(0) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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(isNat(activate(V1)), activate(V2))
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → U131(isNat(activate(V2)))
U121(tt, V2) → ISNAT(activate(V2))
U121(tt, V2) → ACTIVATE(V2)
U211(tt, V1) → U221(isNat(activate(V1)))
U211(tt, V1) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U311(tt, N) → ACTIVATE(N)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
U411(tt, M, N) → PLUS(activate(N), activate(M))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(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)) → AND(isNatKind(activate(V1)), n__isNatKind(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)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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 5 less nodes.

(4) Obligation:

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

U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
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)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(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))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.


ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(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)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
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)  =  U121(x2)
ISNAT(x0, x1)  =  ISNAT(x0)
U111(x0, x1, x2, x3)  =  U111(x2, x3)
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
PLUS(x0, x1, x2)  =  PLUS(x1, x2)
U311(x0, x1, x2)  =  U311(x2)
ISNATKIND(x0, x1)  =  ISNATKIND(x1)
AND(x0, x1, x2)  =  AND(x2)
U211(x0, x1, x2)  =  U211(x0, x2)
U411(x0, x1, x2, x3)  =  U411(x0, x2)

Tags:
U121 has argument tags [0,8,16] and root tag 8
ISNAT has argument tags [16,4] and root tag 8
U111 has argument tags [0,27,20,16] and root tag 12
ACTIVATE has argument tags [14,16] and root tag 2
PLUS has argument tags [13,16,16] and root tag 8
U311 has argument tags [15,30,16] and root tag 4
ISNATKIND has argument tags [8,16] and root tag 2
AND has argument tags [28,4,16] and root tag 2
U211 has argument tags [16,29,12] and root tag 8
U411 has argument tags [16,16,16,0] 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)  =  x1
tt  =  tt
ISNAT(x1)  =  x1
activate(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
U111(x1, x2, x3)  =  x1
and(x1, x2)  =  x2
isNatKind(x1)  =  x1
n__isNatKind(x1)  =  x1
isNat(x1)  =  isNat(x1)
ACTIVATE(x1)  =  x1
PLUS(x1, x2)  =  x2
0  =  0
U311(x1, x2)  =  U311
ISNATKIND(x1)  =  ISNATKIND
AND(x1, x2)  =  AND(x1, x2)
n__and(x1, x2)  =  x2
n__s(x1)  =  x1
U211(x1, x2)  =  x2
s(x1)  =  x1
U411(x1, x2, x3)  =  x3
n__0  =  n__0
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  x2
U11(x1, x2, x3)  =  U11(x1, x2)
U21(x1, x2)  =  U21(x1, x2)
U41(x1, x2, x3)  =  U41(x2, x3)
U12(x1, x2)  =  U12
U22(x1)  =  U22
U13(x1)  =  U13

Recursive path order with status [RPO].
Quasi-Precedence:
[nplus2, plus2, U112, U412] > [isNat1, 0, n0] > [tt, U12, U22, U13]
[nplus2, plus2, U112, U412] > [isNat1, 0, n0] > U212
[nplus2, plus2, U112, U412] > AND2
ISNATKIND > AND2

Status:
tt: multiset
nplus2: multiset
isNat1: [1]
0: multiset
U31^1: multiset
ISNATKIND: []
AND2: [2,1]
n0: multiset
plus2: multiset
U112: multiset
U212: [2,1]
U412: multiset
U12: multiset
U22: multiset
U13: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U12(tt, V2) → U13(isNat(activate(V2)))
plus(X1, X2) → n__plus(X1, X2)
U22(tt) → tt
U13(tt) → tt
s(X) → n__s(X)
0n__0

(6) Obligation:

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

U121(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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 3 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

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

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.


U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U211(x0, x1, x2)  =  U211(x2)
ISNAT(x0, x1)  =  ISNAT(x0, x1)

Tags:
U211 has argument tags [2,0,4] and root tag 0
ISNAT has argument tags [3,4] and root tag 1

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U211(x1, x2)  =  x1
tt  =  tt
ISNAT(x1)  =  x1
activate(x1)  =  x1
n__s(x1)  =  n__s(x1)
isNatKind(x1)  =  isNatKind
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
and(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
n__isNatKind(x1)  =  n__isNatKind
n__and(x1, x2)  =  x2
s(x1)  =  s(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U21(x1, x2)  =  U21
U12(x1, x2)  =  U12
U22(x1)  =  U22
U13(x1)  =  U13

Recursive path order with status [RPO].
Quasi-Precedence:
[n0, 0] > U312
[n0, 0] > isNat1 > [tt, ns1, isNatKind, nisNatKind, s1, U13]
[nplus2, plus2, U413] > U312
[nplus2, plus2, U413] > isNat1 > [tt, ns1, isNatKind, nisNatKind, s1, U13]
U113 > isNat1 > [tt, ns1, isNatKind, nisNatKind, s1, U13]
U21 > isNat1 > [tt, ns1, isNatKind, nisNatKind, s1, U13]
U12 > isNat1 > [tt, ns1, isNatKind, nisNatKind, s1, U13]
U22 > [tt, ns1, isNatKind, nisNatKind, s1, U13]

Status:
tt: multiset
ns1: multiset
isNatKind: multiset
n0: multiset
0: multiset
nplus2: [1,2]
plus2: [1,2]
U312: multiset
isNat1: [1]
nisNatKind: multiset
s1: multiset
U413: [3,2,1]
U113: multiset
U21: multiset
U12: []
U22: []
U13: []


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
0n__0

(11) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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.


PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(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:
PLUS(x0, x1, x2)  =  PLUS(x1, x2)
U411(x0, x1, x2, x3)  =  U411(x0, x1, x3)

Tags:
PLUS has argument tags [4,3,2] and root tag 0
U411 has argument tags [4,7,5,3] and root tag 1

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x2)
s(x1)  =  s(x1)
U411(x1, x2, x3)  =  x2
and(x1, x2)  =  x2
isNat(x1)  =  isNat
n__isNatKind(x1)  =  n__isNatKind
n__and(x1, x2)  =  x2
tt  =  tt
activate(x1)  =  x1
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
U11(x1, x2, x3)  =  U11(x2)
isNatKind(x1)  =  isNatKind
n__s(x1)  =  n__s(x1)
U21(x1, x2)  =  U21
plus(x1, x2)  =  plus(x1, x2)
0  =  0
U31(x1, x2)  =  x2
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U12(x1, x2)  =  U12(x1, x2)
U22(x1)  =  U22
U13(x1)  =  U13(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[n0, 0]
U21 > [isNat, nplus2, U111, plus2, U413, U122] > [s1, ns1] > [nisNatKind, isNatKind] > tt > PLUS1
U21 > [isNat, nplus2, U111, plus2, U413, U122] > U131 > tt > PLUS1
U21 > U22 > tt > PLUS1

Status:
PLUS1: multiset
s1: multiset
isNat: []
nisNatKind: []
tt: multiset
n0: multiset
nplus2: multiset
U111: multiset
isNatKind: []
ns1: multiset
U21: multiset
plus2: multiset
0: multiset
U413: multiset
U122: multiset
U22: multiset
U131: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
0n__0

(16) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
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)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, 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:
ISNATKIND(x0, x1)  =  ISNATKIND(x0)
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
AND(x0, x1, x2)  =  AND(x0, x2)

Tags:
ISNATKIND has argument tags [2,5] and root tag 0
ACTIVATE has argument tags [4,5] and root tag 1
AND has argument tags [4,7,4] and root tag 2

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
n__s(x1)  =  x1
activate(x1)  =  x1
ACTIVATE(x1)  =  x1
n__isNatKind(x1)  =  n__isNatKind(x1)
n__and(x1, x2)  =  n__and(x1, x2)
AND(x1, x2)  =  x2
tt  =  tt
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  U31(x2)
and(x1, x2)  =  and(x1, x2)
isNat(x1)  =  x1
isNatKind(x1)  =  isNatKind(x1)
s(x1)  =  x1
U41(x1, x2, x3)  =  U41(x2, x3)
U11(x1, x2, x3)  =  U11
U21(x1, x2)  =  U21(x1)
U12(x1, x2)  =  U12
U22(x1)  =  U22(x1)
U13(x1)  =  U13(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[ISNATKIND1, nisNatKind1, n0, 0, isNatKind1] > [nand2, and2] > [tt, U131]
[ISNATKIND1, nisNatKind1, n0, 0, isNatKind1] > U311 > [tt, U131]
[nplus2, plus2, U412, U11] > U311 > [tt, U131]
[nplus2, plus2, U412, U11] > U12 > [tt, U131]
U211 > U221 > [tt, U131]

Status:
ISNATKIND1: [1]
nisNatKind1: [1]
nand2: multiset
tt: multiset
n0: multiset
0: multiset
nplus2: multiset
plus2: multiset
U311: [1]
and2: multiset
isNatKind1: [1]
U412: multiset
U11: []
U211: multiset
U12: multiset
U221: multiset
U131: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
0n__0

(21) 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(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(22) 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, x1)

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

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
n__s(x1)  =  n__s(x1)
activate(x1)  =  x1
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
and(x1, x2)  =  x2
isNat(x1)  =  isNat
n__isNatKind(x1)  =  n__isNatKind
tt  =  tt
isNatKind(x1)  =  isNatKind
n__and(x1, x2)  =  x2
s(x1)  =  s(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U11(x1, x2, x3)  =  U11(x1, x2)
U21(x1, x2)  =  U21(x1, x2)
U12(x1, x2)  =  U12(x1, x2)
U22(x1)  =  U22(x1)
U13(x1)  =  U13(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[nplus2, plus2, U312, U413] > [ns1, n0, 0, nisNatKind, isNatKind, s1, U212] > tt > isNat
[nplus2, plus2, U312, U413] > U112 > isNat
U122 > U131 > tt > isNat
U221 > tt > isNat

Status:
ISNATKIND1: multiset
ns1: [1]
n0: multiset
0: multiset
nplus2: multiset
plus2: multiset
U312: multiset
isNat: []
nisNatKind: []
tt: multiset
isNatKind: []
s1: [1]
U413: multiset
U112: multiset
U212: multiset
U122: multiset
U221: [1]
U131: multiset


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
0n__0

(23) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE