22.83/7.07	YES
24.37/7.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
24.37/7.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.37/7.23	
24.37/7.23	
24.37/7.23	Termination w.r.t. Q of the given QTRS could be proven:
24.37/7.23	
24.37/7.23	(0) QTRS
24.37/7.23	(1) QTRS Reverse [EQUIVALENT, 0 ms]
24.37/7.23	(2) QTRS
24.37/7.23	(3) DependencyPairsProof [EQUIVALENT, 17 ms]
24.37/7.23	(4) QDP
24.37/7.23	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
24.37/7.23	(6) AND
24.37/7.23	    (7) QDP
24.37/7.23	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
24.37/7.23	        (9) QDP
24.37/7.23	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
24.37/7.23	        (11) YES
24.37/7.23	    (12) QDP
24.37/7.23	        (13) QDPOrderProof [EQUIVALENT, 185 ms]
24.37/7.23	        (14) QDP
24.37/7.23	        (15) QDPOrderProof [EQUIVALENT, 92 ms]
24.37/7.23	        (16) QDP
24.37/7.23	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
24.37/7.23	        (18) YES
24.37/7.23	
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(0)
24.37/7.23	Obligation:
24.37/7.23	Q restricted rewrite system:
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   a(c(x1)) -> c(c(c(a(b(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(1) QTRS Reverse (EQUIVALENT)
24.37/7.23	We applied the QTRS Reverse Processor [REVERSE].
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(2)
24.37/7.23	Obligation:
24.37/7.23	Q restricted rewrite system:
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(3) DependencyPairsProof (EQUIVALENT)
24.37/7.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(4)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	The TRS P consists of the following rules:
24.37/7.23	
24.37/7.23	   A(x1) -> B(x1)
24.37/7.23	   C(a(x1)) -> B(a(c(c(c(x1)))))
24.37/7.23	   C(a(x1)) -> A(c(c(c(x1))))
24.37/7.23	   C(a(x1)) -> C(c(c(x1)))
24.37/7.23	   C(a(x1)) -> C(c(x1))
24.37/7.23	   C(a(x1)) -> C(x1)
24.37/7.23	   B(b(x1)) -> A(x1)
24.37/7.23	
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(5) DependencyGraphProof (EQUIVALENT)
24.37/7.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(6)
24.37/7.23	Complex Obligation (AND)
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(7)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	The TRS P consists of the following rules:
24.37/7.23	
24.37/7.23	   B(b(x1)) -> A(x1)
24.37/7.23	   A(x1) -> B(x1)
24.37/7.23	
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(8) UsableRulesProof (EQUIVALENT)
24.37/7.23	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.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(9)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	The TRS P consists of the following rules:
24.37/7.23	
24.37/7.23	   B(b(x1)) -> A(x1)
24.37/7.23	   A(x1) -> B(x1)
24.37/7.23	
24.37/7.23	R is empty.
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(10) QDPSizeChangeProof (EQUIVALENT)
24.37/7.23	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. 
24.37/7.23	
24.37/7.23	From the DPs we obtained the following set of size-change graphs:
24.37/7.23	*A(x1) -> B(x1)
24.37/7.23	The graph contains the following edges 1 >= 1
24.37/7.23	
24.37/7.23	
24.37/7.23	*B(b(x1)) -> A(x1)
24.37/7.23	The graph contains the following edges 1 > 1
24.37/7.23	
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(11)
24.37/7.23	YES
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(12)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	The TRS P consists of the following rules:
24.37/7.23	
24.37/7.23	   C(a(x1)) -> C(c(x1))
24.37/7.23	   C(a(x1)) -> C(c(c(x1)))
24.37/7.23	   C(a(x1)) -> C(x1)
24.37/7.23	
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(13) QDPOrderProof (EQUIVALENT)
24.37/7.23	We use the reduction pair processor [LPAR04,JAR06].
24.37/7.23	
24.37/7.23	
24.37/7.23	The following pairs can be oriented strictly and are deleted.
24.37/7.23	
24.37/7.23	   C(a(x1)) -> C(x1)
24.37/7.23	The remaining pairs can at least be oriented weakly.
24.37/7.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(C(x_1)) =  	[[0A]] 	 +  	[[1A, -I, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[1A, 1A, -I], [0A, 0A, 0A], [0A, 1A, 1A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 1A, -I], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	
24.37/7.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.37/7.23	
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(14)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	The TRS P consists of the following rules:
24.37/7.23	
24.37/7.23	   C(a(x1)) -> C(c(x1))
24.37/7.23	   C(a(x1)) -> C(c(c(x1)))
24.37/7.23	
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(15) QDPOrderProof (EQUIVALENT)
24.37/7.23	We use the reduction pair processor [LPAR04,JAR06].
24.37/7.23	
24.37/7.23	
24.37/7.23	The following pairs can be oriented strictly and are deleted.
24.37/7.23	
24.37/7.23	   C(a(x1)) -> C(c(x1))
24.37/7.23	   C(a(x1)) -> C(c(c(x1)))
24.37/7.23	The remaining pairs can at least be oriented weakly.
24.37/7.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(a(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 1A, 1A], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	   <<<
24.37/7.23	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 1A, 1A], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
24.37/7.23	>>>
24.37/7.23	
24.37/7.23	
24.37/7.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.37/7.23	
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	
24.37/7.23	
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(16)
24.37/7.23	Obligation:
24.37/7.23	Q DP problem:
24.37/7.23	P is empty.
24.37/7.23	The TRS R consists of the following rules:
24.37/7.23	
24.37/7.23	   a(x1) -> b(x1)
24.37/7.23	   c(a(x1)) -> b(a(c(c(c(x1)))))
24.37/7.23	   b(b(x1)) -> a(x1)
24.37/7.23	
24.37/7.23	Q is empty.
24.37/7.23	We have to consider all minimal (P,Q,R)-chains.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(17) PisEmptyProof (EQUIVALENT)
24.37/7.23	The TRS P is empty. Hence, there is no (P,Q,R) chain.
24.37/7.23	----------------------------------------
24.37/7.23	
24.37/7.23	(18)
24.37/7.23	YES
24.53/7.30	EOF