Termination w.r.t. Q proof of /tmp/tmpOf58ov/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 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 PisEmptyProof (⇔)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 PisEmptyProof (⇔)

        ↳20 TRUE

    ↳21 QDP

        ↳22 QDPOrderProof (⇔)

        ↳23 QDP

        ↳24 PisEmptyProof (⇔)

        ↳25 TRUE

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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:

U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
U12¹(tt, V2) → ISNAT(activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
U21¹(tt, V1) → ISNAT(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, N) → ACTIVATE(N)
U41¹(tt, M, N) → S(plus(activate(N), activate(M)))
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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 5 less nodes.

(4) Obligation:

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

U12¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
U31¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

U12¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), N)
U31¹(tt, N) → ACTIVATE(N)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, V1, V2) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U12¹(x1, x2) = U12¹(x2)
ISNAT(x1) = ISNAT(x1)
U11¹(x1, x2, x3) = U11¹(x2, x3)
ACTIVATE(x1) = ACTIVATE(x1)
PLUS(x1, x2) = PLUS(x1, x2)
U31¹(x1, x2) = U31¹(x2)
ISNATKIND(x1) = ISNATKIND(x1)
AND(x1, x2) = AND(x2)
U21¹(x1, x2) = U21¹(x2)
U41¹(x1, x2, x3) = U41¹(x2, x3)

Tags:
U12¹ has tags [31,23]
ISNAT has tags [0]
U11¹ has tags [29,0,25]
ACTIVATE has tags [0]
PLUS has tags [17,1]
U31¹ has tags [26,8]
ISNATKIND has tags [0]
AND has tags [27,0]
U21¹ has tags [21,0]
U41¹ has tags [12,1,17]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
tt = tt
activate(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
and(x1, x2) = x2
isNatKind(x1) = x1
n__isNatKind(x1) = x1
isNat(x1) = x1
0 = 0
n__and(x1, x2) = x2
n__s(x1) = x1
s(x1) = x1
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = x2
U11(x1, x2, x3) = x2
U21(x1, x2) = U21
U41(x1, x2, x3) = U41(x2, x3)
U12(x1, x2) = U12(x1, x2)
U22(x1) = U22(x1)
U13(x1) = U13(x1)

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

[0, n_₀] > [tt, U22₁] > [n_{_plus₂}, plus₂, U41₂]
[0, n_₀] > [tt, U22₁] > U12₂
U21 > [tt, U22₁] > [n_{_plus₂}, plus₂, U41₂]
U21 > [tt, U22₁] > U12₂
U13₁ > [tt, U22₁] > [n_{_plus₂}, plus₂, U41₂]
U13₁ > [tt, U22₁] > U12₂

Status:

tt: []
n_{_plus₂}: [2,1]
0: []
n_₀: []
plus₂: [2,1]
U21: []
U41₂: [1,2]
U12₂: [1,2]
U22₁: [1]
U13₁: [1]

The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
0 → n__0

(6) Obligation:

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

ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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 3 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U41¹(x1, x2, x3) = U41¹(x2)
PLUS(x1, x2) = PLUS(x2)

Tags:
U41¹ has tags [6,7,6]
PLUS has tags [1,0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
tt = tt
activate(x1) = x1
s(x1) = s(x1)
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat
n__isNatKind(x1) = x1
n__and(x1, x2) = n__and(x1, x2)
n__0 = n__0
0 = 0
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = U31(x1, x2)
isNatKind(x1) = x1
n__s(x1) = n__s(x1)
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U41(x1, x2, x3) = U41(x1, x2, x3)
U12(x1, x2) = U12
U22(x1) = x1
U13(x1) = U13

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

[isNat, n_{_plus₂}, plus₂, U21, U41₃] > [tt, n_₀, 0, U11, U12, U13] > [s₁, n_{_s₁}] > U31₂
[isNat, n_{_plus₂}, plus₂, U21, U41₃] > [tt, n_₀, 0, U11, U12, U13] > [and₂, n_{_and₂}] > U31₂

Status:

tt: []
s₁: [1]
and₂: [1,2]
isNat: []
n_{_and₂}: [1,2]
n_₀: []
0: []
n_{_plus₂}: [1,2]
plus₂: [1,2]
U31₂: [1,2]
n_{_s₁}: [1]
U11: []
U21: []
U41₃: [3,2,1]
U12: []
U13: []

The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → 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))
and(X1, X2) → n__and(X1, X2)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U12(tt, V2) → U13(isNat(activate(V2)))
plus(X1, X2) → n__plus(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
U22(tt) → tt
U13(tt) → tt
s(X) → n__s(X)
0 → n__0

(11) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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:

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → 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.

ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

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

Tags:
ISNATKIND has tags [3]
ACTIVATE has tags [0]
AND has tags [3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
n__s(x1) = x1
activate(x1) = x1
n__isNatKind(x1) = n__isNatKind(x1)
n__and(x1, x2) = n__and(x1, x2)
tt = tt
n__0 = n__0
0 = 0
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = x2
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat(x1)
isNatKind(x1) = isNatKind(x1)
s(x1) = x1
U41(x1, x2, x3) = U41(x2, x3)
U11(x1, x2, x3) = U11(x1, x2)
U21(x1, x2) = U21(x1, x2)
U12(x1, x2) = U12(x1, x2)
U22(x1) = U22
U13(x1) = x1

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

[n_₀, 0] > [n_{_plus₂}, plus₂, isNat₁, U41₂] > [n_{_isNatKind₁}, n_{_and₂}, and₂, isNatKind₁] > tt
[n_₀, 0] > [n_{_plus₂}, plus₂, isNat₁, U41₂] > U11₂
[n_₀, 0] > [n_{_plus₂}, plus₂, isNat₁, U41₂] > U21₂
U22 > tt

Status:

n_{_isNatKind₁}: [1]
n_{_and₂}: [2,1]
tt: []
n_₀: []
0: []
n_{_plus₂}: [2,1]
plus₂: [2,1]
and₂: [2,1]
isNat₁: [1]
isNatKind₁: [1]
U41₂: [1,2]
U11₂: [2,1]
U21₂: [2,1]
U12₂: [1,2]
U22: []

The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
0 → n__0

(16) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

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

Tags:
ISNATKIND has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
n__s(x1) = n__s(x1)
activate(x1) = x1
n__0 = n__0
0 = 0
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = U31(x1, x2)
and(x1, x2) = x2
isNat(x1) = x1
n__isNatKind(x1) = n__isNatKind
tt = tt
isNatKind(x1) = isNatKind
n__and(x1, x2) = x2
s(x1) = s(x1)
U41(x1, x2, x3) = U41(x1, x2, x3)
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U12(x1, x2) = U12(x1, x2)
U22(x1) = U22
U13(x1) = U13

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

[n_₀, 0, n_{_plus₂}, plus₂, U31₂, n_{_isNatKind}, isNatKind, U41₃] > [tt, U22] > [n_{_s₁}, s₁]
[n_₀, 0, n_{_plus₂}, plus₂, U31₂, n_{_isNatKind}, isNatKind, U41₃] > [tt, U22] > [U11, U12₂, U13]

Status:

n_{_s₁}: [1]
n_₀: []
0: []
n_{_plus₂}: [2,1]
plus₂: [2,1]
U31₂: [1,2]
n_{_isNatKind}: []
tt: []
isNatKind: []
s₁: [1]
U41₃: [2,3,1]
U11: []
U21: []
U12₂: [2,1]
U22: []
U13: []

The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
0 → n__0

(18) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNAT(x1) = ISNAT(x1)
U21¹(x1, x2) = U21¹(x2)

Tags:
ISNAT has tags [0]
U21¹ has tags [3,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
n__s(x1) = n__s(x1)
isNatKind(x1) = x1
activate(x1) = x1
tt = tt
n__0 = n__0
0 = 0
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = U31(x1, x2)
and(x1, x2) = and(x1, x2)
isNat(x1) = x1
n__isNatKind(x1) = x1
n__and(x1, x2) = n__and(x1, x2)
s(x1) = s(x1)
U41(x1, x2, x3) = U41(x1, x2, x3)
U11(x1, x2, x3) = x1
U21(x1, x2) = x1
U12(x1, x2) = U12
U22(x1) = U22
U13(x1) = U13

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

[n_₀, 0] > [tt, n_{_plus₂}, plus₂, U41₃, U12, U22, U13] > [n_{_s₁}, s₁] > [and₂, n_{_and₂}]
[n_₀, 0] > [tt, n_{_plus₂}, plus₂, U41₃, U12, U22, U13] > U31₂

Status:

n_{_s₁}: [1]
tt: []
n_₀: []
0: []
n_{_plus₂}: [2,1]
plus₂: [2,1]
U31₂: [1,2]
and₂: [1,2]
n_{_and₂}: [1,2]
s₁: [1]
U41₃: [2,3,1]
U12: []
U22: []
U13: []

The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
U31(tt, N) → activate(N)
activate(n__isNatKind(X)) → isNatKind(X)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
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))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U21(tt, V1) → U22(isNat(activate(V1)))
U12(tt, V2) → U13(isNat(activate(V2)))
plus(X1, X2) → n__plus(X1, X2)
isNat(n__0) → tt
and(X1, X2) → n__and(X1, X2)
U22(tt) → tt
U13(tt) → tt
s(X) → n__s(X)
0 → n__0

(23) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(24) 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) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) TRUE

(21) Obligation:

(22) QDPOrderProof (EQUIVALENT transformation)

(23) Obligation:

(24) PisEmptyProof (EQUIVALENT transformation)

(25) TRUE