35.59/10.12	YES
35.75/10.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.75/10.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.75/10.22	
35.75/10.22	
35.75/10.22	Termination w.r.t. Q of the given QTRS could be proven:
35.75/10.22	
35.75/10.22	(0) QTRS
35.75/10.22	(1) DependencyPairsProof [EQUIVALENT, 24 ms]
35.75/10.22	(2) QDP
35.75/10.22	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
35.75/10.22	(4) QDP
35.75/10.22	(5) QDPOrderProof [EQUIVALENT, 106 ms]
35.75/10.22	(6) QDP
35.75/10.22	(7) QDPOrderProof [EQUIVALENT, 26 ms]
35.75/10.22	(8) QDP
35.75/10.22	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
35.75/10.22	(10) QDP
35.75/10.22	(11) UsableRulesProof [EQUIVALENT, 0 ms]
35.75/10.22	(12) QDP
35.75/10.22	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
35.75/10.22	(14) YES
35.75/10.22	
35.75/10.22	
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(0)
35.75/10.22	Obligation:
35.75/10.22	Q restricted rewrite system:
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(1) DependencyPairsProof (EQUIVALENT)
35.75/10.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(2)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   A(x1) -> B(c(b(x1)))
35.75/10.22	   A(x1) -> C(b(x1))
35.75/10.22	   A(x1) -> B(x1)
35.75/10.22	   B(b(c(x1))) -> C(a(b(x1)))
35.75/10.22	   B(b(c(x1))) -> A(b(x1))
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(3) DependencyGraphProof (EQUIVALENT)
35.75/10.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(4)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> A(b(x1))
35.75/10.22	   A(x1) -> B(c(b(x1)))
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	   A(x1) -> B(x1)
35.75/10.22	
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(5) QDPOrderProof (EQUIVALENT)
35.75/10.22	We use the reduction pair processor [LPAR04,JAR06].
35.75/10.22	
35.75/10.22	
35.75/10.22	The following pairs can be oriented strictly and are deleted.
35.75/10.22	
35.75/10.22	   A(x1) -> B(c(b(x1)))
35.75/10.22	The remaining pairs can at least be oriented weakly.
35.75/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	
35.75/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.75/10.22	
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	
35.75/10.22	
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(6)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> A(b(x1))
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	   A(x1) -> B(x1)
35.75/10.22	
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(7) QDPOrderProof (EQUIVALENT)
35.75/10.22	We use the reduction pair processor [LPAR04,JAR06].
35.75/10.22	
35.75/10.22	
35.75/10.22	The following pairs can be oriented strictly and are deleted.
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> A(b(x1))
35.75/10.22	The remaining pairs can at least be oriented weakly.
35.75/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	   <<<
35.75/10.22	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
35.75/10.22	>>>
35.75/10.22	
35.75/10.22	
35.75/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.75/10.22	
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(8)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	   A(x1) -> B(x1)
35.75/10.22	
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(9) DependencyGraphProof (EQUIVALENT)
35.75/10.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(10)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	
35.75/10.22	The TRS R consists of the following rules:
35.75/10.22	
35.75/10.22	   a(x1) -> b(c(b(x1)))
35.75/10.22	   b(b(c(x1))) -> c(a(b(x1)))
35.75/10.22	   c(c(x1)) -> x1
35.75/10.22	
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(11) UsableRulesProof (EQUIVALENT)
35.75/10.22	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.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(12)
35.75/10.22	Obligation:
35.75/10.22	Q DP problem:
35.75/10.22	The TRS P consists of the following rules:
35.75/10.22	
35.75/10.22	   B(b(c(x1))) -> B(x1)
35.75/10.22	
35.75/10.22	R is empty.
35.75/10.22	Q is empty.
35.75/10.22	We have to consider all minimal (P,Q,R)-chains.
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(13) QDPSizeChangeProof (EQUIVALENT)
35.75/10.22	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. 
35.75/10.22	
35.75/10.22	From the DPs we obtained the following set of size-change graphs:
35.75/10.22	*B(b(c(x1))) -> B(x1)
35.75/10.22	The graph contains the following edges 1 > 1
35.75/10.22	
35.75/10.22	
35.75/10.22	----------------------------------------
35.75/10.22	
35.75/10.22	(14)
35.75/10.22	YES
36.19/10.38	EOF