20.18/6.07	YES
20.18/6.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.18/6.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.18/6.08	
20.18/6.08	
20.18/6.08	Termination w.r.t. Q of the given QTRS could be proven:
20.18/6.08	
20.18/6.08	(0) QTRS
20.18/6.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.18/6.08	(2) QTRS
20.18/6.08	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
20.18/6.08	(4) QDP
20.18/6.08	(5) QDPOrderProof [EQUIVALENT, 109 ms]
20.18/6.08	(6) QDP
20.18/6.08	(7) UsableRulesProof [EQUIVALENT, 0 ms]
20.18/6.08	(8) QDP
20.18/6.08	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
20.18/6.08	(10) YES
20.18/6.08	
20.18/6.08	
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(0)
20.18/6.08	Obligation:
20.18/6.08	Q restricted rewrite system:
20.18/6.08	The TRS R consists of the following rules:
20.18/6.08	
20.18/6.08	   a(a(b(x1))) -> c(x1)
20.18/6.08	   a(c(x1)) -> b(c(a(a(x1))))
20.18/6.08	   b(c(x1)) -> x1
20.18/6.08	
20.18/6.08	Q is empty.
20.18/6.08	
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(1) QTRS Reverse (EQUIVALENT)
20.18/6.08	We applied the QTRS Reverse Processor [REVERSE].
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(2)
20.18/6.08	Obligation:
20.18/6.08	Q restricted rewrite system:
20.18/6.08	The TRS R consists of the following rules:
20.18/6.08	
20.18/6.08	   b(a(a(x1))) -> c(x1)
20.18/6.08	   c(a(x1)) -> a(a(c(b(x1))))
20.18/6.08	   c(b(x1)) -> x1
20.18/6.08	
20.18/6.08	Q is empty.
20.18/6.08	
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(3) DependencyPairsProof (EQUIVALENT)
20.18/6.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(4)
20.18/6.08	Obligation:
20.18/6.08	Q DP problem:
20.18/6.08	The TRS P consists of the following rules:
20.18/6.08	
20.18/6.08	   B(a(a(x1))) -> C(x1)
20.18/6.08	   C(a(x1)) -> C(b(x1))
20.18/6.08	   C(a(x1)) -> B(x1)
20.18/6.08	
20.18/6.08	The TRS R consists of the following rules:
20.18/6.08	
20.18/6.08	   b(a(a(x1))) -> c(x1)
20.18/6.08	   c(a(x1)) -> a(a(c(b(x1))))
20.18/6.08	   c(b(x1)) -> x1
20.18/6.08	
20.18/6.08	Q is empty.
20.18/6.08	We have to consider all minimal (P,Q,R)-chains.
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(5) QDPOrderProof (EQUIVALENT)
20.18/6.08	We use the reduction pair processor [LPAR04,JAR06].
20.18/6.08	
20.18/6.08	
20.18/6.08	The following pairs can be oriented strictly and are deleted.
20.18/6.08	
20.18/6.08	   C(a(x1)) -> C(b(x1))
20.18/6.08	The remaining pairs can at least be oriented weakly.
20.18/6.08	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.18/6.08	
20.18/6.08	   <<<
20.18/6.08	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
20.18/6.08	>>>
20.18/6.08	
20.18/6.08	   <<<
20.18/6.08	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[-I, 0A, 1A], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
20.18/6.08	>>>
20.18/6.08	
20.18/6.08	   <<<
20.18/6.08	 POL(C(x_1)) =  	[[1A]] 	 +  	[[1A, -I, 0A]] 	* 	x_1
20.18/6.08	>>>
20.18/6.08	
20.18/6.08	   <<<
20.18/6.08	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
20.18/6.08	>>>
20.18/6.08	
20.18/6.08	   <<<
20.18/6.08	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
20.18/6.08	>>>
20.18/6.08	
20.18/6.08	
20.18/6.08	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.18/6.08	
20.18/6.08	   b(a(a(x1))) -> c(x1)
20.18/6.08	   c(a(x1)) -> a(a(c(b(x1))))
20.18/6.08	   c(b(x1)) -> x1
20.18/6.08	
20.18/6.08	
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(6)
20.18/6.08	Obligation:
20.18/6.08	Q DP problem:
20.18/6.08	The TRS P consists of the following rules:
20.18/6.08	
20.18/6.08	   B(a(a(x1))) -> C(x1)
20.18/6.08	   C(a(x1)) -> B(x1)
20.18/6.08	
20.18/6.08	The TRS R consists of the following rules:
20.18/6.08	
20.18/6.08	   b(a(a(x1))) -> c(x1)
20.18/6.08	   c(a(x1)) -> a(a(c(b(x1))))
20.18/6.08	   c(b(x1)) -> x1
20.18/6.08	
20.18/6.08	Q is empty.
20.18/6.08	We have to consider all minimal (P,Q,R)-chains.
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(7) UsableRulesProof (EQUIVALENT)
20.18/6.08	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(8)
20.18/6.08	Obligation:
20.18/6.08	Q DP problem:
20.18/6.08	The TRS P consists of the following rules:
20.18/6.08	
20.18/6.08	   B(a(a(x1))) -> C(x1)
20.18/6.08	   C(a(x1)) -> B(x1)
20.18/6.08	
20.18/6.08	R is empty.
20.18/6.08	Q is empty.
20.18/6.08	We have to consider all minimal (P,Q,R)-chains.
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(9) QDPSizeChangeProof (EQUIVALENT)
20.18/6.08	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
20.18/6.08	
20.18/6.08	From the DPs we obtained the following set of size-change graphs:
20.18/6.08	*C(a(x1)) -> B(x1)
20.18/6.08	The graph contains the following edges 1 > 1
20.18/6.08	
20.18/6.08	
20.18/6.08	*B(a(a(x1))) -> C(x1)
20.18/6.08	The graph contains the following edges 1 > 1
20.18/6.08	
20.18/6.08	
20.18/6.08	----------------------------------------
20.18/6.08	
20.18/6.08	(10)
20.18/6.08	YES
20.48/6.15	EOF