YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_06_SRS/secr9.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 40 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 81 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 QDPOrderProof (⇔, 80 ms)
        ↳11 QDP
        ↳12 PisEmptyProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 QDPOrderProof (⇔, 26 ms)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔, 141 ms)
        ↳18 QDP
        ↳19 DependencyGraphProof (⇔, 0 ms)
        ↳20 AND
            ↳21 QDP
                ↳22 QDPOrderProof (⇔, 161 ms)
                ↳23 QDP
                ↳24 PisEmptyProof (⇔, 0 ms)
                ↳25 YES
            ↳26 QDP
                ↳27 QDPOrderProof (⇔, 7 ms)
                ↳28 QDP
                ↳29 PisEmptyProof (⇔, 0 ms)
                ↳30 YES

(0) Obligation:

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

a(c(x)) → c(b(c(c(a(x)))))
b(b(b(x))) → c(b(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
a(b(x)) → c(c(a(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → c(b(b(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:

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

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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,JAR06].


The following pairs can be oriented strictly and are deleted.


D(d(x)) → D(b(d(b(d(x)))))
D(d(x)) → B(d(b(d(x))))
D(d(x)) → D(b(d(x)))
D(d(x)) → B(d(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 0   
POL(B(x1)) = 0   
POL(C(x1)) = 0   
POL(D(x1)) = x1   
POL(a(x1)) = 0   
POL(b(x1)) = 0   
POL(c(x1)) = 0   
POL(d(x1)) = 1   

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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 2 SCCs with 3 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

A(a(x)) → A(d(a(x)))

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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,JAR06].


The following pairs can be oriented strictly and are deleted.


A(a(x)) → A(d(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A(x1) ) = max{0, 2x1 - 2}

POL( d(x1) ) = 0

POL( a(x1) ) = 2x1 + 2

POL( b(x1) ) = max{0, -2}

POL( c(x1) ) = 0


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

d(d(x)) → d(b(d(b(d(x)))))

(11) Obligation:

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

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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) YES

(14) Obligation:

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

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

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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,JAR06].


The following pairs can be oriented strictly and are deleted.


C(a(x)) → C(b(c(x)))
C(a(x)) → C(c(b(c(x))))
C(a(x)) → B(c(x))
C(a(x)) → C(x)
B(a(x)) → C(c(x))
B(a(x)) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B(x1)) = 1 + x1   
POL(C(x1)) = 1 + x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = x1   
POL(d(x1)) = 0   

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

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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,JAR06].


The following pairs can be oriented strictly and are deleted.


B(b(b(x))) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(B(x1)) = 0A +
[-I,0A,-I]
·x1

POL(b(x1)) =
/0A\
|0A|
\1A/
 +
/-I0A-I\
|-I0A0A|
\0A1A0A/
·x1

POL(c(x1)) =
/0A\
|0A|
\0A/
 +
/1A0A0A\
|0A0A-I|
\0A0A-I/
·x1

POL(C(x1)) = 0A +
[-I,0A,-I]
·x1

POL(a(x1)) =
/0A\
|0A|
\0A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(d(x1)) =
/-I\
|-I|
\-I/
 +
/0A-I-I\
|0A-I0A|
\0A0A0A/
·x1

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

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

(18) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(20) Complex Obligation (AND)

(21) Obligation:

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

B(b(b(x))) → B(c(x))

The TRS R consists of the following rules:

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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,JAR06].


The following pairs can be oriented strictly and are deleted.


B(b(b(x))) → B(c(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(B(x1)) = -I +
[0A,-I,-I]
·x1

POL(b(x1)) =
/0A\
|-I|
\1A/
 +
/-I-I0A\
|-I-I0A|
\0A1A0A/
·x1

POL(c(x1)) =
/0A\
|0A|
\-I/
 +
/-I0A-I\
|0A1A-I|
\0A0A-I/
·x1

POL(a(x1)) =
/0A\
|0A|
\0A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(d(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A1A1A|
\0A0A0A/
·x1

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

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

(23) Obligation:

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

c(a(x)) → a(c(c(b(c(x)))))
b(b(b(x))) → b(c(x))
d(d(x)) → d(b(d(b(d(x)))))
a(a(x)) → a(d(a(x)))
b(a(x)) → a(c(c(x)))
c(c(x)) → c(b(c(b(c(x)))))
c(c(c(x))) → b(b(c(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.

(25) YES

(26) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(27) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


C(c(x)) → C(b(c(x)))
C(c(x)) → C(b(c(b(c(x)))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(C(x1)) = x1   
POL(a(x1)) = 0   
POL(b(x1)) = 0   
POL(c(x1)) = 1   
POL(d(x1)) = 1 + x1   

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

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

(28) Obligation:

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

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

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

(29) PisEmptyProof (EQUIVALENT transformation)

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

(30) YES