162.18/42.18	YES
162.55/42.25	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
162.55/42.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
162.55/42.25	
162.55/42.25	
162.55/42.25	Termination w.r.t. Q of the given QTRS could be proven:
162.55/42.25	
162.55/42.25	(0) QTRS
162.55/42.25	(1) DependencyPairsProof [EQUIVALENT, 32 ms]
162.55/42.25	(2) QDP
162.55/42.25	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
162.55/42.25	(4) AND
162.55/42.25	    (5) QDP
162.55/42.25	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
162.55/42.25	        (7) QDP
162.55/42.25	        (8) QDPSizeChangeProof [EQUIVALENT, 1 ms]
162.55/42.25	        (9) YES
162.55/42.25	    (10) QDP
162.55/42.25	        (11) QDPOrderProof [EQUIVALENT, 30 ms]
162.55/42.25	        (12) QDP
162.55/42.25	        (13) QDPOrderProof [EQUIVALENT, 2671 ms]
162.55/42.25	        (14) QDP
162.55/42.25	        (15) QDPOrderProof [EQUIVALENT, 3127 ms]
162.55/42.25	        (16) QDP
162.55/42.25	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
162.55/42.25	        (18) YES
162.55/42.25	
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(0)
162.55/42.25	Obligation:
162.55/42.25	Q restricted rewrite system:
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(1) DependencyPairsProof (EQUIVALENT)
162.55/42.25	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(2)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(a(x1)))
162.55/42.25	   A(a(b(b(x1)))) -> A(a(x1))
162.55/42.25	   A(a(b(b(x1)))) -> A(x1)
162.55/42.25	   A(c(x1)) -> C(a(x1))
162.55/42.25	   A(c(x1)) -> A(x1)
162.55/42.25	   C(b(x1)) -> C(x1)
162.55/42.25	
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(3) DependencyGraphProof (EQUIVALENT)
162.55/42.25	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(4)
162.55/42.25	Complex Obligation (AND)
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(5)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   C(b(x1)) -> C(x1)
162.55/42.25	
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(6) UsableRulesProof (EQUIVALENT)
162.55/42.25	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.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(7)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   C(b(x1)) -> C(x1)
162.55/42.25	
162.55/42.25	R is empty.
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(8) QDPSizeChangeProof (EQUIVALENT)
162.55/42.25	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. 
162.55/42.25	
162.55/42.25	From the DPs we obtained the following set of size-change graphs:
162.55/42.25	*C(b(x1)) -> C(x1)
162.55/42.25	The graph contains the following edges 1 > 1
162.55/42.25	
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(9)
162.55/42.25	YES
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(10)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(x1))
162.55/42.25	   A(a(b(b(x1)))) -> A(a(a(x1)))
162.55/42.25	   A(a(b(b(x1)))) -> A(x1)
162.55/42.25	   A(c(x1)) -> A(x1)
162.55/42.25	
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(11) QDPOrderProof (EQUIVALENT)
162.55/42.25	We use the reduction pair processor [LPAR04,JAR06].
162.55/42.25	
162.55/42.25	
162.55/42.25	The following pairs can be oriented strictly and are deleted.
162.55/42.25	
162.55/42.25	   A(c(x1)) -> A(x1)
162.55/42.25	The remaining pairs can at least be oriented weakly.
162.55/42.25	Used ordering:  Polynomial interpretation [POLO]:
162.55/42.25	
162.55/42.25	   POL(A(x_1)) = x_1
162.55/42.25	   POL(a(x_1)) = x_1
162.55/42.25	   POL(b(x_1)) = x_1
162.55/42.25	   POL(c(x_1)) = 1 + x_1
162.55/42.25	
162.55/42.25	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(12)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(x1))
162.55/42.25	   A(a(b(b(x1)))) -> A(a(a(x1)))
162.55/42.25	   A(a(b(b(x1)))) -> A(x1)
162.55/42.25	
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(13) QDPOrderProof (EQUIVALENT)
162.55/42.25	We use the reduction pair processor [LPAR04,JAR06].
162.55/42.25	
162.55/42.25	
162.55/42.25	The following pairs can be oriented strictly and are deleted.
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(a(x1)))
162.55/42.25	   A(a(b(b(x1)))) -> A(x1)
162.55/42.25	The remaining pairs can at least be oriented weakly.
162.55/42.25	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [1A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	
162.55/42.25	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(14)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	The TRS P consists of the following rules:
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(x1))
162.55/42.25	
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(15) QDPOrderProof (EQUIVALENT)
162.55/42.25	We use the reduction pair processor [LPAR04,JAR06].
162.55/42.25	
162.55/42.25	
162.55/42.25	The following pairs can be oriented strictly and are deleted.
162.55/42.25	
162.55/42.25	   A(a(b(b(x1)))) -> A(a(x1))
162.55/42.25	The remaining pairs can at least be oriented weakly.
162.55/42.25	Used ordering:  Matrix interpretation [MATRO] to (N^5, +, *, >=, >) :
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(A(x_1)) =  	[[0]] 	 +  	[[0, 1, 1, 0, 0]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(a(x_1)) =  	[[0], [0], [0], [0], [0]] 	 +  	[[0, 1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 0, 0, 0]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(b(x_1)) =  	[[0], [0], [0], [1], [0]] 	 +  	[[0, 0, 0, 1, 1], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, 0, 0, 0], [0, 1, 0, 0, 0]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	   <<<
162.55/42.25	 POL(c(x_1)) =  	[[0], [0], [0], [0], [0]] 	 +  	[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]] 	* 	x_1
162.55/42.25	>>>
162.55/42.25	
162.55/42.25	
162.55/42.25	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(16)
162.55/42.25	Obligation:
162.55/42.25	Q DP problem:
162.55/42.25	P is empty.
162.55/42.25	The TRS R consists of the following rules:
162.55/42.25	
162.55/42.25	   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
162.55/42.25	   a(c(x1)) -> c(a(x1))
162.55/42.25	   c(b(x1)) -> b(c(x1))
162.55/42.25	
162.55/42.25	Q is empty.
162.55/42.25	We have to consider all minimal (P,Q,R)-chains.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(17) PisEmptyProof (EQUIVALENT)
162.55/42.25	The TRS P is empty. Hence, there is no (P,Q,R) chain.
162.55/42.25	----------------------------------------
162.55/42.25	
162.55/42.25	(18)
162.55/42.25	YES
162.75/42.34	EOF