YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_05_SRS/torpa3.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 16 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 0 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 98 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 TRUE

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

B(b(x)) → C(d(x))
B(b(x)) → D(x)
C(c(x)) → D(d(d(x)))
C(c(x)) → D(d(x))
C(c(x)) → D(x)
C(x) → G(x)
D(d(x)) → C(f(x))
D(d(x)) → F(x)
D(d(d(x))) → G(c(x))
D(d(d(x))) → C(x)
F(x) → G(x)
G(x) → D(a(b(x)))
G(x) → B(x)
G(g(x)) → B(c(x))
G(g(x)) → C(x)

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

C(c(x)) → D(d(d(x)))
D(d(x)) → C(f(x))
C(c(x)) → D(d(x))
D(d(x)) → F(x)
F(x) → G(x)
G(x) → B(x)
B(b(x)) → C(d(x))
C(c(x)) → D(x)
D(d(d(x))) → G(c(x))
G(g(x)) → B(c(x))
B(b(x)) → D(x)
D(d(d(x))) → C(x)
C(x) → G(x)
G(g(x)) → C(x)

The TRS R consists of the following rules:

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

POL(B(x1)) = 2 + 2·x1   
POL(C(x1)) = 2 + 2·x1   
POL(D(x1)) = 2·x1   
POL(F(x1)) = 3 + 2·x1   
POL(G(x1)) = 2 + 2·x1   
POL(a(x1)) = 0   
POL(b(x1)) = 3 + x1   
POL(c(x1)) = 3 + x1   
POL(d(x1)) = 2 + x1   
POL(f(x1)) = x1   
POL(g(x1)) = 3 + x1   

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

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

(6) Obligation:

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

C(c(x)) → D(d(d(x)))
G(x) → B(x)
D(d(d(x))) → G(c(x))
G(g(x)) → B(c(x))
C(x) → G(x)

The TRS R consists of the following rules:

b(b(x)) → c(d(x))
c(c(x)) → d(d(d(x)))
c(x) → g(x)
d(d(x)) → c(f(x))
d(d(d(x))) → g(c(x))
f(x) → a(g(x))
g(x) → d(a(b(x)))
g(g(x)) → 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 0 SCCs with 5 less nodes.

(8) TRUE