3.41/1.70	YES
3.41/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.70	
3.41/1.70	
3.41/1.70	Termination w.r.t. Q of the given QTRS could be proven:
3.41/1.70	
3.41/1.70	(0) QTRS
3.41/1.70	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
3.41/1.70	(2) QDP
3.41/1.70	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
3.41/1.70	(4) TRUE
3.41/1.70	
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(0)
3.41/1.70	Obligation:
3.41/1.70	Q restricted rewrite system:
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   f(f(X, Y), Z) -> f(X, f(Y, Z))
3.41/1.70	   f(X, f(Y, Z)) -> f(Y, Y)
3.41/1.70	
3.41/1.70	The set Q consists of the following terms:
3.41/1.70	
3.41/1.70	   f(f(x0, x1), x2)
3.41/1.70	   f(x0, f(x1, x2))
3.41/1.70	
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(1) DependencyPairsProof (EQUIVALENT)
3.41/1.70	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(2)
3.41/1.70	Obligation:
3.41/1.70	Q DP problem:
3.41/1.70	The TRS P consists of the following rules:
3.41/1.70	
3.41/1.70	   F(f(X, Y), Z) -> F(X, f(Y, Z))
3.41/1.70	   F(f(X, Y), Z) -> F(Y, Z)
3.41/1.70	   F(X, f(Y, Z)) -> F(Y, Y)
3.41/1.70	
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   f(f(X, Y), Z) -> f(X, f(Y, Z))
3.41/1.70	   f(X, f(Y, Z)) -> f(Y, Y)
3.41/1.70	
3.41/1.70	The set Q consists of the following terms:
3.41/1.70	
3.41/1.70	   f(f(x0, x1), x2)
3.41/1.70	   f(x0, f(x1, x2))
3.41/1.70	
3.41/1.70	We have to consider all minimal (P,Q,R)-chains.
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(3) DependencyGraphProof (EQUIVALENT)
3.41/1.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(4)
3.41/1.70	TRUE
3.41/1.73	EOF