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

(0) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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__AND(tt, X) → MARK(X)
A__PLUS(N, 0) → MARK(N)
A__PLUS(N, s(M)) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → MARK(N)
A__PLUS(N, s(M)) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → 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(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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

(3) 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)) → MARK(M)
MARK(s(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:
A__AND(x1, x2)  =  A__AND(x2)
MARK(x1)  =  MARK(x1)
A__PLUS(x1, x2)  =  A__PLUS(x1, x2)

Tags:
A__AND has tags [4,1]
MARK has tags [1]
A__PLUS has tags [1,0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__plus(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = 1 + x₁   
POL(tt) = 0   

The following usable rules [FROCOS05] were oriented:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

(4) Obligation:

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

A__AND(tt, X) → MARK(X)
A__PLUS(N, 0) → MARK(N)
A__PLUS(N, s(M)) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → MARK(N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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__PLUS(N, 0) → MARK(N)
A__PLUS(N, s(M)) → MARK(N)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)

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

Tags:
A__AND has tags [5,0]
MARK has tags [0]
A__PLUS has tags [5,5]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 0   
POL(tt) = 0   

The following usable rules [FROCOS05] were oriented:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

(6) Obligation:

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

A__AND(tt, X) → MARK(X)
A__PLUS(N, s(M)) → A__PLUS(mark(N), mark(M))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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:

A__PLUS(N, s(M)) → A__PLUS(mark(N), mark(M))

The TRS R consists of the following rules:

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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.

A__PLUS(N, s(M)) → 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(x1, x2)  =  A__PLUS(x2)

Tags:
A__PLUS has tags [1,0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 1 + x₁   
POL(tt) = 0   

The following usable rules [FROCOS05] were oriented:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

(11) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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:

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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.

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)

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

Tags:
MARK has tags [0]
A__AND has tags [3,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(a__and(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__plus(x₁, x₂)) = x₁   
POL(and(x₁, x₂)) = 1 + x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = x₁   
POL(s(x₁)) = x₁   
POL(tt) = 0   

The following usable rules [FROCOS05] were oriented:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → mark(N)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

(16) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)

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.

(18) TRUE