29.75/8.60	YES
32.77/9.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.77/9.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.77/9.39	
32.77/9.39	
32.77/9.39	Termination w.r.t. Q of the given QTRS could be proven:
32.77/9.39	
32.77/9.39	(0) QTRS
32.77/9.39	(1) QTRSRRRProof [EQUIVALENT, 49 ms]
32.77/9.39	(2) QTRS
32.77/9.39	(3) QTRSRRRProof [EQUIVALENT, 4 ms]
32.77/9.39	(4) QTRS
32.77/9.39	(5) DependencyPairsProof [EQUIVALENT, 22 ms]
32.77/9.39	(6) QDP
32.77/9.39	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
32.77/9.39	(8) AND
32.77/9.39	    (9) QDP
32.77/9.39	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
32.77/9.39	        (11) QDP
32.77/9.39	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
32.77/9.39	        (13) YES
32.77/9.39	    (14) QDP
32.77/9.39	        (15) MRRProof [EQUIVALENT, 28 ms]
32.77/9.39	        (16) QDP
32.77/9.39	        (17) QDPOrderProof [EQUIVALENT, 291 ms]
32.77/9.39	        (18) QDP
32.77/9.39	        (19) PisEmptyProof [EQUIVALENT, 0 ms]
32.77/9.39	        (20) YES
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(0)
32.77/9.39	Obligation:
32.77/9.39	Q restricted rewrite system:
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   C(x1) -> c(x1)
32.77/9.39	   c(c(x1)) -> x1
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	   c(C(x1)) -> x1
32.77/9.39	   C(c(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(1) QTRSRRRProof (EQUIVALENT)
32.77/9.39	Used ordering:
32.77/9.39	Polynomial interpretation [POLO]:
32.77/9.39	
32.77/9.39	   POL(B(x_1)) = x_1
32.77/9.39	   POL(C(x_1)) = 1 + x_1
32.77/9.39	   POL(b(x_1)) = x_1
32.77/9.39	   POL(c(x_1)) = x_1
32.77/9.39	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
32.77/9.39	
32.77/9.39	   C(x1) -> c(x1)
32.77/9.39	   c(C(x1)) -> x1
32.77/9.39	   C(c(x1)) -> x1
32.77/9.39	
32.77/9.39	
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(2)
32.77/9.39	Obligation:
32.77/9.39	Q restricted rewrite system:
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   c(c(x1)) -> x1
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(3) QTRSRRRProof (EQUIVALENT)
32.77/9.39	Used ordering:
32.77/9.39	Polynomial interpretation [POLO]:
32.77/9.39	
32.77/9.39	   POL(B(x_1)) = x_1
32.77/9.39	   POL(b(x_1)) = x_1
32.77/9.39	   POL(c(x_1)) = 1 + x_1
32.77/9.39	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
32.77/9.39	
32.77/9.39	   c(c(x1)) -> x1
32.77/9.39	
32.77/9.39	
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(4)
32.77/9.39	Obligation:
32.77/9.39	Q restricted rewrite system:
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(5) DependencyPairsProof (EQUIVALENT)
32.77/9.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(6)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	The TRS P consists of the following rules:
32.77/9.39	
32.77/9.39	   B^1(b(x1)) -> B^2(x1)
32.77/9.39	   B^2(B(x1)) -> B^1(x1)
32.77/9.39	   C(B(c(b(c(x1))))) -> B^2(c(b(c(B(c(b(x1)))))))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(c(B(c(b(x1))))))
32.77/9.39	   C(B(c(b(c(x1))))) -> B^1(c(B(c(b(x1)))))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(B(c(b(x1))))
32.77/9.39	   C(B(c(b(c(x1))))) -> B^2(c(b(x1)))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(x1))
32.77/9.39	   C(B(c(b(c(x1))))) -> B^1(x1)
32.77/9.39	
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(7) DependencyGraphProof (EQUIVALENT)
32.77/9.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(8)
32.77/9.39	Complex Obligation (AND)
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(9)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	The TRS P consists of the following rules:
32.77/9.39	
32.77/9.39	   B^2(B(x1)) -> B^1(x1)
32.77/9.39	   B^1(b(x1)) -> B^2(x1)
32.77/9.39	
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(10) UsableRulesProof (EQUIVALENT)
32.77/9.39	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.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(11)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	The TRS P consists of the following rules:
32.77/9.39	
32.77/9.39	   B^2(B(x1)) -> B^1(x1)
32.77/9.39	   B^1(b(x1)) -> B^2(x1)
32.77/9.39	
32.77/9.39	R is empty.
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(12) QDPSizeChangeProof (EQUIVALENT)
32.77/9.39	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. 
32.77/9.39	
32.77/9.39	From the DPs we obtained the following set of size-change graphs:
32.77/9.39	*B^1(b(x1)) -> B^2(x1)
32.77/9.39	The graph contains the following edges 1 > 1
32.77/9.39	
32.77/9.39	
32.77/9.39	*B^2(B(x1)) -> B^1(x1)
32.77/9.39	The graph contains the following edges 1 > 1
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(13)
32.77/9.39	YES
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(14)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	The TRS P consists of the following rules:
32.77/9.39	
32.77/9.39	   C(B(c(b(c(x1))))) -> C(B(c(b(x1))))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(c(B(c(b(x1))))))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(x1))
32.77/9.39	
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(15) MRRProof (EQUIVALENT)
32.77/9.39	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
32.77/9.39	
32.77/9.39	Strictly oriented dependency pairs:
32.77/9.39	
32.77/9.39	   C(B(c(b(c(x1))))) -> C(B(c(b(x1))))
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(x1))
32.77/9.39	
32.77/9.39	
32.77/9.39	Used ordering: Polynomial interpretation [POLO]:
32.77/9.39	
32.77/9.39	   POL(B(x_1)) = x_1
32.77/9.39	   POL(C(x_1)) = x_1
32.77/9.39	   POL(b(x_1)) = x_1
32.77/9.39	   POL(c(x_1)) = 1 + x_1
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(16)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	The TRS P consists of the following rules:
32.77/9.39	
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(c(B(c(b(x1))))))
32.77/9.39	
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(17) QDPOrderProof (EQUIVALENT)
32.77/9.39	We use the reduction pair processor [LPAR04,JAR06].
32.77/9.39	
32.77/9.39	
32.77/9.39	The following pairs can be oriented strictly and are deleted.
32.77/9.39	
32.77/9.39	   C(B(c(b(c(x1))))) -> C(b(c(B(c(b(x1))))))
32.77/9.39	The remaining pairs can at least be oriented weakly.
32.77/9.39	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.77/9.39	
32.77/9.39	   <<<
32.77/9.39	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
32.77/9.39	>>>
32.77/9.39	
32.77/9.39	   <<<
32.77/9.39	 POL(B(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
32.77/9.39	>>>
32.77/9.39	
32.77/9.39	   <<<
32.77/9.39	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] 	* 	x_1
32.77/9.39	>>>
32.77/9.39	
32.77/9.39	   <<<
32.77/9.39	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
32.77/9.39	>>>
32.77/9.39	
32.77/9.39	
32.77/9.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.77/9.39	
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(18)
32.77/9.39	Obligation:
32.77/9.39	Q DP problem:
32.77/9.39	P is empty.
32.77/9.39	The TRS R consists of the following rules:
32.77/9.39	
32.77/9.39	   b(b(x1)) -> B(x1)
32.77/9.39	   B(B(x1)) -> b(x1)
32.77/9.39	   c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1)))))))
32.77/9.39	   b(B(x1)) -> x1
32.77/9.39	   B(b(x1)) -> x1
32.77/9.39	
32.77/9.39	Q is empty.
32.77/9.39	We have to consider all minimal (P,Q,R)-chains.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(19) PisEmptyProof (EQUIVALENT)
32.77/9.39	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.77/9.39	----------------------------------------
32.77/9.39	
32.77/9.39	(20)
32.77/9.39	YES
32.96/9.44	EOF