3.33/1.62	YES
3.33/1.63	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.33/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.33/1.63	
3.33/1.63	
3.33/1.63	Termination w.r.t. Q of the given QTRS could be proven:
3.33/1.63	
3.33/1.63	(0) QTRS
3.33/1.63	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
3.33/1.63	(2) QDP
3.33/1.63	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
3.33/1.63	(4) TRUE
3.33/1.63	
3.33/1.63	
3.33/1.63	----------------------------------------
3.33/1.63	
3.33/1.63	(0)
3.33/1.63	Obligation:
3.33/1.63	Q restricted rewrite system:
3.33/1.63	The TRS R consists of the following rules:
3.33/1.63	
3.33/1.63	   f(0, 1, x) -> f(x, x, x)
3.33/1.63	   f(x, y, z) -> 2
3.33/1.63	   0 -> 2
3.33/1.63	   1 -> 2
3.33/1.63	   g(x, x, y) -> y
3.33/1.63	   g(x, y, y) -> x
3.33/1.63	
3.33/1.63	The set Q consists of the following terms:
3.33/1.63	
3.33/1.63	   f(x0, x1, x2)
3.33/1.63	   0
3.33/1.63	   1
3.33/1.63	   g(x0, x0, x1)
3.33/1.63	   g(x0, x1, x1)
3.33/1.63	
3.33/1.63	
3.33/1.63	----------------------------------------
3.33/1.63	
3.33/1.63	(1) DependencyPairsProof (EQUIVALENT)
3.33/1.63	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.33/1.63	----------------------------------------
3.33/1.63	
3.33/1.63	(2)
3.33/1.63	Obligation:
3.33/1.63	Q DP problem:
3.33/1.63	The TRS P consists of the following rules:
3.33/1.63	
3.33/1.63	   F(0, 1, x) -> F(x, x, x)
3.33/1.63	
3.33/1.63	The TRS R consists of the following rules:
3.33/1.63	
3.33/1.63	   f(0, 1, x) -> f(x, x, x)
3.33/1.63	   f(x, y, z) -> 2
3.33/1.63	   0 -> 2
3.33/1.63	   1 -> 2
3.33/1.63	   g(x, x, y) -> y
3.33/1.63	   g(x, y, y) -> x
3.33/1.63	
3.33/1.63	The set Q consists of the following terms:
3.33/1.63	
3.33/1.63	   f(x0, x1, x2)
3.33/1.63	   0
3.33/1.63	   1
3.33/1.63	   g(x0, x0, x1)
3.33/1.63	   g(x0, x1, x1)
3.33/1.63	
3.33/1.63	We have to consider all minimal (P,Q,R)-chains.
3.33/1.63	----------------------------------------
3.33/1.63	
3.33/1.63	(3) DependencyGraphProof (EQUIVALENT)
3.33/1.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
3.33/1.63	----------------------------------------
3.33/1.63	
3.33/1.63	(4)
3.33/1.63	TRUE
3.33/1.65	EOF