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

(0) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a__U11(x1, x2, x3)  =  a__U11(x1, x2, x3)
tt  =  tt
a__U12(x1, x2)  =  a__U12(x1, x2)
a__isNat(x1)  =  x1
a__U13(x1)  =  x1
a__U21(x1, x2)  =  a__U21(x1, x2)
a__U22(x1)  =  x1
a__U31(x1, x2, x3)  =  a__U31(x1, x2, x3)
a__U32(x1, x2)  =  a__U32(x1, x2)
a__U33(x1)  =  x1
a__U41(x1, x2)  =  a__U41(x1, x2)
mark(x1)  =  x1
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
s(x1)  =  s(x1)
a__plus(x1, x2)  =  a__plus(x1, x2)
a__U61(x1)  =  x1
0  =  0
a__U71(x1, x2, x3)  =  a__U71(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
a__and(x1, x2)  =  a__and(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
a__isNatKind(x1)  =  a__isNatKind(x1)
isNatKind(x1)  =  isNatKind(x1)
x(x1, x2)  =  x(x1, x2)
and(x1, x2)  =  and(x1, x2)
isNat(x1)  =  x1
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2)  =  U12(x1, x2)
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)
U61(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1, x2, x3)

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

[tt, 0] > [a_U713, a_x2, x₂, U71₃] > [a_U313, a_U322, U31₃, U32₂] > [a_U212, a_isNatKind1, isNatKind₁, U21₂]
[tt, 0] > [a_U713, a_x2, x₂, U71₃] > [a_U513, a_plus2, plus₂, U51₃] > [a_U113, U11₃] > [a_U122, U12₂] > [a_U212, a_isNatKind1, isNatKind₁, U21₂]
[tt, 0] > [a_U713, a_x2, x₂, U71₃] > [a_U513, a_plus2, plus₂, U51₃] > [a_U412, U41₂] > [a_U212, a_isNatKind1, isNatKind₁, U21₂]
[tt, 0] > [a_U713, a_x2, x₂, U71₃] > [a_U513, a_plus2, plus₂, U51₃] > s₁ > [a_U212, a_isNatKind1, isNatKind₁, U21₂]
[tt, 0] > [a_U713, a_x2, x₂, U71₃] > [a_U513, a_plus2, plus₂, U51₃] > [a_and2, and₂] > [a_U212, a_isNatKind1, isNatKind₁, U21₂]

Status:

U32₂: multiset
U12₂: multiset
x₂: [2,1]
and₂: multiset
U21₂: [2,1]
a_plus2: [2,1]
U31₃: multiset
tt: multiset
a_U212: [2,1]
s₁: [1]
U51₃: [2,3,1]
plus₂: [2,1]
a_U412: multiset
a_and2: multiset
U11₃: multiset
isNatKind₁: multiset
0: multiset
a_U513: [2,3,1]
a_x2: [2,1]
a_isNatKind1: multiset
a_U113: multiset
U41₂: multiset
a_U122: multiset
a_U322: multiset
a_U713: [2,3,1]
a_U313: multiset
U71₃: [2,3,1]

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

(2) Obligation:

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

a__U13(tt) → tt
a__U22(tt) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__isNat(0) → tt
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U12(x₁, x₂)) = x₁ + x₂   
POL(U13(x₁)) = 1 + x₁   
POL(U21(x₁, x₂)) = x₁ + x₂   
POL(U22(x₁)) = x₁   
POL(U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U32(x₁, x₂)) = x₁ + x₂   
POL(U33(x₁)) = 1 + x₁   
POL(U41(x₁, x₂)) = x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U61(x₁)) = 2 + x₁   
POL(U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U12(x₁, x₂)) = x₁ + x₂   
POL(a__U13(x₁)) = 1 + x₁   
POL(a__U21(x₁, x₂)) = x₁ + x₂   
POL(a__U22(x₁)) = x₁   
POL(a__U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U32(x₁, x₂)) = x₁ + x₂   
POL(a__U33(x₁)) = 1 + x₁   
POL(a__U41(x₁, x₂)) = x₁ + x₂   
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U61(x₁)) = 2 + x₁   
POL(a__U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__isNat(x₁)) = x₁   
POL(a__isNatKind(x₁)) = x₁   
POL(a__plus(x₁, x₂)) = x₁ + x₂   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = x₁   
POL(isNatKind(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = x₁   
POL(tt) = 0   
POL(x(x₁, x₂)) = x₁ + x₂   

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

a__U13(tt) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__isNat(0) → tt

(4) Obligation:

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

a__U22(tt) → tt
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U12(x₁, x₂)) = x₁ + x₂   
POL(U13(x₁)) = x₁   
POL(U21(x₁, x₂)) = x₁ + x₂   
POL(U22(x₁)) = 1 + x₁   
POL(U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U32(x₁, x₂)) = x₁ + x₂   
POL(U33(x₁)) = x₁   
POL(U41(x₁, x₂)) = x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U61(x₁)) = x₁   
POL(U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U12(x₁, x₂)) = x₁ + x₂   
POL(a__U13(x₁)) = x₁   
POL(a__U21(x₁, x₂)) = x₁ + x₂   
POL(a__U22(x₁)) = 1 + x₁   
POL(a__U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U32(x₁, x₂)) = x₁ + x₂   
POL(a__U33(x₁)) = x₁   
POL(a__U41(x₁, x₂)) = x₁ + x₂   
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U61(x₁)) = x₁   
POL(a__U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__isNat(x₁)) = x₁   
POL(a__isNatKind(x₁)) = x₁   
POL(a__plus(x₁, x₂)) = x₁ + x₂   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = x₁   
POL(isNatKind(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = x₁   
POL(tt) = 0   
POL(x(x₁, x₂)) = x₁ + x₂   

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

a__U22(tt) → tt

(6) Obligation:

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

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U12(x₁, x₂)) = 1 + x₁ + x₂   
POL(U13(x₁)) = x₁   
POL(U21(x₁, x₂)) = x₁ + x₂   
POL(U22(x₁)) = x₁   
POL(U31(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃   
POL(U32(x₁, x₂)) = 2 + x₁ + x₂   
POL(U33(x₁)) = 1 + x₁   
POL(U41(x₁, x₂)) = x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃   
POL(U61(x₁)) = 1 + x₁   
POL(U71(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃   
POL(a__U11(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃   
POL(a__U12(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(a__U13(x₁)) = x₁   
POL(a__U21(x₁, x₂)) = x₁ + 2·x₂   
POL(a__U22(x₁)) = x₁   
POL(a__U31(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃   
POL(a__U32(x₁, x₂)) = 2 + x₁ + x₂   
POL(a__U33(x₁)) = 1 + x₁   
POL(a__U41(x₁, x₂)) = x₁ + 2·x₂   
POL(a__U51(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃   
POL(a__U61(x₁)) = 2 + x₁   
POL(a__U71(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃   
POL(a__and(x₁, x₂)) = 2 + x₁ + x₂   
POL(a__isNat(x₁)) = 2 + 2·x₁   
POL(a__isNatKind(x₁)) = 1 + 2·x₁   
POL(a__plus(x₁, x₂)) = 2 + x₁ + x₂   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = 2 + x₁ + x₂   
POL(isNat(x₁)) = 1 + x₁   
POL(isNatKind(x₁)) = 1 + x₁   
POL(mark(x₁)) = 2·x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 2 + x₁   
POL(tt) = 1   
POL(x(x₁, x₂)) = x₁ + x₂   

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

mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)

(8) Obligation:

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

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(0) → 0
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U12(x₁, x₂)) = 1 + x₁ + x₂   
POL(U13(x₁)) = x₁   
POL(U21(x₁, x₂)) = x₁ + x₂   
POL(U22(x₁)) = x₁   
POL(U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U32(x₁, x₂)) = x₁ + x₂   
POL(U33(x₁)) = x₁   
POL(U41(x₁, x₂)) = x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U61(x₁)) = 1 + x₁   
POL(U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U12(x₁, x₂)) = x₁ + x₂   
POL(a__U13(x₁)) = x₁   
POL(a__U21(x₁, x₂)) = x₁ + x₂   
POL(a__U22(x₁)) = x₁   
POL(a__U31(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__U32(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U33(x₁)) = 1 + x₁   
POL(a__U41(x₁, x₂)) = x₁ + x₂   
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U61(x₁)) = x₁   
POL(a__U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__and(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__isNat(x₁)) = x₁   
POL(a__isNatKind(x₁)) = 1 + x₁   
POL(a__plus(x₁, x₂)) = x₁ + x₂   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 1 + x₁   
POL(isNatKind(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(x(x₁, x₂)) = x₁ + x₂   

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

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

(10) Obligation:

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

mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(0) → 0
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U13(x₁)) = 1 + 2·x₁   
POL(U21(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(U22(x₁)) = 1 + 2·x₁   
POL(U41(x₁, x₂)) = 1 + x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃   
POL(a__U13(x₁)) = 1 + 2·x₁   
POL(a__U21(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(a__U22(x₁)) = 2 + 2·x₁   
POL(a__U41(x₁, x₂)) = 2 + x₁ + x₂   
POL(a__U51(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃   
POL(a__x(x₁, x₂)) = 2·x₁ + x₂   
POL(mark(x₁)) = 2·x₁   
POL(x(x₁, x₂)) = 2·x₁ + x₂   

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

mark(U13(X)) → a__U13(mark(X))
mark(0) → 0
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)

(12) Obligation:

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

mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__U13(X) → U13(X)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U13(x₁)) = x₁   
POL(U21(x₁, x₂)) = 1 + x₁ + x₂   
POL(U22(x₁)) = 1 + x₁   
POL(U41(x₁, x₂)) = 1 + x₁ + x₂   
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__U13(x₁)) = 1 + x₁   
POL(a__U21(x₁, x₂)) = x₁ + x₂   
POL(a__U22(x₁)) = x₁   
POL(a__U41(x₁, x₂)) = x₁ + x₂   
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(x(x₁, x₂)) = x₁ + x₂   

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

mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
a__U13(X) → U13(X)

(14) Obligation:

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

mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(X1, X2) → x(X1, X2)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

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

mark₁ > a_x2 > x₂

Status:

a_x2: [2,1]
mark₁: [1]
x₂: multiset

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

mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(X1, X2) → x(X1, X2)

(16) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) RisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE