YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 54 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) TRUE


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(x1)) -> C(d(x1))
   B(b(x1)) -> D(x1)
   C(c(x1)) -> D(d(d(x1)))
   C(c(x1)) -> D(d(x1))
   C(c(x1)) -> D(x1)
   C(x1) -> G(x1)
   D(d(x1)) -> C(f(x1))
   D(d(x1)) -> F(x1)
   D(d(d(x1))) -> G(c(x1))
   D(d(d(x1))) -> C(x1)
   F(x1) -> G(x1)
   G(x1) -> D(a(b(x1)))
   G(x1) -> B(x1)
   G(g(x1)) -> B(c(x1))
   G(g(x1)) -> C(x1)

The TRS R consists of the following rules:

   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   C(c(x1)) -> D(d(d(x1)))
   D(d(x1)) -> C(f(x1))
   C(c(x1)) -> D(d(x1))
   D(d(x1)) -> F(x1)
   F(x1) -> G(x1)
   G(x1) -> B(x1)
   B(b(x1)) -> C(d(x1))
   C(c(x1)) -> D(x1)
   D(d(d(x1))) -> G(c(x1))
   G(g(x1)) -> B(c(x1))
   B(b(x1)) -> D(x1)
   D(d(d(x1))) -> C(x1)
   C(x1) -> G(x1)
   G(g(x1)) -> C(x1)

The TRS R consists of the following rules:

   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   D(d(x1)) -> C(f(x1))
   C(c(x1)) -> D(d(x1))
   D(d(x1)) -> F(x1)
   F(x1) -> G(x1)
   B(b(x1)) -> C(d(x1))
   C(c(x1)) -> D(x1)
   B(b(x1)) -> D(x1)
   D(d(d(x1))) -> C(x1)
   G(g(x1)) -> C(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B(x_1)) = 2 + 2*x_1
   POL(C(x_1)) = 2 + 2*x_1
   POL(D(x_1)) = 2*x_1
   POL(F(x_1)) = 3 + 2*x_1
   POL(G(x_1)) = 2 + 2*x_1
   POL(a(x_1)) = 0
   POL(b(x_1)) = 3 + x_1
   POL(c(x_1)) = 3 + x_1
   POL(d(x_1)) = 2 + x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = 3 + x_1

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

   c(c(x1)) -> d(d(d(x1)))
   d(d(x1)) -> c(f(x1))
   c(x1) -> g(x1)
   g(g(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))


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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   C(c(x1)) -> D(d(d(x1)))
   G(x1) -> B(x1)
   D(d(d(x1))) -> G(c(x1))
   G(g(x1)) -> B(c(x1))
   C(x1) -> G(x1)

The TRS R consists of the following rules:

   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.
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(8)
TRUE