Termination w.r.t. Q proof of /tmp/tmpeQtlLB/MYNAT_nokinds

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 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 PisEmptyProof (⇔)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 PisEmptyProof (⇔)

        ↳18 TRUE

(0) Obligation:

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U21(tt, M, N) → MARK(N)
A__U21(tt, M, N) → MARK(M)
A__U41(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U41(tt, M, N) → A__X(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__AND(a__isNat(V1), isNat(V2))
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__U11(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U31(a__isNat(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U41(a__and(a__isNat(M), isNat(N)), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__X(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(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(U31(X)) → A__U31(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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 2 less nodes.

(4) Obligation:

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

MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U11(tt, N) → MARK(N)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U11(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__AND(tt, X) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
A__U21(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__U41(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U41(a__and(a__isNat(M), isNat(N)), M, N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__X(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__X(N, s(M)) → A__ISNAT(M)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
A__U21(tt, M, N) → MARK(M)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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.

MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U11(tt, N) → MARK(N)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__PLUS(N, 0) → A__U11(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__AND(tt, X) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U21(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__U41(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U41(a__and(a__isNat(M), isNat(N)), M, N)
A__U41(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__X(N, s(M)) → A__AND(a__isNat(M), isNat(N))
A__X(N, s(M)) → A__ISNAT(M)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
A__U21(tt, M, N) → MARK(M)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
A__U11(x0, x1, x2) = A__U11(x2)
A__U21(x0, x1, x2, x3) = A__U21(x0, x2)
A__PLUS(x0, x1, x2) = A__PLUS(x0, x2)
A__ISNAT(x0, x1) = A__ISNAT(x1)
A__AND(x0, x1, x2) = A__AND(x2)
A__U41(x0, x1, x2, x3) = A__U41(x0, x2)
A__X(x0, x1, x2) = A__X(x0)

Tags:
MARK has argument tags [16,4] and root tag 4
A__U11 has argument tags [16,16,4] and root tag 7
A__U21 has argument tags [16,0,16,19] and root tag 0
A__PLUS has argument tags [16,0,8] and root tag 0
A__ISNAT has argument tags [0,3] and root tag 0
A__AND has argument tags [24,16,4] and root tag 5
A__U41 has argument tags [22,16,20,7] and root tag 4
A__X has argument tags [0,0,8] and root tag 6

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U11(x1, x2) = U11(x1, x2)
A__U11(x1, x2) = A__U11(x1)
mark(x1) = x1
tt = tt
U21(x1, x2, x3) = U21(x1, x2, x3)
A__U21(x1, x2, x3) = x3
A__PLUS(x1, x2) = x1
0 = 0
a__isNat(x1) = x1
A__ISNAT(x1) = A__ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
A__AND(x1, x2) = A__AND(x1, x2)
isNat(x1) = x1
s(x1) = s(x1)
a__and(x1, x2) = a__and(x1, x2)
U31(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
A__U41(x1, x2, x3) = A__U41(x2, x3)
a__x(x1, x2) = a__x(x1, x2)
x(x1, x2) = x(x1, x2)
A__X(x1, x2) = A__X(x1, x2)
and(x1, x2) = and(x1, x2)
a__U11(x1, x2) = a__U11(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
a__U21(x1, x2, x3) = a__U21(x1, x2, x3)
a__U31(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[A_{_ISNAT₁}, U41₃, A_{_U41₂}, a_{_x₂}, x₂, A_{_X₂}, a_{_U41₃}] > [tt, U21₃, 0, plus₂, a_{_plus₂}, a_{_U21₃}] > MARK₁ > A_{_U11₁}
[A_{_ISNAT₁}, U41₃, A_{_U41₂}, a_{_x₂}, x₂, A_{_X₂}, a_{_U41₃}] > [tt, U21₃, 0, plus₂, a_{_plus₂}, a_{_U21₃}] > [U11₂, a_{_U11₂}]
[A_{_ISNAT₁}, U41₃, A_{_U41₂}, a_{_x₂}, x₂, A_{_X₂}, a_{_U41₃}] > [tt, U21₃, 0, plus₂, a_{_plus₂}, a_{_U21₃}] > [s₁, a_{_and₂}, and₂]
[A_{_ISNAT₁}, U41₃, A_{_U41₂}, a_{_x₂}, x₂, A_{_X₂}, a_{_U41₃}] > A_{_AND₂} > MARK₁ > A_{_U11₁}

Status:

MARK₁: [1]
U11₂: [2,1]
A_{_U11₁}: [1]
tt: []
U21₃: [3,2,1]
0: []
A_{_ISNAT₁}: [1]
plus₂: [1,2]
A_{_AND₂}: [2,1]
s₁: [1]
a_{_and₂}: [1,2]
U41₃: [3,2,1]
A_{_U41₂}: [2,1]
a_{_x₂}: [1,2]
x₂: [1,2]
A_{_X₂}: [1,2]
and₂: [1,2]
a_{_U11₂}: [2,1]
a_{_plus₂}: [1,2]
a_{_U41₃}: [3,2,1]
a_{_U21₃}: [3,2,1]

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__U11(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
a__U41(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U31(X)) → a__U31(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__and(X1, X2) → and(X1, X2)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(X1, X2) → x(X1, X2)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U11(X1, X2) → U11(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U31(tt) → 0
a__U31(X) → U31(X)

(6) Obligation:

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

A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(U31(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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 2 SCCs.

(8) Complex Obligation (AND)

(9) Obligation:

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

MARK(U31(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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.

MARK(U31(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
U31(x1) = U31(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

MARK: []
U31₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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:

A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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.

A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__PLUS(x0, x1, x2) = A__PLUS(x0, x2)
A__U21(x0, x1, x2, x3) = A__U21(x0, x1, x2)

Tags:
A__PLUS has argument tags [0,2,6] and root tag 0
A__U21 has argument tags [0,0,6,4] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS
s(x1) = s(x1)
A__U21(x1, x2, x3) = x2
a__and(x1, x2) = x2
a__isNat(x1) = a__isNat
isNat(x1) = isNat
tt = tt
mark(x1) = x1
0 = 0
U11(x1, x2) = U11(x1, x2)
a__U11(x1, x2) = a__U11(x1, x2)
plus(x1, x2) = plus(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
U41(x1, x2, x3) = U41(x1, x2, x3)
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
a__x(x1, x2) = a__x(x1, x2)
x(x1, x2) = x(x1, x2)
and(x1, x2) = x2
U21(x1, x2, x3) = U21(x1, x2, x3)
a__U21(x1, x2, x3) = a__U21(x1, x2, x3)
U31(x1) = x1
a__U31(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U41₃, a_{_U41₃}, a_{_x₂}, x₂] > [plus₂, a_{_plus₂}, U21₃, a_{_U21₃}] > [s₁, a_{_isNat}, isNat] > [A_{_PLUS}, tt] > 0
[U41₃, a_{_U41₃}, a_{_x₂}, x₂] > [plus₂, a_{_plus₂}, U21₃, a_{_U21₃}] > [U11₂, a_{_U11₂}]

Status:

A_{_PLUS}: []
s₁: [1]
a_{_isNat}: []
isNat: []
tt: []
0: []
U11₂: [1,2]
a_{_U11₂}: [1,2]
plus₂: [1,2]
a_{_plus₂}: [1,2]
U41₃: [3,2,1]
a_{_U41₃}: [3,2,1]
a_{_x₂}: [1,2]
x₂: [1,2]
U21₃: [3,2,1]
a_{_U21₃}: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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.

(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) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) PisEmptyProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) PisEmptyProof (EQUIVALENT transformation)

(18) TRUE