U41, ns1, U11, U14, U15] > [U61^13, U62^12, U63^12, U64^12, PLUS2]
[0, n0]


The following usable rules [FROCOS05] were oriented:

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

(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))
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 3 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: Recursive Path Order [RPO].
Precedence:
ns1 > ACTIVATE1


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
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
U51(x1, x2)  =  x2
isNat(x1)  =  isNat
tt  =  tt
U52(x1, x2)  =  x2
isNatKind(x1)  =  isNatKind
U32(x1)  =  x1
U41(x1)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  x1
n__0  =  n__0
U11(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U12(x1, x2, x3)  =  U12
U13(x1, x2, x3)  =  U13
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  x1
U16(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[nplus2, plus2, U633, U643, U613, U623] > [ISNATKIND1, ns1, isNat, tt, isNatKind, U21, s1, U12, U13, U14]
[0, n0]


The following usable rules [FROCOS05] were oriented:

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

(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)  =  ISNAT(x1)
n__s(x1)  =  n__s(x1)
U211(x1, x2)  =  U211(x2)
isNatKind(x1)  =  isNatKind
activate(x1)  =  x1
tt  =  tt
U221(x1, x2)  =  U221(x2)
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
U51(x1, x2)  =  U51(x2)
isNat(x1)  =  isNat
U52(x1, x2)  =  x2
U32(x1)  =  U32
U41(x1)  =  U41
U23(x1)  =  x1
U31(x1, x2)  =  U31
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
n__0  =  n__0
U11(x1, x2, x3)  =  U11
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U12(x1, x2, x3)  =  U12
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15
U16(x1)  =  U16

Recursive Path Order [RPO].
Precedence:
[ISNAT1, U21^11, U22^11] > [ns1, isNatKind, tt, 0, isNat, U32, U41, U31, n0, U11, s1, U12, U15, U16]
[nplus2, plus2, U511, U633, U643, U613, U623] > [ns1, isNatKind, tt, 0, isNat, U32, U41, U31, n0, U11, s1, U12, U15, U16]


The following usable rules [FROCOS05] were oriented:

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

(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.

(28) TRUE