Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Waldmann_07

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

0 QTRS

↳1 QTRS Reverse (⇔, 0 ms)

↳2 QTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 QDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 QDP

↳7 QDPOrderProof (⇔, 183 ms)

↳8 QDP

↳9 QDPOrderProof (⇔, 124 ms)

↳10 QDP

↳11 QDPOrderProof (⇔, 76 ms)

↳12 QDP

↳13 PisEmptyProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

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

a(x) → x
a(b(x)) → x
a(c(c(x))) → c(c(b(a(c(a(x))))))

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

Q is empty.

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

C(c(a(x))) → A(c(a(b(c(c(x))))))
C(c(a(x))) → C(a(b(c(c(x)))))
C(c(a(x))) → A(b(c(c(x))))
C(c(a(x))) → B(c(c(x)))
C(c(a(x))) → C(c(x))
C(c(a(x))) → C(x)

The TRS R consists of the following rules:

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

C(c(a(x))) → C(c(x))
C(c(a(x))) → C(a(b(c(c(x)))))
C(c(a(x))) → C(x)

The TRS R consists of the following rules:

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

C(c(a(x))) → C(x)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(C(x₁)) = 0A +
[ 0A, 0A, 0A ]

· x₁

POL(c(x₁)) =
/ 0A \

| -I |

\ 0A /

+
/ -I 0A 1A \

| -I -I 0A |

\ 0A 0A 0A /

· x₁

POL(a(x₁)) =
/ 1A \

| 0A |

\ -I /

+
/ 1A 1A 1A \

| 0A 0A 0A |

\ -I 0A 0A /

· x₁

POL(b(x₁)) =
/ 0A \

| -I |

\ -I /

+
/ -I 0A -I \

| -I 0A -I |

\ -I 0A -I /

· x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

c(c(a(x))) → a(c(a(b(c(c(x))))))
b(a(x)) → x
a(x) → x

(8) Obligation:

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

C(c(a(x))) → C(c(x))
C(c(a(x))) → C(a(b(c(c(x)))))

The TRS R consists of the following rules:

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

C(c(a(x))) → C(a(b(c(c(x)))))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(C(x₁)) = 0A +
[ -I, 0A, 0A ]

· x₁

POL(c(x₁)) =
/ 0A \

| 0A |

\ 0A /

+
/ 0A 1A 0A \

| 0A -I 0A |

\ -I 0A -I /

· x₁

POL(a(x₁)) =
/ 1A \

| 0A |

\ 0A /

+
/ 1A 0A 1A \

| -I 0A 0A |

\ 0A 0A 0A /

· x₁

POL(b(x₁)) =
/ 0A \

| -I |

\ -I /

+
/ -I -I 0A \

| -I -I 0A |

\ -I -I 0A /

· x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

c(c(a(x))) → a(c(a(b(c(c(x))))))
b(a(x)) → x
a(x) → x

(10) Obligation:

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

C(c(a(x))) → C(c(x))

The TRS R consists of the following rules:

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

C(c(a(x))) → C(c(x))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(C(x₁)) = -I +
[ 0A, -I, 0A ]

· x₁

POL(c(x₁)) =
/ 0A \

| 0A |

\ 0A /

+
/ 0A 0A 0A \

| 1A -I 0A |

\ 0A -I 0A /

· x₁

POL(a(x₁)) =
/ -I \

| 1A |

\ 0A /

+
/ 0A 0A -I \

| 1A 1A 1A |

\ 0A 0A 0A /

· x₁

POL(b(x₁)) =
/ 0A \

| -I |

\ -I /

+
/ -I -I 0A \

| -I -I 0A |

\ -I -I 0A /

· x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

c(c(a(x))) → a(c(a(b(c(c(x))))))
b(a(x)) → x
a(x) → x

(12) Obligation:

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

a(x) → x
b(a(x)) → x
c(c(a(x))) → a(c(a(b(c(c(x))))))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) YES