24.13/7.08	YES
24.92/7.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
24.92/7.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.92/7.30	
24.92/7.30	
24.92/7.30	Termination w.r.t. Q of the given QTRS could be proven:
24.92/7.30	
24.92/7.30	(0) QTRS
24.92/7.30	(1) QTRS Reverse [EQUIVALENT, 0 ms]
24.92/7.30	(2) QTRS
24.92/7.30	(3) DependencyPairsProof [EQUIVALENT, 18 ms]
24.92/7.30	(4) QDP
24.92/7.30	(5) DependencyGraphProof [EQUIVALENT, 3 ms]
24.92/7.30	(6) AND
24.92/7.30	    (7) QDP
24.92/7.30	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
24.92/7.30	        (9) QDP
24.92/7.30	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
24.92/7.30	        (11) YES
24.92/7.30	    (12) QDP
24.92/7.30	        (13) QDPOrderProof [EQUIVALENT, 82 ms]
24.92/7.30	        (14) QDP
24.92/7.30	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
24.92/7.30	        (16) QDP
24.92/7.30	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
24.92/7.30	        (18) YES
24.92/7.30	
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(0)
24.92/7.30	Obligation:
24.92/7.30	Q restricted rewrite system:
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   b(c(x1)) -> c(c(b(a(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(1) QTRS Reverse (EQUIVALENT)
24.92/7.30	We applied the QTRS Reverse Processor [REVERSE].
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(2)
24.92/7.30	Obligation:
24.92/7.30	Q restricted rewrite system:
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(3) DependencyPairsProof (EQUIVALENT)
24.92/7.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(4)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   A(a(x1)) -> B(x1)
24.92/7.30	   B(x1) -> A(x1)
24.92/7.30	   C(b(x1)) -> A(b(c(c(x1))))
24.92/7.30	   C(b(x1)) -> B(c(c(x1)))
24.92/7.30	   C(b(x1)) -> C(c(x1))
24.92/7.30	   C(b(x1)) -> C(x1)
24.92/7.30	
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(5) DependencyGraphProof (EQUIVALENT)
24.92/7.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(6)
24.92/7.30	Complex Obligation (AND)
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(7)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   B(x1) -> A(x1)
24.92/7.30	   A(a(x1)) -> B(x1)
24.92/7.30	
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(8) UsableRulesProof (EQUIVALENT)
24.92/7.30	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.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(9)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   B(x1) -> A(x1)
24.92/7.30	   A(a(x1)) -> B(x1)
24.92/7.30	
24.92/7.30	R is empty.
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(10) QDPSizeChangeProof (EQUIVALENT)
24.92/7.30	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. 
24.92/7.30	
24.92/7.30	From the DPs we obtained the following set of size-change graphs:
24.92/7.30	*A(a(x1)) -> B(x1)
24.92/7.30	The graph contains the following edges 1 > 1
24.92/7.30	
24.92/7.30	
24.92/7.30	*B(x1) -> A(x1)
24.92/7.30	The graph contains the following edges 1 >= 1
24.92/7.30	
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(11)
24.92/7.30	YES
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(12)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   C(b(x1)) -> C(x1)
24.92/7.30	   C(b(x1)) -> C(c(x1))
24.92/7.30	
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(13) QDPOrderProof (EQUIVALENT)
24.92/7.30	We use the reduction pair processor [LPAR04,JAR06].
24.92/7.30	
24.92/7.30	
24.92/7.30	The following pairs can be oriented strictly and are deleted.
24.92/7.30	
24.92/7.30	   C(b(x1)) -> C(c(x1))
24.92/7.30	The remaining pairs can at least be oriented weakly.
24.92/7.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.92/7.30	
24.92/7.30	   <<<
24.92/7.30	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
24.92/7.30	>>>
24.92/7.30	
24.92/7.30	   <<<
24.92/7.30	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.92/7.30	>>>
24.92/7.30	
24.92/7.30	   <<<
24.92/7.30	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
24.92/7.30	>>>
24.92/7.30	
24.92/7.30	   <<<
24.92/7.30	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 1A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.92/7.30	>>>
24.92/7.30	
24.92/7.30	
24.92/7.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.92/7.30	
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(14)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   C(b(x1)) -> C(x1)
24.92/7.30	
24.92/7.30	The TRS R consists of the following rules:
24.92/7.30	
24.92/7.30	   a(x1) -> x1
24.92/7.30	   a(a(x1)) -> b(x1)
24.92/7.30	   b(x1) -> a(x1)
24.92/7.30	   c(b(x1)) -> a(b(c(c(x1))))
24.92/7.30	
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(15) UsableRulesProof (EQUIVALENT)
24.92/7.30	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.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(16)
24.92/7.30	Obligation:
24.92/7.30	Q DP problem:
24.92/7.30	The TRS P consists of the following rules:
24.92/7.30	
24.92/7.30	   C(b(x1)) -> C(x1)
24.92/7.30	
24.92/7.30	R is empty.
24.92/7.30	Q is empty.
24.92/7.30	We have to consider all minimal (P,Q,R)-chains.
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(17) QDPSizeChangeProof (EQUIVALENT)
24.92/7.30	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. 
24.92/7.30	
24.92/7.30	From the DPs we obtained the following set of size-change graphs:
24.92/7.30	*C(b(x1)) -> C(x1)
24.92/7.30	The graph contains the following edges 1 > 1
24.92/7.30	
24.92/7.30	
24.92/7.30	----------------------------------------
24.92/7.30	
24.92/7.30	(18)
24.92/7.30	YES
25.19/10.23	EOF