22.99/6.77	YES
23.14/6.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.14/6.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.14/6.81	
23.14/6.81	
23.14/6.81	Termination w.r.t. Q of the given QTRS could be proven:
23.14/6.81	
23.14/6.81	(0) QTRS
23.14/6.81	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.14/6.81	(2) QTRS
23.14/6.81	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
23.14/6.81	(4) QDP
23.14/6.81	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
23.14/6.81	(6) QDP
23.14/6.81	(7) QDPOrderProof [EQUIVALENT, 131 ms]
23.14/6.81	(8) QDP
23.14/6.81	(9) UsableRulesProof [EQUIVALENT, 0 ms]
23.14/6.81	(10) QDP
23.14/6.81	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
23.14/6.81	(12) YES
23.14/6.81	
23.14/6.81	
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(0)
23.14/6.81	Obligation:
23.14/6.81	Q restricted rewrite system:
23.14/6.81	The TRS R consists of the following rules:
23.14/6.81	
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	   b(x1) -> x1
23.14/6.81	   b(a(c(x1))) -> c(c(b(a(a(x1)))))
23.14/6.81	
23.14/6.81	Q is empty.
23.14/6.81	
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(1) QTRS Reverse (EQUIVALENT)
23.14/6.81	We applied the QTRS Reverse Processor [REVERSE].
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(2)
23.14/6.81	Obligation:
23.14/6.81	Q restricted rewrite system:
23.14/6.81	The TRS R consists of the following rules:
23.14/6.81	
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	   b(x1) -> x1
23.14/6.81	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
23.14/6.81	
23.14/6.81	Q is empty.
23.14/6.81	
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(3) DependencyPairsProof (EQUIVALENT)
23.14/6.81	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(4)
23.14/6.81	Obligation:
23.14/6.81	Q DP problem:
23.14/6.81	The TRS P consists of the following rules:
23.14/6.81	
23.14/6.81	   A(x1) -> B(x1)
23.14/6.81	   C(a(b(x1))) -> A(a(b(c(c(x1)))))
23.14/6.81	   C(a(b(x1))) -> A(b(c(c(x1))))
23.14/6.81	   C(a(b(x1))) -> B(c(c(x1)))
23.14/6.81	   C(a(b(x1))) -> C(c(x1))
23.14/6.81	   C(a(b(x1))) -> C(x1)
23.14/6.81	
23.14/6.81	The TRS R consists of the following rules:
23.14/6.81	
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	   b(x1) -> x1
23.14/6.81	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
23.14/6.81	
23.14/6.81	Q is empty.
23.14/6.81	We have to consider all minimal (P,Q,R)-chains.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(5) DependencyGraphProof (EQUIVALENT)
23.14/6.81	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(6)
23.14/6.81	Obligation:
23.14/6.81	Q DP problem:
23.14/6.81	The TRS P consists of the following rules:
23.14/6.81	
23.14/6.81	   C(a(b(x1))) -> C(x1)
23.14/6.81	   C(a(b(x1))) -> C(c(x1))
23.14/6.81	
23.14/6.81	The TRS R consists of the following rules:
23.14/6.81	
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	   b(x1) -> x1
23.14/6.81	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
23.14/6.81	
23.14/6.81	Q is empty.
23.14/6.81	We have to consider all minimal (P,Q,R)-chains.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(7) QDPOrderProof (EQUIVALENT)
23.14/6.81	We use the reduction pair processor [LPAR04,JAR06].
23.14/6.81	
23.14/6.81	
23.14/6.81	The following pairs can be oriented strictly and are deleted.
23.14/6.81	
23.14/6.81	   C(a(b(x1))) -> C(c(x1))
23.14/6.81	The remaining pairs can at least be oriented weakly.
23.14/6.81	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.14/6.81	
23.14/6.81	   <<<
23.14/6.81	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
23.14/6.81	>>>
23.14/6.81	
23.14/6.81	   <<<
23.14/6.81	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, 0A, 1A], [1A, 0A, 0A]] 	* 	x_1
23.14/6.81	>>>
23.14/6.81	
23.14/6.81	   <<<
23.14/6.81	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
23.14/6.81	>>>
23.14/6.81	
23.14/6.81	   <<<
23.14/6.81	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.14/6.81	>>>
23.14/6.81	
23.14/6.81	
23.14/6.81	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.14/6.81	
23.14/6.81	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
23.14/6.81	   b(x1) -> x1
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	
23.14/6.81	
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(8)
23.14/6.81	Obligation:
23.14/6.81	Q DP problem:
23.14/6.81	The TRS P consists of the following rules:
23.14/6.81	
23.14/6.81	   C(a(b(x1))) -> C(x1)
23.14/6.81	
23.14/6.81	The TRS R consists of the following rules:
23.14/6.81	
23.14/6.81	   a(x1) -> x1
23.14/6.81	   a(x1) -> b(x1)
23.14/6.81	   b(x1) -> x1
23.14/6.81	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
23.14/6.81	
23.14/6.81	Q is empty.
23.14/6.81	We have to consider all minimal (P,Q,R)-chains.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(9) UsableRulesProof (EQUIVALENT)
23.14/6.81	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.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(10)
23.14/6.81	Obligation:
23.14/6.81	Q DP problem:
23.14/6.81	The TRS P consists of the following rules:
23.14/6.81	
23.14/6.81	   C(a(b(x1))) -> C(x1)
23.14/6.81	
23.14/6.81	R is empty.
23.14/6.81	Q is empty.
23.14/6.81	We have to consider all minimal (P,Q,R)-chains.
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(11) QDPSizeChangeProof (EQUIVALENT)
23.14/6.81	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. 
23.14/6.81	
23.14/6.81	From the DPs we obtained the following set of size-change graphs:
23.14/6.81	*C(a(b(x1))) -> C(x1)
23.14/6.81	The graph contains the following edges 1 > 1
23.14/6.81	
23.14/6.81	
23.14/6.81	----------------------------------------
23.14/6.81	
23.14/6.81	(12)
23.14/6.81	YES
23.39/6.86	EOF