31.83/9.04	YES
32.34/9.09	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.34/9.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.34/9.09	
32.34/9.09	
32.34/9.09	Termination w.r.t. Q of the given QTRS could be proven:
32.34/9.09	
32.34/9.09	(0) QTRS
32.34/9.09	(1) DependencyPairsProof [EQUIVALENT, 19 ms]
32.34/9.09	(2) QDP
32.34/9.09	(3) DependencyGraphProof [EQUIVALENT, 4 ms]
32.34/9.09	(4) QDP
32.34/9.09	(5) QDPOrderProof [EQUIVALENT, 114 ms]
32.34/9.09	(6) QDP
32.34/9.09	(7) QDPOrderProof [EQUIVALENT, 48 ms]
32.34/9.09	(8) QDP
32.34/9.09	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
32.34/9.09	(10) QDP
32.34/9.09	(11) UsableRulesProof [EQUIVALENT, 0 ms]
32.34/9.09	(12) QDP
32.34/9.09	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
32.34/9.09	(14) YES
32.34/9.09	
32.34/9.09	
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(0)
32.34/9.09	Obligation:
32.34/9.09	Q restricted rewrite system:
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(1) DependencyPairsProof (EQUIVALENT)
32.34/9.09	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(2)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   A(x1) -> B(c(x1))
32.34/9.09	   A(x1) -> C(x1)
32.34/9.09	   C(c(b(x1))) -> A(b(c(c(x1))))
32.34/9.09	   C(c(b(x1))) -> B(c(c(x1)))
32.34/9.09	   C(c(b(x1))) -> C(c(x1))
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(3) DependencyGraphProof (EQUIVALENT)
32.34/9.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(4)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   A(x1) -> C(x1)
32.34/9.09	   C(c(b(x1))) -> A(b(c(c(x1))))
32.34/9.09	   C(c(b(x1))) -> C(c(x1))
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(5) QDPOrderProof (EQUIVALENT)
32.34/9.09	We use the reduction pair processor [LPAR04,JAR06].
32.34/9.09	
32.34/9.09	
32.34/9.09	The following pairs can be oriented strictly and are deleted.
32.34/9.09	
32.34/9.09	   C(c(b(x1))) -> C(c(x1))
32.34/9.09	The remaining pairs can at least be oriented weakly.
32.34/9.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, -I], [0A, -I, -I]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(a(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, 1A], [-I, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	
32.34/9.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.34/9.09	
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	
32.34/9.09	
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(6)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   A(x1) -> C(x1)
32.34/9.09	   C(c(b(x1))) -> A(b(c(c(x1))))
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(7) QDPOrderProof (EQUIVALENT)
32.34/9.09	We use the reduction pair processor [LPAR04,JAR06].
32.34/9.09	
32.34/9.09	
32.34/9.09	The following pairs can be oriented strictly and are deleted.
32.34/9.09	
32.34/9.09	   A(x1) -> C(x1)
32.34/9.09	The remaining pairs can at least be oriented weakly.
32.34/9.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [1A, 0A, 1A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	   <<<
32.34/9.09	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, 1A, 0A]] 	* 	x_1
32.34/9.09	>>>
32.34/9.09	
32.34/9.09	
32.34/9.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.34/9.09	
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	
32.34/9.09	
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(8)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   C(c(b(x1))) -> A(b(c(c(x1))))
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(9) DependencyGraphProof (EQUIVALENT)
32.34/9.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(10)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	The TRS R consists of the following rules:
32.34/9.09	
32.34/9.09	   a(x1) -> b(c(x1))
32.34/9.09	   b(b(x1)) -> x1
32.34/9.09	   c(c(b(x1))) -> a(b(c(c(x1))))
32.34/9.09	
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(11) UsableRulesProof (EQUIVALENT)
32.34/9.09	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.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(12)
32.34/9.09	Obligation:
32.34/9.09	Q DP problem:
32.34/9.09	The TRS P consists of the following rules:
32.34/9.09	
32.34/9.09	   C(c(b(x1))) -> C(x1)
32.34/9.09	
32.34/9.09	R is empty.
32.34/9.09	Q is empty.
32.34/9.09	We have to consider all minimal (P,Q,R)-chains.
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(13) QDPSizeChangeProof (EQUIVALENT)
32.34/9.09	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. 
32.34/9.09	
32.34/9.09	From the DPs we obtained the following set of size-change graphs:
32.34/9.09	*C(c(b(x1))) -> C(x1)
32.34/9.09	The graph contains the following edges 1 > 1
32.34/9.09	
32.34/9.09	
32.34/9.09	----------------------------------------
32.34/9.09	
32.34/9.09	(14)
32.34/9.09	YES
32.54/9.19	EOF