42.39/11.78	YES
42.39/11.82	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
42.39/11.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.39/11.82	
42.39/11.82	
42.39/11.82	Termination w.r.t. Q of the given QTRS could be proven:
42.39/11.82	
42.39/11.82	(0) QTRS
42.39/11.82	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
42.39/11.82	(2) QDP
42.39/11.82	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
42.39/11.82	(4) QDP
42.39/11.82	(5) QDPOrderProof [EQUIVALENT, 341 ms]
42.39/11.82	(6) QDP
42.39/11.82	(7) UsableRulesProof [EQUIVALENT, 0 ms]
42.39/11.82	(8) QDP
42.39/11.82	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
42.39/11.82	(10) YES
42.39/11.82	
42.39/11.82	
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(0)
42.39/11.82	Obligation:
42.39/11.82	Q restricted rewrite system:
42.39/11.82	The TRS R consists of the following rules:
42.39/11.82	
42.39/11.82	   a(a(x1)) -> x1
42.39/11.82	   b(b(x1)) -> c(c(c(x1)))
42.39/11.82	   b(c(x1)) -> a(b(b(x1)))
42.39/11.82	
42.39/11.82	Q is empty.
42.39/11.82	
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(1) DependencyPairsProof (EQUIVALENT)
42.39/11.82	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(2)
42.39/11.82	Obligation:
42.39/11.82	Q DP problem:
42.39/11.82	The TRS P consists of the following rules:
42.39/11.82	
42.39/11.82	   B(c(x1)) -> A(b(b(x1)))
42.39/11.82	   B(c(x1)) -> B(b(x1))
42.39/11.82	   B(c(x1)) -> B(x1)
42.39/11.82	
42.39/11.82	The TRS R consists of the following rules:
42.39/11.82	
42.39/11.82	   a(a(x1)) -> x1
42.39/11.82	   b(b(x1)) -> c(c(c(x1)))
42.39/11.82	   b(c(x1)) -> a(b(b(x1)))
42.39/11.82	
42.39/11.82	Q is empty.
42.39/11.82	We have to consider all minimal (P,Q,R)-chains.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(3) DependencyGraphProof (EQUIVALENT)
42.39/11.82	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(4)
42.39/11.82	Obligation:
42.39/11.82	Q DP problem:
42.39/11.82	The TRS P consists of the following rules:
42.39/11.82	
42.39/11.82	   B(c(x1)) -> B(x1)
42.39/11.82	   B(c(x1)) -> B(b(x1))
42.39/11.82	
42.39/11.82	The TRS R consists of the following rules:
42.39/11.82	
42.39/11.82	   a(a(x1)) -> x1
42.39/11.82	   b(b(x1)) -> c(c(c(x1)))
42.39/11.82	   b(c(x1)) -> a(b(b(x1)))
42.39/11.82	
42.39/11.82	Q is empty.
42.39/11.82	We have to consider all minimal (P,Q,R)-chains.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(5) QDPOrderProof (EQUIVALENT)
42.39/11.82	We use the reduction pair processor [LPAR04,JAR06].
42.39/11.82	
42.39/11.82	
42.39/11.82	The following pairs can be oriented strictly and are deleted.
42.39/11.82	
42.39/11.82	   B(c(x1)) -> B(b(x1))
42.39/11.82	The remaining pairs can at least be oriented weakly.
42.39/11.82	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
42.39/11.82	
42.39/11.82	   <<<
42.39/11.82	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -1A]] 	* 	x_1
42.39/11.82	>>>
42.39/11.82	
42.39/11.82	   <<<
42.39/11.82	 POL(c(x_1)) =  	[[2A], [-I], [0A]] 	 +  	[[0A, -1A, 0A], [-I, -1A, -I], [-I, 2A, -I]] 	* 	x_1
42.39/11.82	>>>
42.39/11.82	
42.39/11.82	   <<<
42.39/11.82	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-1A, 1A, -1A], [-1A, 1A, -1A], [-1A, -1A, -1A]] 	* 	x_1
42.39/11.82	>>>
42.39/11.82	
42.39/11.82	   <<<
42.39/11.82	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-1A, -1A, 1A], [-1A, -1A, 1A], [-1A, -1A, -1A]] 	* 	x_1
42.39/11.82	>>>
42.39/11.82	
42.39/11.82	
42.39/11.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.39/11.82	
42.39/11.82	   b(b(x1)) -> c(c(c(x1)))
42.39/11.82	   b(c(x1)) -> a(b(b(x1)))
42.39/11.82	   a(a(x1)) -> x1
42.39/11.82	
42.39/11.82	
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(6)
42.39/11.82	Obligation:
42.39/11.82	Q DP problem:
42.39/11.82	The TRS P consists of the following rules:
42.39/11.82	
42.39/11.82	   B(c(x1)) -> B(x1)
42.39/11.82	
42.39/11.82	The TRS R consists of the following rules:
42.39/11.82	
42.39/11.82	   a(a(x1)) -> x1
42.39/11.82	   b(b(x1)) -> c(c(c(x1)))
42.39/11.82	   b(c(x1)) -> a(b(b(x1)))
42.39/11.82	
42.39/11.82	Q is empty.
42.39/11.82	We have to consider all minimal (P,Q,R)-chains.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(7) UsableRulesProof (EQUIVALENT)
42.39/11.82	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.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(8)
42.39/11.82	Obligation:
42.39/11.82	Q DP problem:
42.39/11.82	The TRS P consists of the following rules:
42.39/11.82	
42.39/11.82	   B(c(x1)) -> B(x1)
42.39/11.82	
42.39/11.82	R is empty.
42.39/11.82	Q is empty.
42.39/11.82	We have to consider all minimal (P,Q,R)-chains.
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(9) QDPSizeChangeProof (EQUIVALENT)
42.39/11.82	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. 
42.39/11.82	
42.39/11.82	From the DPs we obtained the following set of size-change graphs:
42.39/11.82	*B(c(x1)) -> B(x1)
42.39/11.82	The graph contains the following edges 1 > 1
42.39/11.82	
42.39/11.82	
42.39/11.82	----------------------------------------
42.39/11.82	
42.39/11.82	(10)
42.39/11.82	YES
42.84/11.97	EOF