(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
U11(x1, x2, x3)  =  U11(x1, x2, x3)
tt  =  tt
U12(x1, x2)  =  U12(x1, x2)
isNat(x1)  =  x1
activate(x1)  =  x1
U13(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
U61(x1)  =  U61(x1)
0  =  0
U71(x1, x2, x3)  =  U71(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
and(x1, x2)  =  and(x1, x2)
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
isNatKind(x1)  =  x1
n__isNatKind(x1)  =  x1
n__s(x1)  =  n__s(x1)
n__x(x1, x2)  =  n__x(x1, x2)
n__and(x1, x2)  =  n__and(x1, x2)

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

[U71₃, x₂, n_x2] > U31₃ > U32₂
[U71₃, x₂, n_x2] > [U51₃, plus₂, n_plus2] > U11₃ > U12₂
[U71₃, x₂, n_x2] > [U51₃, plus₂, n_plus2] > U41₂
[U71₃, x₂, n_x2] > [U51₃, plus₂, n_plus2] > [s₁, n_s1] > U21₂
[U71₃, x₂, n_x2] > [U51₃, plus₂, n_plus2] > [s₁, n_s1] > [and₂, n_and2]
[U71₃, x₂, n_x2] > U61₁ > [0, n₀] > tt
[U71₃, x₂, n_x2] > U61₁ > [0, n₀] > U41₂
[U71₃, x₂, n_x2] > U61₁ > [0, n₀] > [and₂, n_and2]

Status:

U11₃: multiset
tt: multiset
U12₂: multiset
U21₂: multiset
U31₃: [2,1,3]
U32₂: multiset
U41₂: [2,1]
U51₃: [2,3,1]
s₁: multiset
plus₂: [2,1]
U61₁: multiset
0: multiset
U71₃: [3,2,1]
x₂: [1,2]
and₂: multiset
n₀: multiset
n_plus2: [2,1]
n_s1: multiset
n_x2: [1,2]
n_and2: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U21(tt, V1) → U22(isNat(activate(V1)))
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

U33₁ > n_plus2 > U13₁ > activate₁ > n_and2 > and₂ > U22₁ > n_x2 > x₂ > n_s1 > s₁ > n_isNatKind1 > isNatKind₁ > plus₂ > 0 > n₀ > tt

and weight map:

tt=1
0=1
n__0=1
U13_1=1
U22_1=1
U33_1=0
isNatKind_1=2
n__isNatKind_1=1
s_1=2
n__s_1=1
activate_1=1
plus_2=1
n__plus_2=0
x_2=1
n__x_2=0
and_2=1
n__and_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U13(tt) → tt
U22(tt) → tt
U33(tt) → tt
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.