YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_05_SRS/matchbox1.srs

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

0 QTRS
↳1 QTRSRRRProof (⇔, 31 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 21 ms)
↳4 QDP
↳5 MRRProof (⇔, 88 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 133 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 206 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 TRUE

(0) Obligation:

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

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
t(e(x)) → r(e(x))
w(r(x)) → i(t(x))
e(r(x)) → e(w(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(x))))))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(e(x1)) = 1 + x1   
POL(i(x1)) = x1   
POL(r(x1)) = 2 + x1   
POL(t(x1)) = 3 + x1   
POL(w(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

t(e(x)) → r(e(x))
e(r(x)) → e(w(x))


(2) Obligation:

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

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
w(r(x)) → i(t(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(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:

R(e(x)) → W(r(x))
R(e(x)) → R(x)
I(t(x)) → E(r(x))
I(t(x)) → R(x)
E(w(x)) → R(i(x))
E(w(x)) → I(x)
W(r(x)) → I(t(x))
R(i(t(e(r(x))))) → E(w(r(i(t(e(x))))))
R(i(t(e(r(x))))) → W(r(i(t(e(x)))))
R(i(t(e(r(x))))) → R(i(t(e(x))))
R(i(t(e(r(x))))) → I(t(e(x)))
R(i(t(e(r(x))))) → E(x)

The TRS R consists of the following rules:

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
w(r(x)) → i(t(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(x))))))

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

(5) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

R(e(x)) → R(x)
I(t(x)) → R(x)
E(w(x)) → I(x)
R(i(t(e(r(x))))) → W(r(i(t(e(x)))))
R(i(t(e(r(x))))) → R(i(t(e(x))))
R(i(t(e(r(x))))) → I(t(e(x)))
R(i(t(e(r(x))))) → E(x)


Used ordering: Polynomial interpretation [POLO]:

POL(E(x1)) = 1 + x1   
POL(I(x1)) = x1   
POL(R(x1)) = 2 + x1   
POL(W(x1)) = 1 + x1   
POL(e(x1)) = 1 + x1   
POL(i(x1)) = x1   
POL(r(x1)) = 2 + x1   
POL(t(x1)) = 3 + x1   
POL(w(x1)) = 1 + x1   

(6) Obligation:

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

R(e(x)) → W(r(x))
I(t(x)) → E(r(x))
E(w(x)) → R(i(x))
W(r(x)) → I(t(x))
R(i(t(e(r(x))))) → E(w(r(i(t(e(x))))))

The TRS R consists of the following rules:

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
w(r(x)) → i(t(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(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.


R(i(t(e(r(x))))) → E(w(r(i(t(e(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(R(x1)) = 0A +
[-I,0A,-I]
·x1

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

POL(W(x1)) = -I +
[0A,0A,0A]
·x1

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

POL(I(x1)) = -I +
[0A,0A,0A]
·x1

POL(t(x1)) =
/0A\
|0A|
\0A/
 +
/-I0A0A\
|-I-I-I|
\0A0A0A/
·x1

POL(E(x1)) = 0A +
[-I,-I,0A]
·x1

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

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

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

w(r(x)) → i(t(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
r(e(x)) → w(r(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(x))))))

(8) Obligation:

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

R(e(x)) → W(r(x))
I(t(x)) → E(r(x))
E(w(x)) → R(i(x))
W(r(x)) → I(t(x))

The TRS R consists of the following rules:

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
w(r(x)) → i(t(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(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.


R(e(x)) → W(r(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(R(x1)) = 1A +
[1A,0A,-I]
·x1

POL(e(x1)) =
/0A\
|1A|
\0A/
 +
/0A0A1A\
|-I0A0A|
\-I0A0A/
·x1

POL(W(x1)) = -I +
[0A,0A,0A]
·x1

POL(r(x1)) =
/0A\
|-I|
\0A/
 +
/0A0A1A\
|-I0A0A|
\-I0A0A/
·x1

POL(I(x1)) = -I +
[0A,0A,-I]
·x1

POL(t(x1)) =
/0A\
|0A|
\1A/
 +
/0A0A1A\
|-I0A0A|
\0A1A1A/
·x1

POL(E(x1)) = 0A +
[0A,0A,0A]
·x1

POL(w(x1)) =
/-I\
|1A|
\-I/
 +
/0A0A1A\
|-I0A0A|
\-I0A0A/
·x1

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

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

w(r(x)) → i(t(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
r(e(x)) → w(r(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(x))))))

(10) Obligation:

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

I(t(x)) → E(r(x))
E(w(x)) → R(i(x))
W(r(x)) → I(t(x))

The TRS R consists of the following rules:

r(e(x)) → w(r(x))
i(t(x)) → e(r(x))
e(w(x)) → r(i(x))
w(r(x)) → i(t(x))
r(i(t(e(r(x))))) → e(w(r(i(t(e(x))))))

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(12) TRUE