29.21/8.37	YES
29.83/8.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.83/8.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.83/8.60	
29.83/8.60	
29.83/8.60	Termination w.r.t. Q of the given QTRS could be proven:
29.83/8.60	
29.83/8.60	(0) QTRS
29.83/8.60	(1) DependencyPairsProof [EQUIVALENT, 2 ms]
29.83/8.60	(2) QDP
29.83/8.60	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
29.83/8.60	(4) QDP
29.83/8.60	(5) QDPOrderProof [EQUIVALENT, 174 ms]
29.83/8.60	(6) QDP
29.83/8.60	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
29.83/8.60	(8) AND
29.83/8.60	    (9) QDP
29.83/8.60	        (10) QDPOrderProof [EQUIVALENT, 50 ms]
29.83/8.60	        (11) QDP
29.83/8.60	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
29.83/8.60	        (13) YES
29.83/8.60	    (14) QDP
29.83/8.60	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
29.83/8.60	        (16) QDP
29.83/8.60	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
29.83/8.60	        (18) YES
29.83/8.60	
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(0)
29.83/8.60	Obligation:
29.83/8.60	Q restricted rewrite system:
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(1) DependencyPairsProof (EQUIVALENT)
29.83/8.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(2)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   A(a(x1)) -> A(b(b(c(x1))))
29.83/8.60	   A(a(x1)) -> B(b(c(x1)))
29.83/8.60	   A(a(x1)) -> B(c(x1))
29.83/8.60	   A(a(x1)) -> C(x1)
29.83/8.60	   C(b(x1)) -> A(c(x1))
29.83/8.60	   C(b(x1)) -> C(x1)
29.83/8.60	
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(3) DependencyGraphProof (EQUIVALENT)
29.83/8.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(4)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   A(a(x1)) -> C(x1)
29.83/8.60	   C(b(x1)) -> A(c(x1))
29.83/8.60	   A(a(x1)) -> A(b(b(c(x1))))
29.83/8.60	   C(b(x1)) -> C(x1)
29.83/8.60	
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(5) QDPOrderProof (EQUIVALENT)
29.83/8.60	We use the reduction pair processor [LPAR04,JAR06].
29.83/8.60	
29.83/8.60	
29.83/8.60	The following pairs can be oriented strictly and are deleted.
29.83/8.60	
29.83/8.60	   A(a(x1)) -> C(x1)
29.83/8.60	The remaining pairs can at least be oriented weakly.
29.83/8.60	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	
29.83/8.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.83/8.60	
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(6)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   C(b(x1)) -> A(c(x1))
29.83/8.60	   A(a(x1)) -> A(b(b(c(x1))))
29.83/8.60	   C(b(x1)) -> C(x1)
29.83/8.60	
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(7) DependencyGraphProof (EQUIVALENT)
29.83/8.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(8)
29.83/8.60	Complex Obligation (AND)
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(9)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   A(a(x1)) -> A(b(b(c(x1))))
29.83/8.60	
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(10) QDPOrderProof (EQUIVALENT)
29.83/8.60	We use the reduction pair processor [LPAR04,JAR06].
29.83/8.60	
29.83/8.60	
29.83/8.60	The following pairs can be oriented strictly and are deleted.
29.83/8.60	
29.83/8.60	   A(a(x1)) -> A(b(b(c(x1))))
29.83/8.60	The remaining pairs can at least be oriented weakly.
29.83/8.60	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 1A, 1A], [-I, -I, -I]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, -I, 1A]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	   <<<
29.83/8.60	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, 1A], [-I, -I, -I]] 	* 	x_1
29.83/8.60	>>>
29.83/8.60	
29.83/8.60	
29.83/8.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.83/8.60	
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(11)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	P is empty.
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(12) PisEmptyProof (EQUIVALENT)
29.83/8.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(13)
29.83/8.60	YES
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(14)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   C(b(x1)) -> C(x1)
29.83/8.60	
29.83/8.60	The TRS R consists of the following rules:
29.83/8.60	
29.83/8.60	   a(a(x1)) -> a(b(b(c(x1))))
29.83/8.60	   b(a(x1)) -> x1
29.83/8.60	   c(b(x1)) -> a(c(x1))
29.83/8.60	
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(15) UsableRulesProof (EQUIVALENT)
29.83/8.60	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.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(16)
29.83/8.60	Obligation:
29.83/8.60	Q DP problem:
29.83/8.60	The TRS P consists of the following rules:
29.83/8.60	
29.83/8.60	   C(b(x1)) -> C(x1)
29.83/8.60	
29.83/8.60	R is empty.
29.83/8.60	Q is empty.
29.83/8.60	We have to consider all minimal (P,Q,R)-chains.
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(17) QDPSizeChangeProof (EQUIVALENT)
29.83/8.60	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. 
29.83/8.60	
29.83/8.60	From the DPs we obtained the following set of size-change graphs:
29.83/8.60	*C(b(x1)) -> C(x1)
29.83/8.60	The graph contains the following edges 1 > 1
29.83/8.60	
29.83/8.60	
29.83/8.60	----------------------------------------
29.83/8.60	
29.83/8.60	(18)
29.83/8.60	YES
30.28/8.81	EOF