23.73/6.94	YES
23.98/6.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.98/6.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.98/6.96	
23.98/6.96	
23.98/6.96	Termination w.r.t. Q of the given QTRS could be proven:
23.98/6.96	
23.98/6.96	(0) QTRS
23.98/6.96	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
23.98/6.96	(2) QDP
23.98/6.96	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
23.98/6.96	(4) QDP
23.98/6.96	(5) QDPOrderProof [EQUIVALENT, 134 ms]
23.98/6.96	(6) QDP
23.98/6.96	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
23.98/6.96	(8) QDP
23.98/6.96	(9) QDPOrderProof [EQUIVALENT, 0 ms]
23.98/6.96	(10) QDP
23.98/6.96	(11) UsableRulesProof [EQUIVALENT, 0 ms]
23.98/6.96	(12) QDP
23.98/6.96	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
23.98/6.96	(14) YES
23.98/6.96	
23.98/6.96	
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(0)
23.98/6.96	Obligation:
23.98/6.96	Q restricted rewrite system:
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(1) DependencyPairsProof (EQUIVALENT)
23.98/6.96	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(2)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   A(x1) -> B(c(x1))
23.98/6.96	   A(x1) -> C(x1)
23.98/6.96	   C(c(b(x1))) -> B(c(a(c(x1))))
23.98/6.96	   C(c(b(x1))) -> C(a(c(x1)))
23.98/6.96	   C(c(b(x1))) -> A(c(x1))
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(3) DependencyGraphProof (EQUIVALENT)
23.98/6.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(4)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   A(x1) -> C(x1)
23.98/6.96	   C(c(b(x1))) -> C(a(c(x1)))
23.98/6.96	   C(c(b(x1))) -> A(c(x1))
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(5) QDPOrderProof (EQUIVALENT)
23.98/6.96	We use the reduction pair processor [LPAR04,JAR06].
23.98/6.96	
23.98/6.96	
23.98/6.96	The following pairs can be oriented strictly and are deleted.
23.98/6.96	
23.98/6.96	   A(x1) -> C(x1)
23.98/6.96	The remaining pairs can at least be oriented weakly.
23.98/6.96	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(A(x_1)) =  	[[1A]] 	 +  	[[1A, 1A, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, -I], [0A, -I, -I]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 1A, 1A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(a(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	
23.98/6.96	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.98/6.96	
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	
23.98/6.96	
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(6)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   C(c(b(x1))) -> C(a(c(x1)))
23.98/6.96	   C(c(b(x1))) -> A(c(x1))
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(7) DependencyGraphProof (EQUIVALENT)
23.98/6.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(8)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	   C(c(b(x1))) -> C(a(c(x1)))
23.98/6.96	
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(9) QDPOrderProof (EQUIVALENT)
23.98/6.96	We use the reduction pair processor [LPAR04,JAR06].
23.98/6.96	
23.98/6.96	
23.98/6.96	The following pairs can be oriented strictly and are deleted.
23.98/6.96	
23.98/6.96	   C(c(b(x1))) -> C(a(c(x1)))
23.98/6.96	The remaining pairs can at least be oriented weakly.
23.98/6.96	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	   <<<
23.98/6.96	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
23.98/6.96	>>>
23.98/6.96	
23.98/6.96	
23.98/6.96	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.98/6.96	
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	
23.98/6.96	
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(10)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	
23.98/6.96	The TRS R consists of the following rules:
23.98/6.96	
23.98/6.96	   a(x1) -> b(c(x1))
23.98/6.96	   b(b(x1)) -> x1
23.98/6.96	   c(c(b(x1))) -> b(c(a(c(x1))))
23.98/6.96	
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(11) UsableRulesProof (EQUIVALENT)
23.98/6.96	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.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(12)
23.98/6.96	Obligation:
23.98/6.96	Q DP problem:
23.98/6.96	The TRS P consists of the following rules:
23.98/6.96	
23.98/6.96	   C(c(b(x1))) -> C(x1)
23.98/6.96	
23.98/6.96	R is empty.
23.98/6.96	Q is empty.
23.98/6.96	We have to consider all minimal (P,Q,R)-chains.
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(13) QDPSizeChangeProof (EQUIVALENT)
23.98/6.96	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. 
23.98/6.96	
23.98/6.96	From the DPs we obtained the following set of size-change graphs:
23.98/6.96	*C(c(b(x1))) -> C(x1)
23.98/6.96	The graph contains the following edges 1 > 1
23.98/6.96	
23.98/6.96	
23.98/6.96	----------------------------------------
23.98/6.96	
23.98/6.96	(14)
23.98/6.96	YES
24.20/7.11	EOF