117.13/30.66	YES
117.13/30.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
117.13/30.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
117.13/30.67	
117.13/30.67	
117.13/30.67	Termination w.r.t. Q of the given QTRS could be proven:
117.13/30.67	
117.13/30.67	(0) QTRS
117.13/30.67	(1) QTRS Reverse [EQUIVALENT, 0 ms]
117.13/30.67	(2) QTRS
117.13/30.67	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
117.13/30.67	(4) QDP
117.13/30.67	(5) QDPOrderProof [EQUIVALENT, 1031 ms]
117.13/30.67	(6) QDP
117.13/30.67	(7) QDPOrderProof [EQUIVALENT, 1358 ms]
117.13/30.67	(8) QDP
117.13/30.67	(9) QDPOrderProof [EQUIVALENT, 891 ms]
117.13/30.67	(10) QDP
117.13/30.67	(11) QDPOrderProof [EQUIVALENT, 181 ms]
117.13/30.67	(12) QDP
117.13/30.67	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
117.13/30.67	(14) TRUE
117.13/30.67	
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(0)
117.13/30.67	Obligation:
117.13/30.67	Q restricted rewrite system:
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(1) QTRS Reverse (EQUIVALENT)
117.13/30.67	We applied the QTRS Reverse Processor [REVERSE].
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(2)
117.13/30.67	Obligation:
117.13/30.67	Q restricted rewrite system:
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(3) DependencyPairsProof (EQUIVALENT)
117.13/30.67	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(4)
117.13/30.67	Obligation:
117.13/30.67	Q DP problem:
117.13/30.67	The TRS P consists of the following rules:
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(b(x1))))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(x1)))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(x1))
117.13/30.67	   C(c(c(c(x1)))) -> B(x1)
117.13/30.67	   B(b(x1)) -> C(b(c(x1)))
117.13/30.67	   B(b(x1)) -> B(c(x1))
117.13/30.67	   B(b(x1)) -> C(x1)
117.13/30.67	
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	We have to consider all minimal (P,Q,R)-chains.
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(5) QDPOrderProof (EQUIVALENT)
117.13/30.67	We use the reduction pair processor [LPAR04,JAR06].
117.13/30.67	
117.13/30.67	
117.13/30.67	The following pairs can be oriented strictly and are deleted.
117.13/30.67	
117.13/30.67	   B(b(x1)) -> C(b(c(x1)))
117.13/30.67	The remaining pairs can at least be oriented weakly.
117.13/30.67	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-1A, 0A, -1A], [-1A, 1A, 0A], [-1A, -I, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-1A, -I, 0A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-1A, -I, 1A], [0A, -1A, -1A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	
117.13/30.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
117.13/30.67	
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(6)
117.13/30.67	Obligation:
117.13/30.67	Q DP problem:
117.13/30.67	The TRS P consists of the following rules:
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(b(x1))))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(x1)))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(x1))
117.13/30.67	   C(c(c(c(x1)))) -> B(x1)
117.13/30.67	   B(b(x1)) -> B(c(x1))
117.13/30.67	   B(b(x1)) -> C(x1)
117.13/30.67	
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	We have to consider all minimal (P,Q,R)-chains.
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(7) QDPOrderProof (EQUIVALENT)
117.13/30.67	We use the reduction pair processor [LPAR04,JAR06].
117.13/30.67	
117.13/30.67	
117.13/30.67	The following pairs can be oriented strictly and are deleted.
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(x1)))
117.13/30.67	   C(c(c(c(x1)))) -> B(x1)
117.13/30.67	The remaining pairs can at least be oriented weakly.
117.13/30.67	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-1A, -I, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-1A, 0A, -I], [-1A, 1A, 0A], [-1A, -I, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, -1A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(b(x_1)) =  	[[0A], [-1A], [0A]] 	 +  	[[-I, -I, 0A], [-1A, -I, 1A], [0A, -1A, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	
117.13/30.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
117.13/30.67	
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(8)
117.13/30.67	Obligation:
117.13/30.67	Q DP problem:
117.13/30.67	The TRS P consists of the following rules:
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(b(x1))))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(x1))
117.13/30.67	   B(b(x1)) -> B(c(x1))
117.13/30.67	   B(b(x1)) -> C(x1)
117.13/30.67	
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	We have to consider all minimal (P,Q,R)-chains.
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(9) QDPOrderProof (EQUIVALENT)
117.13/30.67	We use the reduction pair processor [LPAR04,JAR06].
117.13/30.67	
117.13/30.67	
117.13/30.67	The following pairs can be oriented strictly and are deleted.
117.13/30.67	
117.13/30.67	   B(b(x1)) -> B(c(x1))
117.13/30.67	The remaining pairs can at least be oriented weakly.
117.13/30.67	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(C(x_1)) =  	[[1A]] 	 +  	[[-1A, 0A, 0A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(c(x_1)) =  	[[0A], [0A], [-1A]] 	 +  	[[-I, -1A, -1A], [-1A, 1A, -I], [-I, -1A, 0A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(b(x_1)) =  	[[1A], [-1A], [-1A]] 	 +  	[[-1A, 0A, 0A], [0A, -I, -I], [0A, -1A, -I]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	
117.13/30.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
117.13/30.67	
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(10)
117.13/30.67	Obligation:
117.13/30.67	Q DP problem:
117.13/30.67	The TRS P consists of the following rules:
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(b(x1))))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(x1))
117.13/30.67	   B(b(x1)) -> C(x1)
117.13/30.67	
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	We have to consider all minimal (P,Q,R)-chains.
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(11) QDPOrderProof (EQUIVALENT)
117.13/30.67	We use the reduction pair processor [LPAR04,JAR06].
117.13/30.67	
117.13/30.67	
117.13/30.67	The following pairs can be oriented strictly and are deleted.
117.13/30.67	
117.13/30.67	   C(c(c(c(x1)))) -> B(b(b(b(x1))))
117.13/30.67	   C(c(c(c(x1)))) -> B(b(x1))
117.13/30.67	The remaining pairs can at least be oriented weakly.
117.13/30.67	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, -1A, -1A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-1A, 0A, -I], [2A, -I, -1A], [-I, -1A, -1A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -1A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	   <<<
117.13/30.67	 POL(b(x_1)) =  	[[0A], [1A], [2A]] 	 +  	[[-I, -I, -1A], [-1A, -I, 0A], [1A, 0A, 0A]] 	* 	x_1
117.13/30.67	>>>
117.13/30.67	
117.13/30.67	
117.13/30.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
117.13/30.67	
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	
117.13/30.67	
117.13/30.67	----------------------------------------
117.13/30.67	
117.13/30.67	(12)
117.13/30.67	Obligation:
117.13/30.67	Q DP problem:
117.13/30.67	The TRS P consists of the following rules:
117.13/30.67	
117.13/30.67	   B(b(x1)) -> C(x1)
117.13/30.67	
117.13/30.67	The TRS R consists of the following rules:
117.13/30.67	
117.13/30.67	   c(c(c(c(x1)))) -> b(b(b(b(x1))))
117.13/30.67	   b(b(x1)) -> x1
117.13/30.67	   b(b(x1)) -> c(b(c(x1)))
117.13/30.67	
117.13/30.67	Q is empty.
117.13/30.67	We have to consider all minimal (P,Q,R)-chains.
117.13/30.68	----------------------------------------
117.13/30.68	
117.13/30.68	(13) DependencyGraphProof (EQUIVALENT)
117.13/30.68	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
117.13/30.68	----------------------------------------
117.13/30.68	
117.13/30.68	(14)
117.13/30.68	TRUE
117.42/30.76	EOF