26.80/7.67	YES
27.67/7.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.67/7.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.67/7.91	
27.67/7.91	
27.67/7.91	Termination w.r.t. Q of the given QTRS could be proven:
27.67/7.91	
27.67/7.91	(0) QTRS
27.67/7.91	(1) QTRS Reverse [EQUIVALENT, 0 ms]
27.67/7.91	(2) QTRS
27.67/7.91	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
27.67/7.91	(4) QDP
27.67/7.91	(5) DependencyGraphProof [EQUIVALENT, 5 ms]
27.67/7.91	(6) AND
27.67/7.91	    (7) QDP
27.67/7.91	        (8) UsableRulesProof [EQUIVALENT, 3 ms]
27.67/7.91	        (9) QDP
27.67/7.91	        (10) QDPSizeChangeProof [EQUIVALENT, 2 ms]
27.67/7.91	        (11) YES
27.67/7.91	    (12) QDP
27.67/7.91	        (13) QDPOrderProof [EQUIVALENT, 87 ms]
27.67/7.91	        (14) QDP
27.67/7.91	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
27.67/7.91	        (16) YES
27.67/7.91	
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(0)
27.67/7.91	Obligation:
27.67/7.91	Q restricted rewrite system:
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   a(c(x1)) -> c(c(a(b(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(1) QTRS Reverse (EQUIVALENT)
27.67/7.91	We applied the QTRS Reverse Processor [REVERSE].
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(2)
27.67/7.91	Obligation:
27.67/7.91	Q restricted rewrite system:
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(3) DependencyPairsProof (EQUIVALENT)
27.67/7.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(4)
27.67/7.91	Obligation:
27.67/7.91	Q DP problem:
27.67/7.91	The TRS P consists of the following rules:
27.67/7.91	
27.67/7.91	   A(x1) -> B(x1)
27.67/7.91	   C(a(x1)) -> B(a(c(c(x1))))
27.67/7.91	   C(a(x1)) -> A(c(c(x1)))
27.67/7.91	   C(a(x1)) -> C(c(x1))
27.67/7.91	   C(a(x1)) -> C(x1)
27.67/7.91	   B(b(x1)) -> A(x1)
27.67/7.91	
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	We have to consider all minimal (P,Q,R)-chains.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(5) DependencyGraphProof (EQUIVALENT)
27.67/7.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(6)
27.67/7.91	Complex Obligation (AND)
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(7)
27.67/7.91	Obligation:
27.67/7.91	Q DP problem:
27.67/7.91	The TRS P consists of the following rules:
27.67/7.91	
27.67/7.91	   B(b(x1)) -> A(x1)
27.67/7.91	   A(x1) -> B(x1)
27.67/7.91	
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	We have to consider all minimal (P,Q,R)-chains.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(8) UsableRulesProof (EQUIVALENT)
27.67/7.91	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.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(9)
27.67/7.91	Obligation:
27.67/7.91	Q DP problem:
27.67/7.91	The TRS P consists of the following rules:
27.67/7.91	
27.67/7.91	   B(b(x1)) -> A(x1)
27.67/7.91	   A(x1) -> B(x1)
27.67/7.91	
27.67/7.91	R is empty.
27.67/7.91	Q is empty.
27.67/7.91	We have to consider all minimal (P,Q,R)-chains.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(10) QDPSizeChangeProof (EQUIVALENT)
27.67/7.91	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. 
27.67/7.91	
27.67/7.91	From the DPs we obtained the following set of size-change graphs:
27.67/7.91	*A(x1) -> B(x1)
27.67/7.91	The graph contains the following edges 1 >= 1
27.67/7.91	
27.67/7.91	
27.67/7.91	*B(b(x1)) -> A(x1)
27.67/7.91	The graph contains the following edges 1 > 1
27.67/7.91	
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(11)
27.67/7.91	YES
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(12)
27.67/7.91	Obligation:
27.67/7.91	Q DP problem:
27.67/7.91	The TRS P consists of the following rules:
27.67/7.91	
27.67/7.91	   C(a(x1)) -> C(x1)
27.67/7.91	   C(a(x1)) -> C(c(x1))
27.67/7.91	
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	We have to consider all minimal (P,Q,R)-chains.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(13) QDPOrderProof (EQUIVALENT)
27.67/7.91	We use the reduction pair processor [LPAR04,JAR06].
27.67/7.91	
27.67/7.91	
27.67/7.91	The following pairs can be oriented strictly and are deleted.
27.67/7.91	
27.67/7.91	   C(a(x1)) -> C(x1)
27.67/7.91	   C(a(x1)) -> C(c(x1))
27.67/7.91	The remaining pairs can at least be oriented weakly.
27.67/7.91	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.67/7.91	
27.67/7.91	   <<<
27.67/7.91	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
27.67/7.91	>>>
27.67/7.91	
27.67/7.91	   <<<
27.67/7.91	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
27.67/7.91	>>>
27.67/7.91	
27.67/7.91	   <<<
27.67/7.91	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
27.67/7.91	>>>
27.67/7.91	
27.67/7.91	   <<<
27.67/7.91	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
27.67/7.91	>>>
27.67/7.91	
27.67/7.91	
27.67/7.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.67/7.91	
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	
27.67/7.91	
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(14)
27.67/7.91	Obligation:
27.67/7.91	Q DP problem:
27.67/7.91	P is empty.
27.67/7.91	The TRS R consists of the following rules:
27.67/7.91	
27.67/7.91	   a(x1) -> b(x1)
27.67/7.91	   c(a(x1)) -> b(a(c(c(x1))))
27.67/7.91	   b(b(x1)) -> a(x1)
27.67/7.91	
27.67/7.91	Q is empty.
27.67/7.91	We have to consider all minimal (P,Q,R)-chains.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(15) PisEmptyProof (EQUIVALENT)
27.67/7.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.67/7.91	----------------------------------------
27.67/7.91	
27.67/7.91	(16)
27.67/7.91	YES
30.12/9.19	EOF