12.89/4.25	YES
12.89/4.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.89/4.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.89/4.28	
12.89/4.28	
12.89/4.28	Termination w.r.t. Q of the given QTRS could be proven:
12.89/4.28	
12.89/4.28	(0) QTRS
12.89/4.28	(1) QTRS Reverse [EQUIVALENT, 0 ms]
12.89/4.28	(2) QTRS
12.89/4.28	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
12.89/4.28	(4) QDP
12.89/4.28	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
12.89/4.28	(6) QDP
12.89/4.28	(7) QDPOrderProof [EQUIVALENT, 114 ms]
12.89/4.28	(8) QDP
12.89/4.28	(9) UsableRulesProof [EQUIVALENT, 0 ms]
12.89/4.28	(10) QDP
12.89/4.28	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.89/4.28	(12) YES
12.89/4.28	
12.89/4.28	
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(0)
12.89/4.28	Obligation:
12.89/4.28	Q restricted rewrite system:
12.89/4.28	The TRS R consists of the following rules:
12.89/4.28	
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	   b(c(c(x1))) -> c(c(a(c(b(x1)))))
12.89/4.28	
12.89/4.28	Q is empty.
12.89/4.28	
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(1) QTRS Reverse (EQUIVALENT)
12.89/4.28	We applied the QTRS Reverse Processor [REVERSE].
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(2)
12.89/4.28	Obligation:
12.89/4.28	Q restricted rewrite system:
12.89/4.28	The TRS R consists of the following rules:
12.89/4.28	
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
12.89/4.28	
12.89/4.28	Q is empty.
12.89/4.28	
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(3) DependencyPairsProof (EQUIVALENT)
12.89/4.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(4)
12.89/4.28	Obligation:
12.89/4.28	Q DP problem:
12.89/4.28	The TRS P consists of the following rules:
12.89/4.28	
12.89/4.28	   A(x1) -> B(x1)
12.89/4.28	   C(c(b(x1))) -> B(c(a(c(c(x1)))))
12.89/4.28	   C(c(b(x1))) -> C(a(c(c(x1))))
12.89/4.28	   C(c(b(x1))) -> A(c(c(x1)))
12.89/4.28	   C(c(b(x1))) -> C(c(x1))
12.89/4.28	   C(c(b(x1))) -> C(x1)
12.89/4.28	
12.89/4.28	The TRS R consists of the following rules:
12.89/4.28	
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
12.89/4.28	
12.89/4.28	Q is empty.
12.89/4.28	We have to consider all minimal (P,Q,R)-chains.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(5) DependencyGraphProof (EQUIVALENT)
12.89/4.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(6)
12.89/4.28	Obligation:
12.89/4.28	Q DP problem:
12.89/4.28	The TRS P consists of the following rules:
12.89/4.28	
12.89/4.28	   C(c(b(x1))) -> C(c(x1))
12.89/4.28	   C(c(b(x1))) -> C(a(c(c(x1))))
12.89/4.28	   C(c(b(x1))) -> C(x1)
12.89/4.28	
12.89/4.28	The TRS R consists of the following rules:
12.89/4.28	
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
12.89/4.28	
12.89/4.28	Q is empty.
12.89/4.28	We have to consider all minimal (P,Q,R)-chains.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(7) QDPOrderProof (EQUIVALENT)
12.89/4.28	We use the reduction pair processor [LPAR04,JAR06].
12.89/4.28	
12.89/4.28	
12.89/4.28	The following pairs can be oriented strictly and are deleted.
12.89/4.28	
12.89/4.28	   C(c(b(x1))) -> C(c(x1))
12.89/4.28	   C(c(b(x1))) -> C(a(c(c(x1))))
12.89/4.28	The remaining pairs can at least be oriented weakly.
12.89/4.28	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.89/4.28	
12.89/4.28	   <<<
12.89/4.28	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
12.89/4.28	>>>
12.89/4.28	
12.89/4.28	   <<<
12.89/4.28	 POL(c(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
12.89/4.28	>>>
12.89/4.28	
12.89/4.28	   <<<
12.89/4.28	 POL(b(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, -I, -I], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
12.89/4.28	>>>
12.89/4.28	
12.89/4.28	   <<<
12.89/4.28	 POL(a(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, -I, -I], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
12.89/4.28	>>>
12.89/4.28	
12.89/4.28	
12.89/4.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.89/4.28	
12.89/4.28	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	
12.89/4.28	
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(8)
12.89/4.28	Obligation:
12.89/4.28	Q DP problem:
12.89/4.28	The TRS P consists of the following rules:
12.89/4.28	
12.89/4.28	   C(c(b(x1))) -> C(x1)
12.89/4.28	
12.89/4.28	The TRS R consists of the following rules:
12.89/4.28	
12.89/4.28	   a(x1) -> b(x1)
12.89/4.28	   b(b(x1)) -> x1
12.89/4.28	   c(c(b(x1))) -> b(c(a(c(c(x1)))))
12.89/4.28	
12.89/4.28	Q is empty.
12.89/4.28	We have to consider all minimal (P,Q,R)-chains.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(9) UsableRulesProof (EQUIVALENT)
12.89/4.28	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.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(10)
12.89/4.28	Obligation:
12.89/4.28	Q DP problem:
12.89/4.28	The TRS P consists of the following rules:
12.89/4.28	
12.89/4.28	   C(c(b(x1))) -> C(x1)
12.89/4.28	
12.89/4.28	R is empty.
12.89/4.28	Q is empty.
12.89/4.28	We have to consider all minimal (P,Q,R)-chains.
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(11) QDPSizeChangeProof (EQUIVALENT)
12.89/4.28	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. 
12.89/4.28	
12.89/4.28	From the DPs we obtained the following set of size-change graphs:
12.89/4.28	*C(c(b(x1))) -> C(x1)
12.89/4.28	The graph contains the following edges 1 > 1
12.89/4.28	
12.89/4.28	
12.89/4.28	----------------------------------------
12.89/4.28	
12.89/4.28	(12)
12.89/4.28	YES
13.57/4.41	EOF