66.83/18.15	YES
67.58/18.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
67.58/18.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
67.58/18.29	
67.58/18.29	
67.58/18.29	Termination w.r.t. Q of the given QTRS could be proven:
67.58/18.29	
67.58/18.29	(0) QTRS
67.58/18.29	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
67.58/18.29	(2) QDP
67.58/18.29	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
67.58/18.29	(4) QDP
67.58/18.29	(5) QDPOrderProof [EQUIVALENT, 81 ms]
67.58/18.29	(6) QDP
67.58/18.29	(7) QDPOrderProof [EQUIVALENT, 0 ms]
67.58/18.29	(8) QDP
67.58/18.29	(9) UsableRulesProof [EQUIVALENT, 0 ms]
67.58/18.29	(10) QDP
67.58/18.29	(11) QDPSizeChangeProof [EQUIVALENT, 1 ms]
67.58/18.29	(12) YES
67.58/18.29	
67.58/18.29	
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(0)
67.58/18.29	Obligation:
67.58/18.29	Q restricted rewrite system:
67.58/18.29	The TRS R consists of the following rules:
67.58/18.29	
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	
67.58/18.29	Q is empty.
67.58/18.29	
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(1) DependencyPairsProof (EQUIVALENT)
67.58/18.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(2)
67.58/18.29	Obligation:
67.58/18.29	Q DP problem:
67.58/18.29	The TRS P consists of the following rules:
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> A(c(a(c(c(x1)))))
67.58/18.29	   C(c(a(x1))) -> C(a(c(c(x1))))
67.58/18.29	   C(c(a(x1))) -> A(c(c(x1)))
67.58/18.29	   C(c(a(x1))) -> C(c(x1))
67.58/18.29	   C(c(a(x1))) -> C(x1)
67.58/18.29	
67.58/18.29	The TRS R consists of the following rules:
67.58/18.29	
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	
67.58/18.29	Q is empty.
67.58/18.29	We have to consider all minimal (P,Q,R)-chains.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(3) DependencyGraphProof (EQUIVALENT)
67.58/18.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(4)
67.58/18.29	Obligation:
67.58/18.29	Q DP problem:
67.58/18.29	The TRS P consists of the following rules:
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(c(x1))
67.58/18.29	   C(c(a(x1))) -> C(a(c(c(x1))))
67.58/18.29	   C(c(a(x1))) -> C(x1)
67.58/18.29	
67.58/18.29	The TRS R consists of the following rules:
67.58/18.29	
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	
67.58/18.29	Q is empty.
67.58/18.29	We have to consider all minimal (P,Q,R)-chains.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(5) QDPOrderProof (EQUIVALENT)
67.58/18.29	We use the reduction pair processor [LPAR04,JAR06].
67.58/18.29	
67.58/18.29	
67.58/18.29	The following pairs can be oriented strictly and are deleted.
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(a(c(c(x1))))
67.58/18.29	The remaining pairs can at least be oriented weakly.
67.58/18.29	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [0A, 1A, 0A], [-I, -I, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	
67.58/18.29	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
67.58/18.29	
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	
67.58/18.29	
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(6)
67.58/18.29	Obligation:
67.58/18.29	Q DP problem:
67.58/18.29	The TRS P consists of the following rules:
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(c(x1))
67.58/18.29	   C(c(a(x1))) -> C(x1)
67.58/18.29	
67.58/18.29	The TRS R consists of the following rules:
67.58/18.29	
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	
67.58/18.29	Q is empty.
67.58/18.29	We have to consider all minimal (P,Q,R)-chains.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(7) QDPOrderProof (EQUIVALENT)
67.58/18.29	We use the reduction pair processor [LPAR04,JAR06].
67.58/18.29	
67.58/18.29	
67.58/18.29	The following pairs can be oriented strictly and are deleted.
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(c(x1))
67.58/18.29	The remaining pairs can at least be oriented weakly.
67.58/18.29	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	   <<<
67.58/18.29	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, -I, 0A], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1
67.58/18.29	>>>
67.58/18.29	
67.58/18.29	
67.58/18.29	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
67.58/18.29	
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	
67.58/18.29	
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(8)
67.58/18.29	Obligation:
67.58/18.29	Q DP problem:
67.58/18.29	The TRS P consists of the following rules:
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(x1)
67.58/18.29	
67.58/18.29	The TRS R consists of the following rules:
67.58/18.29	
67.58/18.29	   a(x1) -> b(x1)
67.58/18.29	   a(b(x1)) -> x1
67.58/18.29	   c(c(a(x1))) -> a(c(a(c(c(x1)))))
67.58/18.29	
67.58/18.29	Q is empty.
67.58/18.29	We have to consider all minimal (P,Q,R)-chains.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(9) UsableRulesProof (EQUIVALENT)
67.58/18.29	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.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(10)
67.58/18.29	Obligation:
67.58/18.29	Q DP problem:
67.58/18.29	The TRS P consists of the following rules:
67.58/18.29	
67.58/18.29	   C(c(a(x1))) -> C(x1)
67.58/18.29	
67.58/18.29	R is empty.
67.58/18.29	Q is empty.
67.58/18.29	We have to consider all minimal (P,Q,R)-chains.
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(11) QDPSizeChangeProof (EQUIVALENT)
67.58/18.29	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. 
67.58/18.29	
67.58/18.29	From the DPs we obtained the following set of size-change graphs:
67.58/18.29	*C(c(a(x1))) -> C(x1)
67.58/18.29	The graph contains the following edges 1 > 1
67.58/18.29	
67.58/18.29	
67.58/18.29	----------------------------------------
67.58/18.29	
67.58/18.29	(12)
67.58/18.29	YES
67.74/18.37	EOF