19.53/5.93	YES
19.91/6.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.91/6.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.91/6.10	
19.91/6.10	
19.91/6.10	Termination w.r.t. Q of the given QTRS could be proven:
19.91/6.10	
19.91/6.10	(0) QTRS
19.91/6.10	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.91/6.10	(2) QTRS
19.91/6.10	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
19.91/6.10	(4) QDP
19.91/6.10	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
19.91/6.10	(6) QDP
19.91/6.10	(7) QDPOrderProof [EQUIVALENT, 138 ms]
19.91/6.10	(8) QDP
19.91/6.10	(9) UsableRulesProof [EQUIVALENT, 0 ms]
19.91/6.10	(10) QDP
19.91/6.10	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
19.91/6.10	(12) YES
19.91/6.10	
19.91/6.10	
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(0)
19.91/6.10	Obligation:
19.91/6.10	Q restricted rewrite system:
19.91/6.10	The TRS R consists of the following rules:
19.91/6.10	
19.91/6.10	   a(x1) -> x1
19.91/6.10	   a(a(b(x1))) -> b(b(a(a(x1))))
19.91/6.10	   c(b(x1)) -> c(a(x1))
19.91/6.10	
19.91/6.10	Q is empty.
19.91/6.10	
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(1) QTRS Reverse (EQUIVALENT)
19.91/6.10	We applied the QTRS Reverse Processor [REVERSE].
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(2)
19.91/6.10	Obligation:
19.91/6.10	Q restricted rewrite system:
19.91/6.10	The TRS R consists of the following rules:
19.91/6.10	
19.91/6.10	   a(x1) -> x1
19.91/6.10	   b(a(a(x1))) -> a(a(b(b(x1))))
19.91/6.10	   b(c(x1)) -> a(c(x1))
19.91/6.10	
19.91/6.10	Q is empty.
19.91/6.10	
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(3) DependencyPairsProof (EQUIVALENT)
19.91/6.10	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(4)
19.91/6.10	Obligation:
19.91/6.10	Q DP problem:
19.91/6.10	The TRS P consists of the following rules:
19.91/6.10	
19.91/6.10	   B(a(a(x1))) -> A(a(b(b(x1))))
19.91/6.10	   B(a(a(x1))) -> A(b(b(x1)))
19.91/6.10	   B(a(a(x1))) -> B(b(x1))
19.91/6.10	   B(a(a(x1))) -> B(x1)
19.91/6.10	   B(c(x1)) -> A(c(x1))
19.91/6.10	
19.91/6.10	The TRS R consists of the following rules:
19.91/6.10	
19.91/6.10	   a(x1) -> x1
19.91/6.10	   b(a(a(x1))) -> a(a(b(b(x1))))
19.91/6.10	   b(c(x1)) -> a(c(x1))
19.91/6.10	
19.91/6.10	Q is empty.
19.91/6.10	We have to consider all minimal (P,Q,R)-chains.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(5) DependencyGraphProof (EQUIVALENT)
19.91/6.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(6)
19.91/6.10	Obligation:
19.91/6.10	Q DP problem:
19.91/6.10	The TRS P consists of the following rules:
19.91/6.10	
19.91/6.10	   B(a(a(x1))) -> B(x1)
19.91/6.10	   B(a(a(x1))) -> B(b(x1))
19.91/6.10	
19.91/6.10	The TRS R consists of the following rules:
19.91/6.10	
19.91/6.10	   a(x1) -> x1
19.91/6.10	   b(a(a(x1))) -> a(a(b(b(x1))))
19.91/6.10	   b(c(x1)) -> a(c(x1))
19.91/6.10	
19.91/6.10	Q is empty.
19.91/6.10	We have to consider all minimal (P,Q,R)-chains.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(7) QDPOrderProof (EQUIVALENT)
19.91/6.10	We use the reduction pair processor [LPAR04,JAR06].
19.91/6.10	
19.91/6.10	
19.91/6.10	The following pairs can be oriented strictly and are deleted.
19.91/6.10	
19.91/6.10	   B(a(a(x1))) -> B(b(x1))
19.91/6.10	The remaining pairs can at least be oriented weakly.
19.91/6.10	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.91/6.10	
19.91/6.10	   <<<
19.91/6.10	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
19.91/6.10	>>>
19.91/6.10	
19.91/6.10	   <<<
19.91/6.10	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [-I, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
19.91/6.10	>>>
19.91/6.10	
19.91/6.10	   <<<
19.91/6.10	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
19.91/6.10	>>>
19.91/6.10	
19.91/6.10	   <<<
19.91/6.10	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
19.91/6.10	>>>
19.91/6.10	
19.91/6.10	
19.91/6.10	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.91/6.10	
19.91/6.10	   b(a(a(x1))) -> a(a(b(b(x1))))
19.91/6.10	   b(c(x1)) -> a(c(x1))
19.91/6.10	   a(x1) -> x1
19.91/6.10	
19.91/6.10	
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(8)
19.91/6.10	Obligation:
19.91/6.10	Q DP problem:
19.91/6.10	The TRS P consists of the following rules:
19.91/6.10	
19.91/6.10	   B(a(a(x1))) -> B(x1)
19.91/6.10	
19.91/6.10	The TRS R consists of the following rules:
19.91/6.10	
19.91/6.10	   a(x1) -> x1
19.91/6.10	   b(a(a(x1))) -> a(a(b(b(x1))))
19.91/6.10	   b(c(x1)) -> a(c(x1))
19.91/6.10	
19.91/6.10	Q is empty.
19.91/6.10	We have to consider all minimal (P,Q,R)-chains.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(9) UsableRulesProof (EQUIVALENT)
19.91/6.10	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.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(10)
19.91/6.10	Obligation:
19.91/6.10	Q DP problem:
19.91/6.10	The TRS P consists of the following rules:
19.91/6.10	
19.91/6.10	   B(a(a(x1))) -> B(x1)
19.91/6.10	
19.91/6.10	R is empty.
19.91/6.10	Q is empty.
19.91/6.10	We have to consider all minimal (P,Q,R)-chains.
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(11) QDPSizeChangeProof (EQUIVALENT)
19.91/6.10	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. 
19.91/6.10	
19.91/6.10	From the DPs we obtained the following set of size-change graphs:
19.91/6.10	*B(a(a(x1))) -> B(x1)
19.91/6.10	The graph contains the following edges 1 > 1
19.91/6.10	
19.91/6.10	
19.91/6.10	----------------------------------------
19.91/6.10	
19.91/6.10	(12)
19.91/6.10	YES
20.31/6.20	EOF