YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_07_SRS/x04.srs

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

0 QTRS
↳1 QTRSRRRProof (⇔, 26 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 35 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 QDPOrderProof (⇔, 275 ms)
        ↳11 QDP
        ↳12 QDPOrderProof (⇔, 139 ms)
        ↳13 QDP
        ↳14 QDPOrderProof (⇔, 135 ms)
        ↳15 QDP
        ↳16 QDPOrderProof (⇔, 38 ms)
        ↳17 QDP
        ↳18 DependencyGraphProof (⇔, 0 ms)
        ↳19 QDP
        ↳20 QDPOrderProof (⇔, 33 ms)
        ↳21 QDP
        ↳22 PisEmptyProof (⇔, 0 ms)
        ↳23 YES
    ↳24 QDP
        ↳25 UsableRulesProof (⇔, 0 ms)
        ↳26 QDP
        ↳27 QDPOrderProof (⇔, 0 ms)
        ↳28 QDP
        ↳29 PisEmptyProof (⇔, 0 ms)
        ↳30 YES
    ↳31 QDP
        ↳32 QDPOrderProof (⇔, 0 ms)
        ↳33 QDP
        ↳34 QDPOrderProof (⇔, 0 ms)
        ↳35 QDP
        ↳36 PisEmptyProof (⇔, 0 ms)
        ↳37 YES

(0) Obligation:

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

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a(x1)) = x1   
POL(b(x1)) = x1   
POL(c(x1)) = 1 + x1   
POL(d(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

c(b(x)) → b(a(d(x)))


(2) Obligation:

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

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

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

The TRS R consists of the following rules:

a(a(x)) → a(b(a(b(a(x)))))
c(a(x)) → a(b(a(a(c(x)))))
b(b(b(x))) → a(b(x))
c(b(x)) → a(a(c(x)))
d(d(x)) → d(b(d(b(d(x)))))
c(c(x)) → c(d(c(x)))
a(a(a(x))) → a(b(b(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 3 SCCs with 11 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(8) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(9) Obligation:

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

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

The TRS R consists of the following rules:

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

POL(A(x1)) = 0A +
[0A,-I,-I]
·x1

POL(a(x1)) =
/-I\
|0A|
\0A/
 +
/0A0A-I\
|1A1A0A|
\0A0A-I/
·x1

POL(b(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I0A\
|-I-I0A|
\1A0A0A/
·x1

POL(B(x1)) = -I +
[0A,0A,-I]
·x1

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

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

(11) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

POL(A(x1)) = 0A +
[-I,0A,0A]
·x1

POL(a(x1)) =
/0A\
|1A|
\0A/
 +
/-I0A-I\
|0A0A0A|
\-I-I-I/
·x1

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

POL(B(x1)) = -I +
[0A,0A,0A]
·x1

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

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

(13) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

POL(A(x1)) = 0A +
[0A,0A,-I]
·x1

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

POL(b(x1)) =
/0A\
|0A|
\1A/
 +
/-I0A0A\
|-I0A0A|
\0A0A0A/
·x1

POL(B(x1)) = 1A +
[0A,0A,0A]
·x1

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

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

(15) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(16) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(A(x1)) = 0A +
[-I,-I,0A]
·x1

POL(a(x1)) =
/0A\
|0A|
\1A/
 +
/-I-I-I\
|0A0A1A|
\0A0A1A/
·x1

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

POL(B(x1)) = 0A +
[0A,-I,-I]
·x1

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

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

(17) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(18) DependencyGraphProof (EQUIVALENT transformation)

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

(19) Obligation:

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

A(a(a(x))) → A(b(b(x)))

The TRS R consists of the following rules:

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

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

(20) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(A(x1)) = 0A +
[-I,-I,0A]
·x1

POL(a(x1)) =
/0A\
|1A|
\0A/
 +
/-I-I0A\
|0A0A0A|
\0A0A1A/
·x1

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

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

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

(21) Obligation:

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

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

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

(22) PisEmptyProof (EQUIVALENT transformation)

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

(23) YES

(24) Obligation:

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

D(d(x)) → D(b(d(x)))
D(d(x)) → D(b(d(b(d(x)))))

The TRS R consists of the following rules:

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

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

(25) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(26) Obligation:

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

D(d(x)) → D(b(d(x)))
D(d(x)) → D(b(d(b(d(x)))))

The TRS R consists of the following rules:

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


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

POL(D(x1)) = x1   
POL(a(x1)) = x1   
POL(b(x1)) = 0   
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))) → a(b(x))
a(a(x)) → a(b(a(b(a(x)))))
a(a(a(x))) → a(b(b(x)))

(28) Obligation:

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

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

(31) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(32) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

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

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)))))

(33) Obligation:

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

C(c(x)) → C(d(c(x)))

The TRS R consists of the following rules:

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

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

(34) 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(d(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)) = x1   
POL(c(x1)) = 1 + x1   
POL(d(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)))))

(35) Obligation:

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

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

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

(36) PisEmptyProof (EQUIVALENT transformation)

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

(37) YES