18.54/5.76	YES
19.76/6.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.76/6.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.76/6.05	
19.76/6.05	
19.76/6.05	Termination w.r.t. Q of the given QTRS could be proven:
19.76/6.05	
19.76/6.05	(0) QTRS
19.76/6.05	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
19.76/6.05	(2) QDP
19.76/6.05	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
19.76/6.05	(4) QDP
19.76/6.05	(5) QDPOrderProof [EQUIVALENT, 140 ms]
19.76/6.05	(6) QDP
19.76/6.05	(7) QDPOrderProof [EQUIVALENT, 79 ms]
19.76/6.05	(8) QDP
19.76/6.05	(9) UsableRulesProof [EQUIVALENT, 0 ms]
19.76/6.05	(10) QDP
19.76/6.05	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
19.76/6.05	(12) YES
19.76/6.05	
19.76/6.05	
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(0)
19.76/6.05	Obligation:
19.76/6.05	Q restricted rewrite system:
19.76/6.05	The TRS R consists of the following rules:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	Q is empty.
19.76/6.05	
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(1) DependencyPairsProof (EQUIVALENT)
19.76/6.05	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(2)
19.76/6.05	Obligation:
19.76/6.05	Q DP problem:
19.76/6.05	The TRS P consists of the following rules:
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> C(a(c(a(a(x1)))))
19.76/6.05	   A(b(c(x1))) -> A(c(a(a(x1))))
19.76/6.05	   A(b(c(x1))) -> C(a(a(x1)))
19.76/6.05	   A(b(c(x1))) -> A(a(x1))
19.76/6.05	   A(b(c(x1))) -> A(x1)
19.76/6.05	
19.76/6.05	The TRS R consists of the following rules:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	Q is empty.
19.76/6.05	We have to consider all minimal (P,Q,R)-chains.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(3) DependencyGraphProof (EQUIVALENT)
19.76/6.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(4)
19.76/6.05	Obligation:
19.76/6.05	Q DP problem:
19.76/6.05	The TRS P consists of the following rules:
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(a(x1))
19.76/6.05	   A(b(c(x1))) -> A(c(a(a(x1))))
19.76/6.05	   A(b(c(x1))) -> A(x1)
19.76/6.05	
19.76/6.05	The TRS R consists of the following rules:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	Q is empty.
19.76/6.05	We have to consider all minimal (P,Q,R)-chains.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(5) QDPOrderProof (EQUIVALENT)
19.76/6.05	We use the reduction pair processor [LPAR04,JAR06].
19.76/6.05	
19.76/6.05	
19.76/6.05	The following pairs can be oriented strictly and are deleted.
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(c(a(a(x1))))
19.76/6.05	The remaining pairs can at least be oriented weakly.
19.76/6.05	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	
19.76/6.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(6)
19.76/6.05	Obligation:
19.76/6.05	Q DP problem:
19.76/6.05	The TRS P consists of the following rules:
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(a(x1))
19.76/6.05	   A(b(c(x1))) -> A(x1)
19.76/6.05	
19.76/6.05	The TRS R consists of the following rules:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	Q is empty.
19.76/6.05	We have to consider all minimal (P,Q,R)-chains.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(7) QDPOrderProof (EQUIVALENT)
19.76/6.05	We use the reduction pair processor [LPAR04,JAR06].
19.76/6.05	
19.76/6.05	
19.76/6.05	The following pairs can be oriented strictly and are deleted.
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(a(x1))
19.76/6.05	The remaining pairs can at least be oriented weakly.
19.76/6.05	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	   <<<
19.76/6.05	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
19.76/6.05	>>>
19.76/6.05	
19.76/6.05	
19.76/6.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(8)
19.76/6.05	Obligation:
19.76/6.05	Q DP problem:
19.76/6.05	The TRS P consists of the following rules:
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(x1)
19.76/6.05	
19.76/6.05	The TRS R consists of the following rules:
19.76/6.05	
19.76/6.05	   a(x1) -> b(x1)
19.76/6.05	   a(b(c(x1))) -> c(a(c(a(a(x1)))))
19.76/6.05	   c(x1) -> b(x1)
19.76/6.05	
19.76/6.05	Q is empty.
19.76/6.05	We have to consider all minimal (P,Q,R)-chains.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(9) UsableRulesProof (EQUIVALENT)
19.76/6.05	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.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(10)
19.76/6.05	Obligation:
19.76/6.05	Q DP problem:
19.76/6.05	The TRS P consists of the following rules:
19.76/6.05	
19.76/6.05	   A(b(c(x1))) -> A(x1)
19.76/6.05	
19.76/6.05	R is empty.
19.76/6.05	Q is empty.
19.76/6.05	We have to consider all minimal (P,Q,R)-chains.
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(11) QDPSizeChangeProof (EQUIVALENT)
19.76/6.05	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. 
19.76/6.05	
19.76/6.05	From the DPs we obtained the following set of size-change graphs:
19.76/6.05	*A(b(c(x1))) -> A(x1)
19.76/6.05	The graph contains the following edges 1 > 1
19.76/6.05	
19.76/6.05	
19.76/6.05	----------------------------------------
19.76/6.05	
19.76/6.05	(12)
19.76/6.05	YES
20.14/6.16	EOF