YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z118.srs

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

0 QTRS
↳1 QTRSRRRProof (⇔, 51 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 2 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 8 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 TRUE

(0) Obligation:

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

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a(x1)) = 6 + x1   
POL(b(x1)) = 9 + x1   
POL(c(x1)) = 13 + x1   
POL(d(x1)) = x1   
POL(g(x1)) = 6 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

b(b(b(x))) → c(d(c(x)))
c(d(x)) → g(g(x))


(2) Obligation:

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

a(x) → g(d(x))
b(b(x)) → a(g(g(x)))
g(g(g(x))) → 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(x) → G(d(x))
B(b(x)) → A(g(g(x)))
B(b(x)) → G(g(x))
B(b(x)) → G(x)
G(g(g(x))) → B(b(x))
G(g(g(x))) → B(x)

The TRS R consists of the following rules:

a(x) → g(d(x))
b(b(x)) → a(g(g(x)))
g(g(g(x))) → 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 1 SCC with 2 less nodes.

(6) Obligation:

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

B(b(x)) → G(g(x))
G(g(g(x))) → B(b(x))
B(b(x)) → G(x)
G(g(g(x))) → B(x)

The TRS R consists of the following rules:

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


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

POL(B(x1)) = 1 + x1   
POL(G(x1)) = 1 + x1   
POL(a(x1)) = 1   
POL(b(x1)) = 1 + x1   
POL(d(x1)) = 0   
POL(g(x1)) = 1 + x1   

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

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

(8) Obligation:

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

B(b(x)) → G(g(x))

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(10) TRUE