44.67/12.58	YES
44.89/12.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
44.89/12.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
44.89/12.61	
44.89/12.61	
44.89/12.61	Termination w.r.t. Q of the given QTRS could be proven:
44.89/12.61	
44.89/12.61	(0) QTRS
44.89/12.61	(1) QTRS Reverse [EQUIVALENT, 0 ms]
44.89/12.61	(2) QTRS
44.89/12.61	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
44.89/12.61	(4) QDP
44.89/12.61	(5) DependencyGraphProof [EQUIVALENT, 7 ms]
44.89/12.61	(6) QDP
44.89/12.61	(7) QDPOrderProof [EQUIVALENT, 78 ms]
44.89/12.61	(8) QDP
44.89/12.61	(9) QDPOrderProof [EQUIVALENT, 31 ms]
44.89/12.61	(10) QDP
44.89/12.61	(11) UsableRulesProof [EQUIVALENT, 0 ms]
44.89/12.61	(12) QDP
44.89/12.61	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
44.89/12.61	(14) YES
44.89/12.61	
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(0)
44.89/12.61	Obligation:
44.89/12.61	Q restricted rewrite system:
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(b(c(b(x1))))
44.89/12.61	   c(b(b(x1))) -> a(c(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(1) QTRS Reverse (EQUIVALENT)
44.89/12.61	We applied the QTRS Reverse Processor [REVERSE].
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(2)
44.89/12.61	Obligation:
44.89/12.61	Q restricted rewrite system:
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(3) DependencyPairsProof (EQUIVALENT)
44.89/12.61	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(4)
44.89/12.61	Obligation:
44.89/12.61	Q DP problem:
44.89/12.61	The TRS P consists of the following rules:
44.89/12.61	
44.89/12.61	   A(x1) -> B(c(b(b(x1))))
44.89/12.61	   A(x1) -> C(b(b(x1)))
44.89/12.61	   A(x1) -> B(b(x1))
44.89/12.61	   A(x1) -> B(x1)
44.89/12.61	   B(b(c(x1))) -> C(a(x1))
44.89/12.61	   B(b(c(x1))) -> A(x1)
44.89/12.61	
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	We have to consider all minimal (P,Q,R)-chains.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(5) DependencyGraphProof (EQUIVALENT)
44.89/12.61	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(6)
44.89/12.61	Obligation:
44.89/12.61	Q DP problem:
44.89/12.61	The TRS P consists of the following rules:
44.89/12.61	
44.89/12.61	   B(b(c(x1))) -> A(x1)
44.89/12.61	   A(x1) -> B(c(b(b(x1))))
44.89/12.61	   A(x1) -> B(b(x1))
44.89/12.61	   A(x1) -> B(x1)
44.89/12.61	
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	We have to consider all minimal (P,Q,R)-chains.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(7) QDPOrderProof (EQUIVALENT)
44.89/12.61	We use the reduction pair processor [LPAR04,JAR06].
44.89/12.61	
44.89/12.61	
44.89/12.61	The following pairs can be oriented strictly and are deleted.
44.89/12.61	
44.89/12.61	   A(x1) -> B(b(x1))
44.89/12.61	The remaining pairs can at least be oriented weakly.
44.89/12.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [-I, -I, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(A(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	
44.89/12.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.89/12.61	
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(8)
44.89/12.61	Obligation:
44.89/12.61	Q DP problem:
44.89/12.61	The TRS P consists of the following rules:
44.89/12.61	
44.89/12.61	   B(b(c(x1))) -> A(x1)
44.89/12.61	   A(x1) -> B(c(b(b(x1))))
44.89/12.61	   A(x1) -> B(x1)
44.89/12.61	
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	We have to consider all minimal (P,Q,R)-chains.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(9) QDPOrderProof (EQUIVALENT)
44.89/12.61	We use the reduction pair processor [LPAR04,JAR06].
44.89/12.61	
44.89/12.61	
44.89/12.61	The following pairs can be oriented strictly and are deleted.
44.89/12.61	
44.89/12.61	   A(x1) -> B(c(b(b(x1))))
44.89/12.61	The remaining pairs can at least be oriented weakly.
44.89/12.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [0A, 0A, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 1A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	   <<<
44.89/12.61	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
44.89/12.61	>>>
44.89/12.61	
44.89/12.61	
44.89/12.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.89/12.61	
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(10)
44.89/12.61	Obligation:
44.89/12.61	Q DP problem:
44.89/12.61	The TRS P consists of the following rules:
44.89/12.61	
44.89/12.61	   B(b(c(x1))) -> A(x1)
44.89/12.61	   A(x1) -> B(x1)
44.89/12.61	
44.89/12.61	The TRS R consists of the following rules:
44.89/12.61	
44.89/12.61	   a(x1) -> b(c(b(b(x1))))
44.89/12.61	   b(b(c(x1))) -> c(a(x1))
44.89/12.61	   c(c(x1)) -> x1
44.89/12.61	
44.89/12.61	Q is empty.
44.89/12.61	We have to consider all minimal (P,Q,R)-chains.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(11) UsableRulesProof (EQUIVALENT)
44.89/12.61	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.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(12)
44.89/12.61	Obligation:
44.89/12.61	Q DP problem:
44.89/12.61	The TRS P consists of the following rules:
44.89/12.61	
44.89/12.61	   B(b(c(x1))) -> A(x1)
44.89/12.61	   A(x1) -> B(x1)
44.89/12.61	
44.89/12.61	R is empty.
44.89/12.61	Q is empty.
44.89/12.61	We have to consider all minimal (P,Q,R)-chains.
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(13) QDPSizeChangeProof (EQUIVALENT)
44.89/12.61	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. 
44.89/12.61	
44.89/12.61	From the DPs we obtained the following set of size-change graphs:
44.89/12.61	*A(x1) -> B(x1)
44.89/12.61	The graph contains the following edges 1 >= 1
44.89/12.61	
44.89/12.61	
44.89/12.61	*B(b(c(x1))) -> A(x1)
44.89/12.61	The graph contains the following edges 1 > 1
44.89/12.61	
44.89/12.61	
44.89/12.61	----------------------------------------
44.89/12.61	
44.89/12.61	(14)
44.89/12.61	YES
45.23/12.71	EOF