40.30/11.09	YES
40.45/11.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.45/11.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.45/11.13	
40.45/11.13	
40.45/11.13	Termination w.r.t. Q of the given QTRS could be proven:
40.45/11.13	
40.45/11.13	(0) QTRS
40.45/11.13	(1) DependencyPairsProof [EQUIVALENT, 10 ms]
40.45/11.13	(2) QDP
40.45/11.13	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
40.45/11.13	(4) QDP
40.45/11.13	(5) QDPOrderProof [EQUIVALENT, 117 ms]
40.45/11.13	(6) QDP
40.45/11.13	(7) QDPOrderProof [EQUIVALENT, 63 ms]
40.45/11.13	(8) QDP
40.45/11.13	(9) UsableRulesProof [EQUIVALENT, 0 ms]
40.45/11.13	(10) QDP
40.45/11.13	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
40.45/11.13	(12) YES
40.45/11.13	
40.45/11.13	
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(0)
40.45/11.13	Obligation:
40.45/11.13	Q restricted rewrite system:
40.45/11.13	The TRS R consists of the following rules:
40.45/11.13	
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	
40.45/11.13	Q is empty.
40.45/11.13	
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(1) DependencyPairsProof (EQUIVALENT)
40.45/11.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(2)
40.45/11.13	Obligation:
40.45/11.13	Q DP problem:
40.45/11.13	The TRS P consists of the following rules:
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(a(c(c(x1))))
40.45/11.13	   C(c(b(x1))) -> A(c(c(x1)))
40.45/11.13	   C(c(b(x1))) -> C(c(x1))
40.45/11.13	   C(c(b(x1))) -> C(x1)
40.45/11.13	
40.45/11.13	The TRS R consists of the following rules:
40.45/11.13	
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	
40.45/11.13	Q is empty.
40.45/11.13	We have to consider all minimal (P,Q,R)-chains.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(3) DependencyGraphProof (EQUIVALENT)
40.45/11.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(4)
40.45/11.13	Obligation:
40.45/11.13	Q DP problem:
40.45/11.13	The TRS P consists of the following rules:
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(c(x1))
40.45/11.13	   C(c(b(x1))) -> C(a(c(c(x1))))
40.45/11.13	   C(c(b(x1))) -> C(x1)
40.45/11.13	
40.45/11.13	The TRS R consists of the following rules:
40.45/11.13	
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	
40.45/11.13	Q is empty.
40.45/11.13	We have to consider all minimal (P,Q,R)-chains.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(5) QDPOrderProof (EQUIVALENT)
40.45/11.13	We use the reduction pair processor [LPAR04,JAR06].
40.45/11.13	
40.45/11.13	
40.45/11.13	The following pairs can be oriented strictly and are deleted.
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(a(c(c(x1))))
40.45/11.13	The remaining pairs can at least be oriented weakly.
40.45/11.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	
40.45/11.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.45/11.13	
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	
40.45/11.13	
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(6)
40.45/11.13	Obligation:
40.45/11.13	Q DP problem:
40.45/11.13	The TRS P consists of the following rules:
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(c(x1))
40.45/11.13	   C(c(b(x1))) -> C(x1)
40.45/11.13	
40.45/11.13	The TRS R consists of the following rules:
40.45/11.13	
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	
40.45/11.13	Q is empty.
40.45/11.13	We have to consider all minimal (P,Q,R)-chains.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(7) QDPOrderProof (EQUIVALENT)
40.45/11.13	We use the reduction pair processor [LPAR04,JAR06].
40.45/11.13	
40.45/11.13	
40.45/11.13	The following pairs can be oriented strictly and are deleted.
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(c(x1))
40.45/11.13	The remaining pairs can at least be oriented weakly.
40.45/11.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, -I, -I], [0A, 1A, 1A], [-I, -I, -I]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	   <<<
40.45/11.13	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, -I]] 	* 	x_1
40.45/11.13	>>>
40.45/11.13	
40.45/11.13	
40.45/11.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.45/11.13	
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	
40.45/11.13	
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(8)
40.45/11.13	Obligation:
40.45/11.13	Q DP problem:
40.45/11.13	The TRS P consists of the following rules:
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(x1)
40.45/11.13	
40.45/11.13	The TRS R consists of the following rules:
40.45/11.13	
40.45/11.13	   a(x1) -> b(x1)
40.45/11.13	   a(b(x1)) -> x1
40.45/11.13	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
40.45/11.13	
40.45/11.13	Q is empty.
40.45/11.13	We have to consider all minimal (P,Q,R)-chains.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(9) UsableRulesProof (EQUIVALENT)
40.45/11.13	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.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(10)
40.45/11.13	Obligation:
40.45/11.13	Q DP problem:
40.45/11.13	The TRS P consists of the following rules:
40.45/11.13	
40.45/11.13	   C(c(b(x1))) -> C(x1)
40.45/11.13	
40.45/11.13	R is empty.
40.45/11.13	Q is empty.
40.45/11.13	We have to consider all minimal (P,Q,R)-chains.
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(11) QDPSizeChangeProof (EQUIVALENT)
40.45/11.13	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. 
40.45/11.13	
40.45/11.13	From the DPs we obtained the following set of size-change graphs:
40.45/11.13	*C(c(b(x1))) -> C(x1)
40.45/11.13	The graph contains the following edges 1 > 1
40.45/11.13	
40.45/11.13	
40.45/11.13	----------------------------------------
40.45/11.13	
40.45/11.13	(12)
40.45/11.13	YES
40.53/11.21	EOF