33.20/9.45	YES
33.41/9.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.41/9.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.41/9.53	
33.41/9.53	
33.41/9.53	Termination w.r.t. Q of the given QTRS could be proven:
33.41/9.53	
33.41/9.53	(0) QTRS
33.41/9.53	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.41/9.53	(2) QTRS
33.41/9.53	(3) DependencyPairsProof [EQUIVALENT, 6 ms]
33.41/9.53	(4) QDP
33.41/9.53	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
33.41/9.53	(6) QDP
33.41/9.53	(7) QDPOrderProof [EQUIVALENT, 84 ms]
33.41/9.53	(8) QDP
33.41/9.53	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
33.41/9.53	(10) QDP
33.41/9.53	(11) UsableRulesProof [EQUIVALENT, 0 ms]
33.41/9.53	(12) QDP
33.41/9.53	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.41/9.53	(14) YES
33.41/9.53	
33.41/9.53	
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(0)
33.41/9.53	Obligation:
33.41/9.53	Q restricted rewrite system:
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   a(b(x1)) -> b(a(c(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> b(a(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(1) QTRS Reverse (EQUIVALENT)
33.41/9.53	We applied the QTRS Reverse Processor [REVERSE].
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(2)
33.41/9.53	Obligation:
33.41/9.53	Q restricted rewrite system:
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(3) DependencyPairsProof (EQUIVALENT)
33.41/9.53	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(4)
33.41/9.53	Obligation:
33.41/9.53	Q DP problem:
33.41/9.53	The TRS P consists of the following rules:
33.41/9.53	
33.41/9.53	   B(a(x1)) -> C(a(b(x1)))
33.41/9.53	   B(a(x1)) -> A(b(x1))
33.41/9.53	   B(a(x1)) -> B(x1)
33.41/9.53	   C(c(x1)) -> A(b(x1))
33.41/9.53	   C(c(x1)) -> B(x1)
33.41/9.53	
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	We have to consider all minimal (P,Q,R)-chains.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(5) DependencyGraphProof (EQUIVALENT)
33.41/9.53	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(6)
33.41/9.53	Obligation:
33.41/9.53	Q DP problem:
33.41/9.53	The TRS P consists of the following rules:
33.41/9.53	
33.41/9.53	   C(c(x1)) -> B(x1)
33.41/9.53	   B(a(x1)) -> C(a(b(x1)))
33.41/9.53	   B(a(x1)) -> B(x1)
33.41/9.53	
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	We have to consider all minimal (P,Q,R)-chains.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(7) QDPOrderProof (EQUIVALENT)
33.41/9.53	We use the reduction pair processor [LPAR04,JAR06].
33.41/9.53	
33.41/9.53	
33.41/9.53	The following pairs can be oriented strictly and are deleted.
33.41/9.53	
33.41/9.53	   C(c(x1)) -> B(x1)
33.41/9.53	The remaining pairs can at least be oriented weakly.
33.41/9.53	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.41/9.53	
33.41/9.53	   <<<
33.41/9.53	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
33.41/9.53	>>>
33.41/9.53	
33.41/9.53	   <<<
33.41/9.53	 POL(c(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 1A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
33.41/9.53	>>>
33.41/9.53	
33.41/9.53	   <<<
33.41/9.53	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
33.41/9.53	>>>
33.41/9.53	
33.41/9.53	   <<<
33.41/9.53	 POL(a(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
33.41/9.53	>>>
33.41/9.53	
33.41/9.53	   <<<
33.41/9.53	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
33.41/9.53	>>>
33.41/9.53	
33.41/9.53	
33.41/9.53	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.41/9.53	
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   a(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(8)
33.41/9.53	Obligation:
33.41/9.53	Q DP problem:
33.41/9.53	The TRS P consists of the following rules:
33.41/9.53	
33.41/9.53	   B(a(x1)) -> C(a(b(x1)))
33.41/9.53	   B(a(x1)) -> B(x1)
33.41/9.53	
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	We have to consider all minimal (P,Q,R)-chains.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(9) DependencyGraphProof (EQUIVALENT)
33.41/9.53	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(10)
33.41/9.53	Obligation:
33.41/9.53	Q DP problem:
33.41/9.53	The TRS P consists of the following rules:
33.41/9.53	
33.41/9.53	   B(a(x1)) -> B(x1)
33.41/9.53	
33.41/9.53	The TRS R consists of the following rules:
33.41/9.53	
33.41/9.53	   a(x1) -> x1
33.41/9.53	   b(a(x1)) -> c(a(b(x1)))
33.41/9.53	   b(x1) -> x1
33.41/9.53	   c(c(x1)) -> a(b(x1))
33.41/9.53	
33.41/9.53	Q is empty.
33.41/9.53	We have to consider all minimal (P,Q,R)-chains.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(11) UsableRulesProof (EQUIVALENT)
33.41/9.53	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.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(12)
33.41/9.53	Obligation:
33.41/9.53	Q DP problem:
33.41/9.53	The TRS P consists of the following rules:
33.41/9.53	
33.41/9.53	   B(a(x1)) -> B(x1)
33.41/9.53	
33.41/9.53	R is empty.
33.41/9.53	Q is empty.
33.41/9.53	We have to consider all minimal (P,Q,R)-chains.
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(13) QDPSizeChangeProof (EQUIVALENT)
33.41/9.53	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. 
33.41/9.53	
33.41/9.53	From the DPs we obtained the following set of size-change graphs:
33.41/9.53	*B(a(x1)) -> B(x1)
33.41/9.53	The graph contains the following edges 1 > 1
33.41/9.53	
33.41/9.53	
33.41/9.53	----------------------------------------
33.41/9.53	
33.41/9.53	(14)
33.41/9.53	YES
33.96/9.67	EOF