31.43/8.91	YES
31.71/8.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.71/8.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.71/8.95	
31.71/8.95	
31.71/8.95	Termination w.r.t. Q of the given QTRS could be proven:
31.71/8.95	
31.71/8.95	(0) QTRS
31.71/8.95	(1) QTRS Reverse [EQUIVALENT, 0 ms]
31.71/8.95	(2) QTRS
31.71/8.95	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
31.71/8.95	(4) QDP
31.71/8.95	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
31.71/8.95	(6) QDP
31.71/8.95	(7) QDPOrderProof [EQUIVALENT, 112 ms]
31.71/8.95	(8) QDP
31.71/8.95	(9) QDPOrderProof [EQUIVALENT, 0 ms]
31.71/8.95	(10) QDP
31.71/8.95	(11) UsableRulesProof [EQUIVALENT, 0 ms]
31.71/8.95	(12) QDP
31.71/8.95	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
31.71/8.95	(14) YES
31.71/8.95	
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(0)
31.71/8.95	Obligation:
31.71/8.95	Q restricted rewrite system:
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   b(c(c(x1))) -> c(c(b(c(b(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(1) QTRS Reverse (EQUIVALENT)
31.71/8.95	We applied the QTRS Reverse Processor [REVERSE].
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(2)
31.71/8.95	Obligation:
31.71/8.95	Q restricted rewrite system:
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(3) DependencyPairsProof (EQUIVALENT)
31.71/8.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(4)
31.71/8.95	Obligation:
31.71/8.95	Q DP problem:
31.71/8.95	The TRS P consists of the following rules:
31.71/8.95	
31.71/8.95	   B(b(x1)) -> A(x1)
31.71/8.95	   C(c(b(x1))) -> B(c(b(c(c(x1)))))
31.71/8.95	   C(c(b(x1))) -> C(b(c(c(x1))))
31.71/8.95	   C(c(b(x1))) -> B(c(c(x1)))
31.71/8.95	   C(c(b(x1))) -> C(c(x1))
31.71/8.95	   C(c(b(x1))) -> C(x1)
31.71/8.95	
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	We have to consider all minimal (P,Q,R)-chains.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(5) DependencyGraphProof (EQUIVALENT)
31.71/8.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(6)
31.71/8.95	Obligation:
31.71/8.95	Q DP problem:
31.71/8.95	The TRS P consists of the following rules:
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(c(x1))
31.71/8.95	   C(c(b(x1))) -> C(b(c(c(x1))))
31.71/8.95	   C(c(b(x1))) -> C(x1)
31.71/8.95	
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	We have to consider all minimal (P,Q,R)-chains.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(7) QDPOrderProof (EQUIVALENT)
31.71/8.95	We use the reduction pair processor [LPAR04,JAR06].
31.71/8.95	
31.71/8.95	
31.71/8.95	The following pairs can be oriented strictly and are deleted.
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(b(c(c(x1))))
31.71/8.95	The remaining pairs can at least be oriented weakly.
31.71/8.95	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, -I, 0A], [1A, 1A, -I], [-I, -I, 0A]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	
31.71/8.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.71/8.95	
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   a(x1) -> x1
31.71/8.95	
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(8)
31.71/8.95	Obligation:
31.71/8.95	Q DP problem:
31.71/8.95	The TRS P consists of the following rules:
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(c(x1))
31.71/8.95	   C(c(b(x1))) -> C(x1)
31.71/8.95	
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	We have to consider all minimal (P,Q,R)-chains.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(9) QDPOrderProof (EQUIVALENT)
31.71/8.95	We use the reduction pair processor [LPAR04,JAR06].
31.71/8.95	
31.71/8.95	
31.71/8.95	The following pairs can be oriented strictly and are deleted.
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(c(x1))
31.71/8.95	The remaining pairs can at least be oriented weakly.
31.71/8.95	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	   <<<
31.71/8.95	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, -I, -I], [-I, 1A, -I], [-I, -I, 0A]] 	* 	x_1
31.71/8.95	>>>
31.71/8.95	
31.71/8.95	
31.71/8.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.71/8.95	
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   a(x1) -> x1
31.71/8.95	
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(10)
31.71/8.95	Obligation:
31.71/8.95	Q DP problem:
31.71/8.95	The TRS P consists of the following rules:
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(x1)
31.71/8.95	
31.71/8.95	The TRS R consists of the following rules:
31.71/8.95	
31.71/8.95	   a(x1) -> x1
31.71/8.95	   b(b(x1)) -> a(x1)
31.71/8.95	   c(c(b(x1))) -> b(c(b(c(c(x1)))))
31.71/8.95	
31.71/8.95	Q is empty.
31.71/8.95	We have to consider all minimal (P,Q,R)-chains.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(11) UsableRulesProof (EQUIVALENT)
31.71/8.95	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.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(12)
31.71/8.95	Obligation:
31.71/8.95	Q DP problem:
31.71/8.95	The TRS P consists of the following rules:
31.71/8.95	
31.71/8.95	   C(c(b(x1))) -> C(x1)
31.71/8.95	
31.71/8.95	R is empty.
31.71/8.95	Q is empty.
31.71/8.95	We have to consider all minimal (P,Q,R)-chains.
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(13) QDPSizeChangeProof (EQUIVALENT)
31.71/8.95	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. 
31.71/8.95	
31.71/8.95	From the DPs we obtained the following set of size-change graphs:
31.71/8.95	*C(c(b(x1))) -> C(x1)
31.71/8.95	The graph contains the following edges 1 > 1
31.71/8.95	
31.71/8.95	
31.71/8.95	----------------------------------------
31.71/8.95	
31.71/8.95	(14)
31.71/8.95	YES
32.04/9.17	EOF