12.98/4.14	YES
13.21/4.17	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.21/4.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.21/4.17	
13.21/4.17	
13.21/4.17	Termination w.r.t. Q of the given QTRS could be proven:
13.21/4.17	
13.21/4.17	(0) QTRS
13.21/4.17	(1) DependencyPairsProof [EQUIVALENT, 31 ms]
13.21/4.17	(2) QDP
13.21/4.17	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
13.21/4.17	(4) QDP
13.21/4.17	(5) QDPOrderProof [EQUIVALENT, 94 ms]
13.21/4.17	(6) QDP
13.21/4.17	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
13.21/4.17	(8) TRUE
13.21/4.17	
13.21/4.17	
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(0)
13.21/4.17	Obligation:
13.21/4.17	Q restricted rewrite system:
13.21/4.17	The TRS R consists of the following rules:
13.21/4.17	
13.21/4.17	   b(b(x1)) -> c(d(x1))
13.21/4.17	   c(c(x1)) -> d(d(d(x1)))
13.21/4.17	   c(x1) -> g(x1)
13.21/4.17	   d(d(x1)) -> c(f(x1))
13.21/4.17	   d(d(d(x1))) -> g(c(x1))
13.21/4.17	   f(x1) -> a(g(x1))
13.21/4.17	   g(x1) -> d(a(b(x1)))
13.21/4.17	   g(g(x1)) -> b(c(x1))
13.21/4.17	
13.21/4.17	Q is empty.
13.21/4.17	
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(1) DependencyPairsProof (EQUIVALENT)
13.21/4.17	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(2)
13.21/4.17	Obligation:
13.21/4.17	Q DP problem:
13.21/4.17	The TRS P consists of the following rules:
13.21/4.17	
13.21/4.17	   B(b(x1)) -> C(d(x1))
13.21/4.17	   B(b(x1)) -> D(x1)
13.21/4.17	   C(c(x1)) -> D(d(d(x1)))
13.21/4.17	   C(c(x1)) -> D(d(x1))
13.21/4.17	   C(c(x1)) -> D(x1)
13.21/4.17	   C(x1) -> G(x1)
13.21/4.17	   D(d(x1)) -> C(f(x1))
13.21/4.17	   D(d(x1)) -> F(x1)
13.21/4.17	   D(d(d(x1))) -> G(c(x1))
13.21/4.17	   D(d(d(x1))) -> C(x1)
13.21/4.17	   F(x1) -> G(x1)
13.21/4.17	   G(x1) -> D(a(b(x1)))
13.21/4.17	   G(x1) -> B(x1)
13.21/4.17	   G(g(x1)) -> B(c(x1))
13.21/4.17	   G(g(x1)) -> C(x1)
13.21/4.17	
13.21/4.17	The TRS R consists of the following rules:
13.21/4.17	
13.21/4.17	   b(b(x1)) -> c(d(x1))
13.21/4.17	   c(c(x1)) -> d(d(d(x1)))
13.21/4.17	   c(x1) -> g(x1)
13.21/4.17	   d(d(x1)) -> c(f(x1))
13.21/4.17	   d(d(d(x1))) -> g(c(x1))
13.21/4.17	   f(x1) -> a(g(x1))
13.21/4.17	   g(x1) -> d(a(b(x1)))
13.21/4.17	   g(g(x1)) -> b(c(x1))
13.21/4.17	
13.21/4.17	Q is empty.
13.21/4.17	We have to consider all minimal (P,Q,R)-chains.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(3) DependencyGraphProof (EQUIVALENT)
13.21/4.17	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(4)
13.21/4.17	Obligation:
13.21/4.17	Q DP problem:
13.21/4.17	The TRS P consists of the following rules:
13.21/4.17	
13.21/4.17	   C(c(x1)) -> D(d(d(x1)))
13.21/4.17	   D(d(x1)) -> C(f(x1))
13.21/4.17	   C(c(x1)) -> D(d(x1))
13.21/4.17	   D(d(x1)) -> F(x1)
13.21/4.17	   F(x1) -> G(x1)
13.21/4.17	   G(x1) -> B(x1)
13.21/4.17	   B(b(x1)) -> C(d(x1))
13.21/4.17	   C(c(x1)) -> D(x1)
13.21/4.17	   D(d(d(x1))) -> G(c(x1))
13.21/4.17	   G(g(x1)) -> B(c(x1))
13.21/4.17	   B(b(x1)) -> D(x1)
13.21/4.17	   D(d(d(x1))) -> C(x1)
13.21/4.17	   C(x1) -> G(x1)
13.21/4.17	   G(g(x1)) -> C(x1)
13.21/4.17	
13.21/4.17	The TRS R consists of the following rules:
13.21/4.17	
13.21/4.17	   b(b(x1)) -> c(d(x1))
13.21/4.17	   c(c(x1)) -> d(d(d(x1)))
13.21/4.17	   c(x1) -> g(x1)
13.21/4.17	   d(d(x1)) -> c(f(x1))
13.21/4.17	   d(d(d(x1))) -> g(c(x1))
13.21/4.17	   f(x1) -> a(g(x1))
13.21/4.17	   g(x1) -> d(a(b(x1)))
13.21/4.17	   g(g(x1)) -> b(c(x1))
13.21/4.17	
13.21/4.17	Q is empty.
13.21/4.17	We have to consider all minimal (P,Q,R)-chains.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(5) QDPOrderProof (EQUIVALENT)
13.21/4.17	We use the reduction pair processor [LPAR04,JAR06].
13.21/4.17	
13.21/4.17	
13.21/4.17	The following pairs can be oriented strictly and are deleted.
13.21/4.17	
13.21/4.17	   D(d(x1)) -> C(f(x1))
13.21/4.17	   C(c(x1)) -> D(d(x1))
13.21/4.17	   D(d(x1)) -> F(x1)
13.21/4.17	   F(x1) -> G(x1)
13.21/4.17	   B(b(x1)) -> C(d(x1))
13.21/4.17	   C(c(x1)) -> D(x1)
13.21/4.17	   B(b(x1)) -> D(x1)
13.21/4.17	   D(d(d(x1))) -> C(x1)
13.21/4.17	   G(g(x1)) -> C(x1)
13.21/4.17	The remaining pairs can at least be oriented weakly.
13.21/4.17	Used ordering:  Polynomial interpretation [POLO]:
13.21/4.17	
13.21/4.17	   POL(B(x_1)) = 2 + 2*x_1
13.21/4.17	   POL(C(x_1)) = 2 + 2*x_1
13.21/4.17	   POL(D(x_1)) = 2*x_1
13.21/4.17	   POL(F(x_1)) = 3 + 2*x_1
13.21/4.17	   POL(G(x_1)) = 2 + 2*x_1
13.21/4.17	   POL(a(x_1)) = 0
13.21/4.17	   POL(b(x_1)) = 3 + x_1
13.21/4.17	   POL(c(x_1)) = 3 + x_1
13.21/4.17	   POL(d(x_1)) = 2 + x_1
13.21/4.17	   POL(f(x_1)) = x_1
13.21/4.17	   POL(g(x_1)) = 3 + x_1
13.21/4.17	
13.21/4.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.21/4.17	
13.21/4.17	   c(c(x1)) -> d(d(d(x1)))
13.21/4.17	   d(d(x1)) -> c(f(x1))
13.21/4.17	   c(x1) -> g(x1)
13.21/4.17	   g(g(x1)) -> b(c(x1))
13.21/4.17	   b(b(x1)) -> c(d(x1))
13.21/4.17	   d(d(d(x1))) -> g(c(x1))
13.21/4.17	   f(x1) -> a(g(x1))
13.21/4.17	   g(x1) -> d(a(b(x1)))
13.21/4.17	
13.21/4.17	
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(6)
13.21/4.17	Obligation:
13.21/4.17	Q DP problem:
13.21/4.17	The TRS P consists of the following rules:
13.21/4.17	
13.21/4.17	   C(c(x1)) -> D(d(d(x1)))
13.21/4.17	   G(x1) -> B(x1)
13.21/4.17	   D(d(d(x1))) -> G(c(x1))
13.21/4.17	   G(g(x1)) -> B(c(x1))
13.21/4.17	   C(x1) -> G(x1)
13.21/4.17	
13.21/4.17	The TRS R consists of the following rules:
13.21/4.17	
13.21/4.17	   b(b(x1)) -> c(d(x1))
13.21/4.17	   c(c(x1)) -> d(d(d(x1)))
13.21/4.17	   c(x1) -> g(x1)
13.21/4.17	   d(d(x1)) -> c(f(x1))
13.21/4.17	   d(d(d(x1))) -> g(c(x1))
13.21/4.17	   f(x1) -> a(g(x1))
13.21/4.17	   g(x1) -> d(a(b(x1)))
13.21/4.17	   g(g(x1)) -> b(c(x1))
13.21/4.17	
13.21/4.17	Q is empty.
13.21/4.17	We have to consider all minimal (P,Q,R)-chains.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(7) DependencyGraphProof (EQUIVALENT)
13.21/4.17	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.
13.21/4.17	----------------------------------------
13.21/4.17	
13.21/4.17	(8)
13.21/4.17	TRUE
13.50/4.31	EOF