Termination w.r.t. Q proof of /tmp/tmpcF93Gb/PEANO_complete

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 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 PisEmptyProof (⇔)

↳14 TRUE

(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(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:

A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__U12(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, N) → MARK(N)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U41(tt, M, N) → MARK(N)
A__U41(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → A__U22(mark(X))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(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 4 less nodes.

(4) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(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.

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2) = A__U12(x2)
tt = tt
A__ISNAT(x1) = A__ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x1, x2, x3)
a__and(x1, x2) = a__and(x1, x2)
a__isNatKind(x1) = x1
isNatKind(x1) = x1
a__isNat(x1) = a__isNat(x1)
A__AND(x1, x2) = A__AND(x1, x2)
MARK(x1) = MARK(x1)
U11(x1, x2, x3) = U11(x1, x2, x3)
mark(x1) = x1
U12(x1, x2) = U12(x1, x2)
isNat(x1) = isNat(x1)
A__ISNATKIND(x1) = A__ISNATKIND(x1)
s(x1) = s(x1)
A__U21(x1, x2) = A__U21(x2)
U13(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1) = x1
U31(x1, x2) = U31(x1, x2)
A__U31(x1, x2) = A__U31(x2)
U41(x1, x2, x3) = U41(x1, x2, x3)
A__U41(x1, x2, x3) = A__U41(x2, x3)
A__PLUS(x1, x2) = A__PLUS(x1, x2)
0 = 0
and(x1, x2) = and(x1, x2)
a__U11(x1, x2, x3) = a__U11(x1, x2, x3)
a__U12(x1, x2) = a__U12(x1, x2)
a__U13(x1) = x1
a__U21(x1, x2) = a__U21(x1, x2)
a__U22(x1) = x1
a__U31(x1, x2) = a__U31(x1, x2)
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)

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

[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > [U11₃, a_{_U11₃}] > A_{_U11₃} > [A_{_U12₁}, A_{_ISNAT₁}, A_{_U21₁}] > [a_{_and₂}, A_{_AND₂}, MARK₁, A_{_ISNATKIND₁}, A_{_U31₁}, and₂]
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > [U11₃, a_{_U11₃}] > [U12₂, a_{_U12₂}] > [a_{_isNat₁}, isNat₁, 0] > [A_{_U12₁}, A_{_ISNAT₁}, A_{_U21₁}] > [a_{_and₂}, A_{_AND₂}, MARK₁, A_{_ISNATKIND₁}, A_{_U31₁}, and₂]
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > [U11₃, a_{_U11₃}] > [U12₂, a_{_U12₂}] > [a_{_isNat₁}, isNat₁, 0] > tt
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > s₁ > [U21₂, a_{_U21₂}] > [a_{_isNat₁}, isNat₁, 0] > [A_{_U12₁}, A_{_ISNAT₁}, A_{_U21₁}] > [a_{_and₂}, A_{_AND₂}, MARK₁, A_{_ISNATKIND₁}, A_{_U31₁}, and₂]
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > s₁ > [U21₂, a_{_U21₂}] > [a_{_isNat₁}, isNat₁, 0] > tt
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > s₁ > [A_{_U41₂}, A_{_PLUS₂}] > [a_{_isNat₁}, isNat₁, 0] > [A_{_U12₁}, A_{_ISNAT₁}, A_{_U21₁}] > [a_{_and₂}, A_{_AND₂}, MARK₁, A_{_ISNATKIND₁}, A_{_U31₁}, and₂]
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > s₁ > [A_{_U41₂}, A_{_PLUS₂}] > [a_{_isNat₁}, isNat₁, 0] > tt
[plus₂, U41₃, a_{_U41₃}, a_{_plus₂}] > [U31₂, a_{_U31₂}] > [a_{_and₂}, A_{_AND₂}, MARK₁, A_{_ISNATKIND₁}, A_{_U31₁}, and₂]

Status:

A_{_U12₁}: multiset
tt: multiset
A_{_ISNAT₁}: multiset
plus₂: [1,2]
A_{_U11₃}: [3,2,1]
a_{_and₂}: multiset
a_{_isNat₁}: [1]
A_{_AND₂}: multiset
MARK₁: multiset
U11₃: [3,2,1]
U12₂: [2,1]
isNat₁: [1]
A_{_ISNATKIND₁}: multiset
s₁: [1]
A_{_U21₁}: multiset
U21₂: multiset
U31₂: multiset
A_{_U31₁}: multiset
U41₃: [3,2,1]
A_{_U41₂}: multiset
A_{_PLUS₂}: multiset
0: multiset
and₂: multiset
a_{_U11₃}: [3,2,1]
a_{_U12₂}: [2,1]
a_{_U21₂}: multiset
a_{_U31₂}: multiset
a_{_U41₃}: [3,2,1]
a_{_plus₂}: [1,2]

The following usable rules [FROCOS05] were oriented:

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

(6) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U13(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
A__U31(tt, N) → MARK(N)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(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 1 SCC with 5 less nodes.

(8) Obligation:

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

MARK(U22(X)) → MARK(X)
MARK(U13(X)) → MARK(X)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U13(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U22(x1) = x1
U13(x1) = U13(x1)
a__U11(x1, x2, x3) = a__U11(x2, x3)
tt = tt
a__U12(x1, x2) = a__U12(x1, x2)
a__isNat(x1) = x1
a__U13(x1) = a__U13(x1)
a__U21(x1, x2) = x2
a__U22(x1) = x1
a__U31(x1, x2) = a__U31(x2)
mark(x1) = x1
a__U41(x1, x2, x3) = a__U41(x2, x3)
s(x1) = x1
a__plus(x1, x2) = a__plus(x1, x2)
a__and(x1, x2) = x2
0 = 0
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = a__isNatKind(x1)
isNatKind(x1) = isNatKind(x1)
and(x1, x2) = x2
isNat(x1) = x1
U11(x1, x2, x3) = U11(x2, x3)
U12(x1, x2) = U12(x1, x2)
U21(x1, x2) = x2
U31(x1, x2) = U31(x2)
U41(x1, x2, x3) = U41(x2, x3)

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

MARK₁ > [a_{_isNatKind₁}, isNatKind₁]
[a_{_U11₂}, a_{_U41₂}, a_{_plus₂}, plus₂, U11₂, U41₂] > [tt, a_{_U12₂}, U12₂] > [U13₁, a_{_U13₁}] > [a_{_isNatKind₁}, isNatKind₁]
[a_{_U11₂}, a_{_U41₂}, a_{_plus₂}, plus₂, U11₂, U41₂] > [a_{_U31₁}, U31₁] > [a_{_isNatKind₁}, isNatKind₁]
0 > [tt, a_{_U12₂}, U12₂] > [U13₁, a_{_U13₁}] > [a_{_isNatKind₁}, isNatKind₁]
0 > [a_{_U31₁}, U31₁] > [a_{_isNatKind₁}, isNatKind₁]

Status:

MARK₁: multiset
U13₁: multiset
a_{_U11₂}: multiset
tt: multiset
a_{_U12₂}: multiset
a_{_U13₁}: multiset
a_{_U31₁}: [1]
a_{_U41₂}: multiset
a_{_plus₂}: multiset
0: multiset
plus₂: multiset
a_{_isNatKind₁}: multiset
isNatKind₁: multiset
U11₂: multiset
U12₂: multiset
U31₁: [1]
U41₂: multiset

The following usable rules [FROCOS05] were oriented:

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

(10) Obligation:

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

MARK(U22(X)) → MARK(X)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U22(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = x1
U22(x1) = U22(x1)
a__U11(x1, x2, x3) = a__U11(x1, x2)
tt = tt
a__U12(x1, x2) = a__U12(x1)
a__isNat(x1) = x1
a__U13(x1) = a__U13
a__U21(x1, x2) = a__U21(x2)
a__U22(x1) = a__U22(x1)
a__U31(x1, x2) = a__U31(x1, x2)
mark(x1) = x1
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
s(x1) = s(x1)
a__plus(x1, x2) = a__plus(x1, x2)
a__and(x1, x2) = x2
0 = 0
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = a__isNatKind
isNatKind(x1) = isNatKind
and(x1, x2) = x2
isNat(x1) = x1
U11(x1, x2, x3) = U11(x1, x2)
U12(x1, x2) = U12(x1)
U13(x1) = U13
U21(x1, x2) = U21(x2)
U31(x1, x2) = U31(x1, x2)
U41(x1, x2, x3) = U41(x1, x2, x3)

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

[a_{_U41₃}, a_{_plus₂}, plus₂, U41₃] > [a_{_U11₂}, U11₂] > [a_{_U12₁}, a_{_U13}, U12₁, U13] > [tt, a_{_isNatKind}, isNatKind]
[a_{_U41₃}, a_{_plus₂}, plus₂, U41₃] > [a_{_U31₂}, U31₂] > [tt, a_{_isNatKind}, isNatKind]
[a_{_U41₃}, a_{_plus₂}, plus₂, U41₃] > s₁ > [U22₁, a_{_U21₁}, a_{_U22₁}, U21₁] > [tt, a_{_isNatKind}, isNatKind]
0 > [a_{_U31₂}, U31₂] > [tt, a_{_isNatKind}, isNatKind]

Status:

U22₁: multiset
a_{_U11₂}: multiset
tt: multiset
a_{_U12₁}: multiset
a_{_U13}: multiset
a_{_U21₁}: multiset
a_{_U22₁}: multiset
a_{_U31₂}: multiset
a_{_U41₃}: [3,2,1]
s₁: multiset
a_{_plus₂}: [1,2]
0: multiset
plus₂: [1,2]
a_{_isNatKind}: multiset
isNatKind: multiset
U11₂: multiset
U12₁: multiset
U13: multiset
U21₁: multiset
U31₂: multiset
U41₃: [3,2,1]

The following usable rules [FROCOS05] were oriented:

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

(12) Obligation:

Q DP problem:
P is empty.
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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE