34.57/9.70	YES
35.38/9.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.38/9.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.38/9.91	
35.38/9.91	
35.38/9.91	Termination w.r.t. Q of the given QTRS could be proven:
35.38/9.91	
35.38/9.91	(0) QTRS
35.38/9.91	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
35.38/9.91	(2) QDP
35.38/9.91	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
35.38/9.91	(4) QDP
35.38/9.91	(5) QDPOrderProof [EQUIVALENT, 226 ms]
35.38/9.91	(6) QDP
35.38/9.91	(7) UsableRulesProof [EQUIVALENT, 0 ms]
35.38/9.91	(8) QDP
35.38/9.91	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
35.38/9.91	(10) YES
35.38/9.91	
35.38/9.91	
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(0)
35.38/9.91	Obligation:
35.38/9.91	Q restricted rewrite system:
35.38/9.91	The TRS R consists of the following rules:
35.38/9.91	
35.38/9.91	   a(b(x1)) -> x1
35.38/9.91	   a(c(x1)) -> c(c(x1))
35.38/9.91	   b(c(x1)) -> a(a(b(b(x1))))
35.38/9.91	
35.38/9.91	Q is empty.
35.38/9.91	
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(1) DependencyPairsProof (EQUIVALENT)
35.38/9.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(2)
35.38/9.91	Obligation:
35.38/9.91	Q DP problem:
35.38/9.91	The TRS P consists of the following rules:
35.38/9.91	
35.38/9.91	   B(c(x1)) -> A(a(b(b(x1))))
35.38/9.91	   B(c(x1)) -> A(b(b(x1)))
35.38/9.91	   B(c(x1)) -> B(b(x1))
35.38/9.91	   B(c(x1)) -> B(x1)
35.38/9.91	
35.38/9.91	The TRS R consists of the following rules:
35.38/9.91	
35.38/9.91	   a(b(x1)) -> x1
35.38/9.91	   a(c(x1)) -> c(c(x1))
35.38/9.91	   b(c(x1)) -> a(a(b(b(x1))))
35.38/9.91	
35.38/9.91	Q is empty.
35.38/9.91	We have to consider all minimal (P,Q,R)-chains.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(3) DependencyGraphProof (EQUIVALENT)
35.38/9.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(4)
35.38/9.91	Obligation:
35.38/9.91	Q DP problem:
35.38/9.91	The TRS P consists of the following rules:
35.38/9.91	
35.38/9.91	   B(c(x1)) -> B(x1)
35.38/9.91	   B(c(x1)) -> B(b(x1))
35.38/9.91	
35.38/9.91	The TRS R consists of the following rules:
35.38/9.91	
35.38/9.91	   a(b(x1)) -> x1
35.38/9.91	   a(c(x1)) -> c(c(x1))
35.38/9.91	   b(c(x1)) -> a(a(b(b(x1))))
35.38/9.91	
35.38/9.91	Q is empty.
35.38/9.91	We have to consider all minimal (P,Q,R)-chains.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(5) QDPOrderProof (EQUIVALENT)
35.38/9.91	We use the reduction pair processor [LPAR04,JAR06].
35.38/9.91	
35.38/9.91	
35.38/9.91	The following pairs can be oriented strictly and are deleted.
35.38/9.91	
35.38/9.91	   B(c(x1)) -> B(b(x1))
35.38/9.91	The remaining pairs can at least be oriented weakly.
35.38/9.91	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
35.38/9.91	
35.38/9.91	   <<<
35.38/9.91	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -1A, -1A]] 	* 	x_1
35.38/9.91	>>>
35.38/9.91	
35.38/9.91	   <<<
35.38/9.91	 POL(c(x_1)) =  	[[2A], [2A], [-1A]] 	 +  	[[0A, 1A, 1A], [0A, -1A, 0A], [-1A, 0A, -1A]] 	* 	x_1
35.38/9.91	>>>
35.38/9.91	
35.38/9.91	   <<<
35.38/9.91	 POL(b(x_1)) =  	[[0A], [-1A], [-1A]] 	 +  	[[-I, 0A, 0A], [-1A, -1A, -I], [-1A, -1A, -I]] 	* 	x_1
35.38/9.91	>>>
35.38/9.91	
35.38/9.91	   <<<
35.38/9.91	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 1A, -1A], [0A, -1A, -1A], [0A, -1A, -1A]] 	* 	x_1
35.38/9.91	>>>
35.38/9.91	
35.38/9.91	
35.38/9.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.38/9.91	
35.38/9.91	   b(c(x1)) -> a(a(b(b(x1))))
35.38/9.91	   a(b(x1)) -> x1
35.38/9.91	   a(c(x1)) -> c(c(x1))
35.38/9.91	
35.38/9.91	
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(6)
35.38/9.91	Obligation:
35.38/9.91	Q DP problem:
35.38/9.91	The TRS P consists of the following rules:
35.38/9.91	
35.38/9.91	   B(c(x1)) -> B(x1)
35.38/9.91	
35.38/9.91	The TRS R consists of the following rules:
35.38/9.91	
35.38/9.91	   a(b(x1)) -> x1
35.38/9.91	   a(c(x1)) -> c(c(x1))
35.38/9.91	   b(c(x1)) -> a(a(b(b(x1))))
35.38/9.91	
35.38/9.91	Q is empty.
35.38/9.91	We have to consider all minimal (P,Q,R)-chains.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(7) UsableRulesProof (EQUIVALENT)
35.38/9.91	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.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(8)
35.38/9.91	Obligation:
35.38/9.91	Q DP problem:
35.38/9.91	The TRS P consists of the following rules:
35.38/9.91	
35.38/9.91	   B(c(x1)) -> B(x1)
35.38/9.91	
35.38/9.91	R is empty.
35.38/9.91	Q is empty.
35.38/9.91	We have to consider all minimal (P,Q,R)-chains.
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(9) QDPSizeChangeProof (EQUIVALENT)
35.38/9.91	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. 
35.38/9.91	
35.38/9.91	From the DPs we obtained the following set of size-change graphs:
35.38/9.91	*B(c(x1)) -> B(x1)
35.38/9.91	The graph contains the following edges 1 > 1
35.38/9.91	
35.38/9.91	
35.38/9.91	----------------------------------------
35.38/9.91	
35.38/9.91	(10)
35.38/9.91	YES
35.84/10.05	EOF