Termination w.r.t. Q proof of Transformed_CSR_04_MYNAT_complete

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

0 QTRS

↳1 QTRSRRRProof (⇔, 274 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 0 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 1 ms)

↳6 QTRS

↳7 QTRSRRRProof (⇔, 0 ms)

↳8 QTRS

↳9 QTRSRRRProof (⇔, 22 ms)

↳10 QTRS

↳11 RisEmptyProof (⇔, 0 ms)

↳12 YES

(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) = a__isNat(x1)
a__U13(x1) = a__U13(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) = isNat(x1)
U11(x1, x2, x3) = U11(x1, x2, x3)
U12(x1, x2) = U12(x1, x2)
U13(x1) = U13(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:

[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U31₃}, U31₃] > [a_{_U32₂}, U32₂] > [tt, a_{_isNat₁}, isNat₁] > 0
[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U51₃}, a_{_plus₂}, plus₂, U51₃] > [a_{_U11₃}, U11₃] > [a_{_U12₂}, U12₂] > [a_{_U13₁}, U13₁] > [tt, a_{_isNat₁}, isNat₁] > 0
[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U51₃}, a_{_plus₂}, plus₂, U51₃] > [a_{_U41₂}, U41₂]
[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U51₃}, a_{_plus₂}, plus₂, U51₃] > s₁ > [a_{_U21₂}, U21₂] > [tt, a_{_isNat₁}, isNat₁] > 0
[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U51₃}, a_{_plus₂}, plus₂, U51₃] > s₁ > [a_{_isNatKind₁}, isNatKind₁] > [tt, a_{_isNat₁}, isNat₁] > 0
[a_{_U71₃}, a_{_x₂}, x₂, U71₃] > [a_{_U51₃}, a_{_plus₂}, plus₂, U51₃] > s₁ > [a_{_isNatKind₁}, isNatKind₁] > [a_{_and₂}, and₂]

Status:

a_{_U11₃}: [1,2,3]
tt: multiset
a_{_U12₂}: [2,1]
a_{_isNat₁}: [1]
a_{_U13₁}: [1]
a_{_U21₂}: multiset
a_{_U31₃}: multiset
a_{_U32₂}: multiset
a_{_U41₂}: multiset
a_{_U51₃}: [2,3,1]
s₁: [1]
a_{_plus₂}: [2,1]
0: multiset
a_{_U71₃}: [2,3,1]
a_{_x₂}: [2,1]
a_{_and₂}: multiset
plus₂: [2,1]
a_{_isNatKind₁}: multiset
isNatKind₁: multiset
x₂: [2,1]
and₂: multiset
isNat₁: [1]
U11₃: [1,2,3]
U12₂: [2,1]
U13₁: [1]
U21₂: multiset
U31₃: multiset
U32₂: multiset
U41₂: multiset
U51₃: [2,3,1]
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__U13(tt) → tt
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__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)

(2) Obligation:

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

a__U22(tt) → tt
a__U33(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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 2
POL(U11(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 1 + x₁ + x₂
POL(U13(x₁)) = 2 + x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U32(x₁, x₂)) = 1 + x₁ + x₂
POL(U33(x₁)) = 1 + 2·x₁
POL(U41(x₁, x₂)) = 1 + x₁ + x₂
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U61(x₁)) = 1 + x₁
POL(U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(a__U12(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U13(x₁)) = 2 + x₁
POL(a__U21(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U22(x₁)) = 2 + x₁
POL(a__U31(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(a__U32(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U33(x₁)) = 2 + 2·x₁
POL(a__U41(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U51(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(a__U61(x₁)) = 2 + x₁
POL(a__U71(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(a__and(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__isNat(x₁)) = 2 + 2·x₁
POL(a__isNatKind(x₁)) = 2 + x₁
POL(a__plus(x₁, x₂)) = 2 + x₁ + x₂
POL(a__x(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(and(x₁, x₂)) = 1 + 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)) = 1 + x₁
POL(tt) = 1
POL(x(x₁, x₂)) = 1 + x₁ + 2·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
a__U33(tt) → tt
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
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)

(4) Obligation:

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

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
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)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U13(X) → U13(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 2·x₁ + x₂
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁)) = 2·x₁
POL(U31(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U32(x₁, x₂)) = 2·x₁ + x₂
POL(U33(x₁)) = 2·x₁
POL(U41(x₁, x₂)) = 2·x₁ + x₂
POL(U51(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U61(x₁)) = 2·x₁
POL(U71(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + 2·x₃
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U13(x₁)) = 2 + 2·x₁
POL(a__U21(x₁, x₂)) = x₁ + x₂
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·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 + 2·x₂
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(plus(x₁, x₂)) = 2·x₁ + 2·x₂
POL(x(x₁, x₂)) = 2·x₁ + 2·x₂

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

a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U13(X) → U13(X)

(6) Obligation:

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

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
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)
a__U31(X1, X2, X3) → U31(X1, X2, X3)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U12(x₁, x₂)) = 2·x₁ + x₂
POL(U21(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁)) = 2·x₁
POL(U31(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U32(x₁, x₂)) = 2·x₁ + x₂
POL(U33(x₁)) = 2·x₁
POL(U41(x₁, x₂)) = 2·x₁ + x₂
POL(U51(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U61(x₁)) = 2·x₁
POL(U71(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U21(x₁, x₂)) = 2·x₁ + x₂
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + 2·x₃
POL(a__U32(x₁, x₂)) = x₁ + x₂
POL(a__U33(x₁)) = 2·x₁
POL(a__U41(x₁, x₂)) = x₁ + x₂
POL(a__U51(x₁, x₂, x₃)) = 2·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 + 2·x₂
POL(a__x(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(plus(x₁, x₂)) = 2·x₁ + 2·x₂
POL(x(x₁, x₂)) = 2·x₁ + 2·x₂

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

a__U31(X1, X2, X3) → U31(X1, X2, X3)

(8) Obligation:

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

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
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)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

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

a_{_U41₂} > a_{_U33₁} > mark₁ > a_{_plus₂} > U32₂ > U61₁ > a_{_isNatKind₁} > U12₂ > isNatKind₁ > x₂ > plus₂ > a_{_U51₃} > a_{_U22₁} > a_{_U32₂} > a_{_U12₂} > isNat₁ > a_{_U21₂} > and₂ > a_{_U61₁} > U71₃ > a_{_U71₃} > U41₂ > U33₁ > a_{_x₂} > a_{_isNat₁} > U51₃ > a_{_and₂} > U22₁ > U21₂

and weight map:

mark_1=4
isNat_1=1
a__isNat_1=5
U22_1=2
a__U22_1=1
U33_1=2
a__U33_1=1
U61_1=2
a__U61_1=1
isNatKind_1=1
a__isNatKind_1=5
U12_2=1
a__U12_2=0
U21_2=1
a__U21_2=0
U32_2=1
a__U32_2=0
U41_2=1
a__U41_2=0
U51_3=1
a__U51_3=0
plus_2=4
a__plus_2=0
U71_3=1
a__U71_3=0
x_2=4
a__x_2=0
and_2=4
a__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:

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
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)

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) QTRSRRRProof (EQUIVALENT transformation)

(10) Obligation:

(11) RisEmptyProof (EQUIVALENT transformation)

(12) YES