Termination w.r.t. Q proof of /tmp/tmp_6Elcu/Ex1_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:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__s(X)) → S(activate(X))
IF(true, X, Y) → ACTIVATE(X)
DIFF(X, Y) → LEQ(X, Y)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__p(X)) → P(activate(X))
ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__0) → 0¹
LEQ(s(X), s(Y)) → LEQ(X, Y)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(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:

ACTIVATE(n__s(X)) → S(activate(X))
IF(true, X, Y) → ACTIVATE(X)
DIFF(X, Y) → LEQ(X, Y)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__p(X)) → P(activate(X))
ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__0) → 0¹
LEQ(s(X), s(Y)) → LEQ(X, Y)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

LEQ(s(X), s(Y)) → LEQ(X, Y)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

LEQ(s(X), s(Y)) → LEQ(X, Y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

LEQ(s(X), s(Y)) → LEQ(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

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

ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF(true, X, Y) → ACTIVATE(X) we obtained the following new rules:

IF(true, n__0, n__s(n__diff(n__p(y_3), y_4))) → ACTIVATE(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

              ↳ QDP

                ↳ DependencyGraphProof

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

IF(true, n__0, n__s(n__diff(n__p(y_3), y_4))) → ACTIVATE(n__0)
ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__diff(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(DIFF(x₁, x₂)) = 1 + x₁ + x₂
POL(IF(x₁, x₂, x₃)) = x₃
POL(activate(x₁)) = x₁
POL(diff(x₁, x₂)) = 1 + x₁ + x₂
POL(false) = 0
POL(if(x₁, x₂, x₃)) = x₂ + x₃
POL(leq(x₁, x₂)) = 0
POL(n__0) = 0
POL(n__diff(x₁, x₂)) = 1 + x₁ + x₂
POL(n__p(x₁)) = x₁
POL(n__s(x₁)) = x₁
POL(p(x₁)) = x₁
POL(s(x₁)) = x₁
POL(true) = 0

The following usable rules [17] were oriented:

p(0) → 0
p(s(X)) → X
0 → n__0
activate(n__0) → 0
p(X) → n__p(X)
diff(X1, X2) → n__diff(X1, X2)
s(X) → n__s(X)
activate(X) → X
activate(n__p(X)) → p(activate(X))
if(false, X, Y) → activate(Y)
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
if(true, X, Y) → activate(X)
activate(n__s(X)) → s(activate(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

ACTIVATE(n__p(X)) → ACTIVATE(X)
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__p(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(DIFF(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃)) = x₃
POL(activate(x₁)) = 0
POL(diff(x₁, x₂)) = 0
POL(false) = 0
POL(if(x₁, x₂, x₃)) = 0
POL(leq(x₁, x₂)) = 0
POL(n__0) = 0
POL(n__diff(x₁, x₂)) = 0
POL(n__p(x₁)) = 1 + x₁
POL(n__s(x₁)) = x₁
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ Instantiation

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

ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF(false, X, Y) → ACTIVATE(Y) we obtained the following new rules:

IF(false, n__0, n__s(n__diff(n__p(y_3), y_4))) → ACTIVATE(n__s(n__diff(n__p(y_3), y_4)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Instantiation

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

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

IF(false, n__0, n__s(n__diff(n__p(y_3), y_4))) → ACTIVATE(n__s(n__diff(n__p(y_3), y_4)))
ACTIVATE(n__diff(X1, X2)) → DIFF(activate(X1), activate(X2))
DIFF(X, Y) → IF(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
0 → n__0
s(X) → n__s(X)
diff(X1, X2) → n__diff(X1, X2)
p(X) → n__p(X)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2))
activate(n__p(X)) → p(activate(X))
activate(X) → X

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