27.44/7.87	YES
27.44/7.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.44/7.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.44/7.91	
27.44/7.91	
27.44/7.91	Termination w.r.t. Q of the given QTRS could be proven:
27.44/7.91	
27.44/7.91	(0) QTRS
27.44/7.91	(1) QTRS Reverse [EQUIVALENT, 0 ms]
27.44/7.91	(2) QTRS
27.44/7.91	(3) DependencyPairsProof [EQUIVALENT, 35 ms]
27.44/7.91	(4) QDP
27.44/7.91	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
27.44/7.91	(6) QDP
27.44/7.91	(7) QDPOrderProof [EQUIVALENT, 128 ms]
27.44/7.91	(8) QDP
27.44/7.91	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
27.44/7.91	(10) QDP
27.44/7.91	(11) UsableRulesProof [EQUIVALENT, 0 ms]
27.44/7.91	(12) QDP
27.44/7.91	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
27.44/7.91	(14) YES
27.44/7.91	
27.44/7.91	
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(0)
27.44/7.91	Obligation:
27.44/7.91	Q restricted rewrite system:
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   a(b(x1)) -> b(a(c(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> b(b(a(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(1) QTRS Reverse (EQUIVALENT)
27.44/7.91	We applied the QTRS Reverse Processor [REVERSE].
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(2)
27.44/7.91	Obligation:
27.44/7.91	Q restricted rewrite system:
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(3) DependencyPairsProof (EQUIVALENT)
27.44/7.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(4)
27.44/7.91	Obligation:
27.44/7.91	Q DP problem:
27.44/7.91	The TRS P consists of the following rules:
27.44/7.91	
27.44/7.91	   B(a(x1)) -> C(a(b(x1)))
27.44/7.91	   B(a(x1)) -> A(b(x1))
27.44/7.91	   B(a(x1)) -> B(x1)
27.44/7.91	   C(c(x1)) -> A(b(b(x1)))
27.44/7.91	   C(c(x1)) -> B(b(x1))
27.44/7.91	   C(c(x1)) -> B(x1)
27.44/7.91	
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	We have to consider all minimal (P,Q,R)-chains.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(5) DependencyGraphProof (EQUIVALENT)
27.44/7.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(6)
27.44/7.91	Obligation:
27.44/7.91	Q DP problem:
27.44/7.91	The TRS P consists of the following rules:
27.44/7.91	
27.44/7.91	   C(c(x1)) -> B(b(x1))
27.44/7.91	   B(a(x1)) -> C(a(b(x1)))
27.44/7.91	   C(c(x1)) -> B(x1)
27.44/7.91	   B(a(x1)) -> B(x1)
27.44/7.91	
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	We have to consider all minimal (P,Q,R)-chains.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(7) QDPOrderProof (EQUIVALENT)
27.44/7.91	We use the reduction pair processor [LPAR04,JAR06].
27.44/7.91	
27.44/7.91	
27.44/7.91	The following pairs can be oriented strictly and are deleted.
27.44/7.91	
27.44/7.91	   B(a(x1)) -> C(a(b(x1)))
27.44/7.91	The remaining pairs can at least be oriented weakly.
27.44/7.91	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.44/7.91	
27.44/7.91	   <<<
27.44/7.91	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
27.44/7.91	>>>
27.44/7.91	
27.44/7.91	   <<<
27.44/7.91	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
27.44/7.91	>>>
27.44/7.91	
27.44/7.91	   <<<
27.44/7.91	 POL(B(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
27.44/7.91	>>>
27.44/7.91	
27.44/7.91	   <<<
27.44/7.91	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
27.44/7.91	>>>
27.44/7.91	
27.44/7.91	   <<<
27.44/7.91	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
27.44/7.91	>>>
27.44/7.91	
27.44/7.91	
27.44/7.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.44/7.91	
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   a(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(8)
27.44/7.91	Obligation:
27.44/7.91	Q DP problem:
27.44/7.91	The TRS P consists of the following rules:
27.44/7.91	
27.44/7.91	   C(c(x1)) -> B(b(x1))
27.44/7.91	   C(c(x1)) -> B(x1)
27.44/7.91	   B(a(x1)) -> B(x1)
27.44/7.91	
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	We have to consider all minimal (P,Q,R)-chains.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(9) DependencyGraphProof (EQUIVALENT)
27.44/7.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(10)
27.44/7.91	Obligation:
27.44/7.91	Q DP problem:
27.44/7.91	The TRS P consists of the following rules:
27.44/7.91	
27.44/7.91	   B(a(x1)) -> B(x1)
27.44/7.91	
27.44/7.91	The TRS R consists of the following rules:
27.44/7.91	
27.44/7.91	   a(x1) -> x1
27.44/7.91	   b(a(x1)) -> c(a(b(x1)))
27.44/7.91	   b(x1) -> x1
27.44/7.91	   c(c(x1)) -> a(b(b(x1)))
27.44/7.91	
27.44/7.91	Q is empty.
27.44/7.91	We have to consider all minimal (P,Q,R)-chains.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(11) UsableRulesProof (EQUIVALENT)
27.44/7.91	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.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(12)
27.44/7.91	Obligation:
27.44/7.91	Q DP problem:
27.44/7.91	The TRS P consists of the following rules:
27.44/7.91	
27.44/7.91	   B(a(x1)) -> B(x1)
27.44/7.91	
27.44/7.91	R is empty.
27.44/7.91	Q is empty.
27.44/7.91	We have to consider all minimal (P,Q,R)-chains.
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(13) QDPSizeChangeProof (EQUIVALENT)
27.44/7.91	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. 
27.44/7.91	
27.44/7.91	From the DPs we obtained the following set of size-change graphs:
27.44/7.91	*B(a(x1)) -> B(x1)
27.44/7.91	The graph contains the following edges 1 > 1
27.44/7.91	
27.44/7.91	
27.44/7.91	----------------------------------------
27.44/7.91	
27.44/7.91	(14)
27.44/7.91	YES
27.81/7.99	EOF