Termination w.r.t. Q proof of TRS/TRCSREx1_GM03

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVATE₁(n__0) -> 0¹
DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))
DIFF₂(X, Y) -> LEQ₂(X, Y)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X2)
LEQ₂(s₁(X), s₁(Y)) -> LEQ₂(X, Y)
IF₃(false, X, Y) -> ACTIVATE₁(Y)
ACTIVATE₁(n__p₁(X)) -> P₁(activate₁(X))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ACTIVATE₁(n__0) -> 0¹
DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))
DIFF₂(X, Y) -> LEQ₂(X, Y)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X2)
LEQ₂(s₁(X), s₁(Y)) -> LEQ₂(X, Y)
IF₃(false, X, Y) -> ACTIVATE₁(Y)
ACTIVATE₁(n__p₁(X)) -> P₁(activate₁(X))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

LEQ₂(s₁(X), s₁(Y)) -> LEQ₂(X, Y)

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

LEQ₂(s₁(X), s₁(Y)) -> LEQ₂(X, Y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s₁(x₁) ) = x₁ + 3

POL( LEQ₂(x₁, x₂) ) = 3x₂ + 3

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X2)
DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__diff₂(X1, X2)) -> ACTIVATE₁(X2)

The remaining pairs can at least be oriented weakly.

ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)
DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( if₃(x₁, ..., x₃) ) = 2x₂ + x₃

POL( n__0 ) = max{0, -3}

POL( n__diff₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( IF₃(x₁, ..., x₃) ) = 2x₂ + 2x₃ + 1

POL( n__s₁(x₁) ) = x₁

POL( 0 ) = max{0, -3}

POL( DIFF₂(x₁, x₂) ) = 2x₁ + 2x₂ + 3

POL( activate₁(x₁) ) = x₁

POL( true ) = max{0, -3}

POL( ACTIVATE₁(x₁) ) = 2x₁ + 1

POL( false ) = 1

POL( leq₂(x₁, x₂) ) = max{0, -3}

POL( s₁(x₁) ) = x₁

POL( n__p₁(x₁) ) = x₁

POL( diff₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( p₁(x₁) ) = x₁

The following usable rules [14] were oriented:

0 -> n__0
activate₁(n__0) -> 0
s₁(X) -> n__s₁(X)
p₁(0) -> 0
activate₁(X) -> X
activate₁(n__s₁(X)) -> s₁(activate₁(X))
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(s₁(X)) -> X
if₃(false, X, Y) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
p₁(X) -> n__p₁(X)
activate₁(n__p₁(X)) -> p₁(activate₁(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)
DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVATE₁(n__p₁(X)) -> ACTIVATE₁(X)

The remaining pairs can at least be oriented weakly.

DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( if₃(x₁, ..., x₃) ) = x₂ + 2x₃

POL( n__0 ) = max{0, -3}

POL( n__diff₂(x₁, x₂) ) = 1

POL( IF₃(x₁, ..., x₃) ) = 2x₂ + 2x₃ + 1

POL( n__s₁(x₁) ) = x₁

POL( 0 ) = 2

POL( DIFF₂(x₁, x₂) ) = 3

POL( activate₁(x₁) ) = 1

POL( true ) = 2

POL( ACTIVATE₁(x₁) ) = 2x₁ + 1

POL( false ) = 0

POL( leq₂(x₁, x₂) ) = x₂ + 1

POL( s₁(x₁) ) = max{0, 3x₁ - 2}

POL( n__p₁(x₁) ) = 3x₁ + 2

POL( diff₂(x₁, x₂) ) = 2x₁ + 3

POL( p₁(x₁) ) = max{0, -2}

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

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

DIFF₂(X, Y) -> IF₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
IF₃(true, X, Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__diff₂(X1, X2)) -> DIFF₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

p₁(0) -> 0
p₁(s₁(X)) -> X
leq₂(0, Y) -> true
leq₂(s₁(X), 0) -> false
leq₂(s₁(X), s₁(Y)) -> leq₂(X, Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
diff₂(X, Y) -> if₃(leq₂(X, Y), n__0, n__s₁(n__diff₂(n__p₁(X), Y)))
0 -> n__0
s₁(X) -> n__s₁(X)
diff₂(X1, X2) -> n__diff₂(X1, X2)
p₁(X) -> n__p₁(X)
activate₁(n__0) -> 0
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__diff₂(X1, X2)) -> diff₂(activate₁(X1), activate₁(X2))
activate₁(n__p₁(X)) -> p₁(activate₁(X))
activate₁(X) -> X

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