3.31/1.72	YES
3.31/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.31/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.31/1.73	
3.31/1.73	
3.31/1.73	Termination w.r.t. Q of the given QTRS could be proven:
3.31/1.73	
3.31/1.73	(0) QTRS
3.31/1.73	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
3.31/1.73	(2) QDP
3.31/1.73	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
3.31/1.73	(4) TRUE
3.31/1.73	
3.31/1.73	
3.31/1.73	----------------------------------------
3.31/1.73	
3.31/1.73	(0)
3.31/1.73	Obligation:
3.31/1.73	Q restricted rewrite system:
3.31/1.73	The TRS R consists of the following rules:
3.31/1.73	
3.31/1.73	   f(g(x), x, y) -> f(y, y, g(y))
3.31/1.73	   g(g(x)) -> g(x)
3.31/1.73	
3.31/1.73	The set Q consists of the following terms:
3.31/1.73	
3.31/1.73	   f(g(x0), x0, x1)
3.31/1.73	   g(g(x0))
3.31/1.73	
3.31/1.73	
3.31/1.73	----------------------------------------
3.31/1.73	
3.31/1.73	(1) DependencyPairsProof (EQUIVALENT)
3.31/1.73	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.31/1.73	----------------------------------------
3.31/1.73	
3.31/1.73	(2)
3.31/1.73	Obligation:
3.31/1.73	Q DP problem:
3.31/1.73	The TRS P consists of the following rules:
3.31/1.73	
3.31/1.73	   F(g(x), x, y) -> F(y, y, g(y))
3.31/1.73	   F(g(x), x, y) -> G(y)
3.31/1.73	
3.31/1.73	The TRS R consists of the following rules:
3.31/1.73	
3.31/1.73	   f(g(x), x, y) -> f(y, y, g(y))
3.31/1.73	   g(g(x)) -> g(x)
3.31/1.73	
3.31/1.73	The set Q consists of the following terms:
3.31/1.73	
3.31/1.73	   f(g(x0), x0, x1)
3.31/1.73	   g(g(x0))
3.31/1.73	
3.31/1.73	We have to consider all minimal (P,Q,R)-chains.
3.31/1.73	----------------------------------------
3.31/1.73	
3.31/1.73	(3) DependencyGraphProof (EQUIVALENT)
3.31/1.73	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
3.31/1.73	----------------------------------------
3.31/1.73	
3.31/1.73	(4)
3.31/1.73	TRUE
3.31/1.76	EOF