34.12/9.53	YES
34.12/9.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.12/9.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.12/9.57	
34.12/9.57	
34.12/9.57	Termination w.r.t. Q of the given QTRS could be proven:
34.12/9.57	
34.12/9.57	(0) QTRS
34.12/9.57	(1) QTRS Reverse [EQUIVALENT, 0 ms]
34.12/9.57	(2) QTRS
34.12/9.57	(3) DependencyPairsProof [EQUIVALENT, 10 ms]
34.12/9.57	(4) QDP
34.12/9.57	(5) DependencyGraphProof [EQUIVALENT, 8 ms]
34.12/9.57	(6) QDP
34.12/9.57	(7) QDPOrderProof [EQUIVALENT, 105 ms]
34.12/9.57	(8) QDP
34.12/9.57	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
34.12/9.57	(10) QDP
34.12/9.57	(11) QDPOrderProof [EQUIVALENT, 24 ms]
34.12/9.57	(12) QDP
34.12/9.57	(13) UsableRulesProof [EQUIVALENT, 0 ms]
34.12/9.57	(14) QDP
34.12/9.57	(15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
34.12/9.57	(16) YES
34.12/9.57	
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(0)
34.12/9.57	Obligation:
34.12/9.57	Q restricted rewrite system:
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> b(c(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   b(c(c(x1))) -> c(c(a(b(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(1) QTRS Reverse (EQUIVALENT)
34.12/9.57	We applied the QTRS Reverse Processor [REVERSE].
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(2)
34.12/9.57	Obligation:
34.12/9.57	Q restricted rewrite system:
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(3) DependencyPairsProof (EQUIVALENT)
34.12/9.57	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(4)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   A(x1) -> C(b(x1))
34.12/9.57	   A(x1) -> B(x1)
34.12/9.57	   C(c(b(x1))) -> B(a(c(c(x1))))
34.12/9.57	   C(c(b(x1))) -> A(c(c(x1)))
34.12/9.57	   C(c(b(x1))) -> C(c(x1))
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(5) DependencyGraphProof (EQUIVALENT)
34.12/9.57	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(6)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> A(c(c(x1)))
34.12/9.57	   A(x1) -> C(b(x1))
34.12/9.57	   C(c(b(x1))) -> C(c(x1))
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(7) QDPOrderProof (EQUIVALENT)
34.12/9.57	We use the reduction pair processor [LPAR04,JAR06].
34.12/9.57	
34.12/9.57	
34.12/9.57	The following pairs can be oriented strictly and are deleted.
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> A(c(c(x1)))
34.12/9.57	The remaining pairs can at least be oriented weakly.
34.12/9.57	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(a(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	
34.12/9.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.12/9.57	
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(8)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   A(x1) -> C(b(x1))
34.12/9.57	   C(c(b(x1))) -> C(c(x1))
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(9) DependencyGraphProof (EQUIVALENT)
34.12/9.57	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(10)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	   C(c(b(x1))) -> C(c(x1))
34.12/9.57	
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(11) QDPOrderProof (EQUIVALENT)
34.12/9.57	We use the reduction pair processor [LPAR04,JAR06].
34.12/9.57	
34.12/9.57	
34.12/9.57	The following pairs can be oriented strictly and are deleted.
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> C(c(x1))
34.12/9.57	The remaining pairs can at least be oriented weakly.
34.12/9.57	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(b(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [1A, 0A, 1A]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	   <<<
34.12/9.57	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
34.12/9.57	>>>
34.12/9.57	
34.12/9.57	
34.12/9.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.12/9.57	
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(12)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	
34.12/9.57	The TRS R consists of the following rules:
34.12/9.57	
34.12/9.57	   a(x1) -> c(b(x1))
34.12/9.57	   b(b(x1)) -> x1
34.12/9.57	   c(c(b(x1))) -> b(a(c(c(x1))))
34.12/9.57	
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(13) UsableRulesProof (EQUIVALENT)
34.12/9.57	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.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(14)
34.12/9.57	Obligation:
34.12/9.57	Q DP problem:
34.12/9.57	The TRS P consists of the following rules:
34.12/9.57	
34.12/9.57	   C(c(b(x1))) -> C(x1)
34.12/9.57	
34.12/9.57	R is empty.
34.12/9.57	Q is empty.
34.12/9.57	We have to consider all minimal (P,Q,R)-chains.
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(15) QDPSizeChangeProof (EQUIVALENT)
34.12/9.57	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. 
34.12/9.57	
34.12/9.57	From the DPs we obtained the following set of size-change graphs:
34.12/9.57	*C(c(b(x1))) -> C(x1)
34.12/9.57	The graph contains the following edges 1 > 1
34.12/9.57	
34.12/9.57	
34.12/9.57	----------------------------------------
34.12/9.57	
34.12/9.57	(16)
34.12/9.57	YES
34.65/9.69	EOF