30.38/8.59	YES
30.38/8.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.38/8.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.38/8.61	
30.38/8.61	
30.38/8.61	Termination w.r.t. Q of the given QTRS could be proven:
30.38/8.61	
30.38/8.61	(0) QTRS
30.38/8.61	(1) QTRS Reverse [EQUIVALENT, 0 ms]
30.38/8.61	(2) QTRS
30.38/8.61	(3) DependencyPairsProof [EQUIVALENT, 6 ms]
30.38/8.61	(4) QDP
30.38/8.61	(5) QDPOrderProof [EQUIVALENT, 99 ms]
30.38/8.61	(6) QDP
30.38/8.61	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
30.38/8.61	(8) AND
30.38/8.61	    (9) QDP
30.38/8.61	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
30.38/8.61	        (11) QDP
30.38/8.61	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
30.38/8.61	        (13) YES
30.38/8.61	    (14) QDP
30.38/8.61	        (15) QDPOrderProof [EQUIVALENT, 0 ms]
30.38/8.61	        (16) QDP
30.38/8.61	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
30.38/8.61	        (18) YES
30.38/8.61	
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(0)
30.38/8.61	Obligation:
30.38/8.61	Q restricted rewrite system:
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   a(b(x1)) -> b(c(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(1) QTRS Reverse (EQUIVALENT)
30.38/8.61	We applied the QTRS Reverse Processor [REVERSE].
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(2)
30.38/8.61	Obligation:
30.38/8.61	Q restricted rewrite system:
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(3) DependencyPairsProof (EQUIVALENT)
30.38/8.61	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(4)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	The TRS P consists of the following rules:
30.38/8.61	
30.38/8.61	   A(x1) -> B(x1)
30.38/8.61	   B(a(x1)) -> C(b(x1))
30.38/8.61	   B(a(x1)) -> B(x1)
30.38/8.61	   C(c(x1)) -> A(c(a(x1)))
30.38/8.61	   C(c(x1)) -> C(a(x1))
30.38/8.61	   C(c(x1)) -> A(x1)
30.38/8.61	
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(5) QDPOrderProof (EQUIVALENT)
30.38/8.61	We use the reduction pair processor [LPAR04,JAR06].
30.38/8.61	
30.38/8.61	
30.38/8.61	The following pairs can be oriented strictly and are deleted.
30.38/8.61	
30.38/8.61	   B(a(x1)) -> C(b(x1))
30.38/8.61	The remaining pairs can at least be oriented weakly.
30.38/8.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 1A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [1A, 1A, 1A], [-I, 0A, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	
30.38/8.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.38/8.61	
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	   a(x1) -> x1
30.38/8.61	
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(6)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	The TRS P consists of the following rules:
30.38/8.61	
30.38/8.61	   A(x1) -> B(x1)
30.38/8.61	   B(a(x1)) -> B(x1)
30.38/8.61	   C(c(x1)) -> A(c(a(x1)))
30.38/8.61	   C(c(x1)) -> C(a(x1))
30.38/8.61	   C(c(x1)) -> A(x1)
30.38/8.61	
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(7) DependencyGraphProof (EQUIVALENT)
30.38/8.61	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(8)
30.38/8.61	Complex Obligation (AND)
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(9)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	The TRS P consists of the following rules:
30.38/8.61	
30.38/8.61	   B(a(x1)) -> B(x1)
30.38/8.61	
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(10) UsableRulesProof (EQUIVALENT)
30.38/8.61	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.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(11)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	The TRS P consists of the following rules:
30.38/8.61	
30.38/8.61	   B(a(x1)) -> B(x1)
30.38/8.61	
30.38/8.61	R is empty.
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(12) QDPSizeChangeProof (EQUIVALENT)
30.38/8.61	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. 
30.38/8.61	
30.38/8.61	From the DPs we obtained the following set of size-change graphs:
30.38/8.61	*B(a(x1)) -> B(x1)
30.38/8.61	The graph contains the following edges 1 > 1
30.38/8.61	
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(13)
30.38/8.61	YES
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(14)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	The TRS P consists of the following rules:
30.38/8.61	
30.38/8.61	   C(c(x1)) -> C(a(x1))
30.38/8.61	
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(15) QDPOrderProof (EQUIVALENT)
30.38/8.61	We use the reduction pair processor [LPAR04,JAR06].
30.38/8.61	
30.38/8.61	
30.38/8.61	The following pairs can be oriented strictly and are deleted.
30.38/8.61	
30.38/8.61	   C(c(x1)) -> C(a(x1))
30.38/8.61	The remaining pairs can at least be oriented weakly.
30.38/8.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(c(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 1A, 1A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 1A, 1A], [-I, -I, 0A]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	   <<<
30.38/8.61	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 0A], [-I, -I, -I]] 	* 	x_1
30.38/8.61	>>>
30.38/8.61	
30.38/8.61	
30.38/8.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(16)
30.38/8.61	Obligation:
30.38/8.61	Q DP problem:
30.38/8.61	P is empty.
30.38/8.61	The TRS R consists of the following rules:
30.38/8.61	
30.38/8.61	   a(x1) -> x1
30.38/8.61	   a(x1) -> b(x1)
30.38/8.61	   b(a(x1)) -> c(b(x1))
30.38/8.61	   c(c(x1)) -> a(c(a(x1)))
30.38/8.61	
30.38/8.61	Q is empty.
30.38/8.61	We have to consider all minimal (P,Q,R)-chains.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(17) PisEmptyProof (EQUIVALENT)
30.38/8.61	The TRS P is empty. Hence, there is no (P,Q,R) chain.
30.38/8.61	----------------------------------------
30.38/8.61	
30.38/8.61	(18)
30.38/8.61	YES
30.92/8.81	EOF