87.94/23.16	YES
88.34/23.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
88.34/23.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
88.34/23.22	
88.34/23.22	
88.34/23.22	Termination w.r.t. Q of the given QTRS could be proven:
88.34/23.22	
88.34/23.22	(0) QTRS
88.34/23.22	(1) QTRS Reverse [EQUIVALENT, 0 ms]
88.34/23.22	(2) QTRS
88.34/23.22	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
88.34/23.22	(4) QDP
88.34/23.22	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
88.34/23.22	(6) QDP
88.34/23.22	(7) QDPOrderProof [EQUIVALENT, 2020 ms]
88.34/23.22	(8) QDP
88.34/23.22	(9) QDPOrderProof [EQUIVALENT, 873 ms]
88.34/23.22	(10) QDP
88.34/23.22	(11) UsableRulesProof [EQUIVALENT, 0 ms]
88.34/23.22	(12) QDP
88.34/23.22	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
88.34/23.22	(14) YES
88.34/23.22	
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(0)
88.34/23.22	Obligation:
88.34/23.22	Q restricted rewrite system:
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   a(a(c(x1))) -> b(c(b(a(x1))))
88.34/23.22	   b(x1) -> d(a(x1))
88.34/23.22	   b(a(c(d(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> c(d(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(1) QTRS Reverse (EQUIVALENT)
88.34/23.22	We applied the QTRS Reverse Processor [REVERSE].
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(2)
88.34/23.22	Obligation:
88.34/23.22	Q restricted rewrite system:
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(3) DependencyPairsProof (EQUIVALENT)
88.34/23.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(4)
88.34/23.22	Obligation:
88.34/23.22	Q DP problem:
88.34/23.22	The TRS P consists of the following rules:
88.34/23.22	
88.34/23.22	   C(a(a(x1))) -> B(c(b(x1)))
88.34/23.22	   C(a(a(x1))) -> C(b(x1))
88.34/23.22	   C(a(a(x1))) -> B(x1)
88.34/23.22	   B(x1) -> D(x1)
88.34/23.22	   B(x1) -> D(c(x1))
88.34/23.22	   B(x1) -> C(x1)
88.34/23.22	
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	We have to consider all minimal (P,Q,R)-chains.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(5) DependencyGraphProof (EQUIVALENT)
88.34/23.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(6)
88.34/23.22	Obligation:
88.34/23.22	Q DP problem:
88.34/23.22	The TRS P consists of the following rules:
88.34/23.22	
88.34/23.22	   B(x1) -> C(x1)
88.34/23.22	   C(a(a(x1))) -> B(c(b(x1)))
88.34/23.22	   C(a(a(x1))) -> C(b(x1))
88.34/23.22	   C(a(a(x1))) -> B(x1)
88.34/23.22	
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	We have to consider all minimal (P,Q,R)-chains.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(7) QDPOrderProof (EQUIVALENT)
88.34/23.22	We use the reduction pair processor [LPAR04,JAR06].
88.34/23.22	
88.34/23.22	
88.34/23.22	The following pairs can be oriented strictly and are deleted.
88.34/23.22	
88.34/23.22	   C(a(a(x1))) -> C(b(x1))
88.34/23.22	The remaining pairs can at least be oriented weakly.
88.34/23.22	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -I]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-1A, -1A, 0A], [1A, 0A, -1A], [-1A, -1A, -I]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, -1A], [1A, 0A, -1A], [0A, -1A, 0A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(b(x_1)) =  	[[1A], [1A], [2A]] 	 +  	[[-1A, -1A, -1A], [0A, -1A, 0A], [1A, 0A, 1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(d(x_1)) =  	[[0A], [-I], [-1A]] 	 +  	[[-I, -I, -1A], [-I, -1A, -1A], [-1A, -I, -I]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	
88.34/23.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
88.34/23.22	
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(8)
88.34/23.22	Obligation:
88.34/23.22	Q DP problem:
88.34/23.22	The TRS P consists of the following rules:
88.34/23.22	
88.34/23.22	   B(x1) -> C(x1)
88.34/23.22	   C(a(a(x1))) -> B(c(b(x1)))
88.34/23.22	   C(a(a(x1))) -> B(x1)
88.34/23.22	
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	We have to consider all minimal (P,Q,R)-chains.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(9) QDPOrderProof (EQUIVALENT)
88.34/23.22	We use the reduction pair processor [LPAR04,JAR06].
88.34/23.22	
88.34/23.22	
88.34/23.22	The following pairs can be oriented strictly and are deleted.
88.34/23.22	
88.34/23.22	   C(a(a(x1))) -> B(c(b(x1)))
88.34/23.22	The remaining pairs can at least be oriented weakly.
88.34/23.22	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 1A, -1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 1A, -1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(c(x_1)) =  	[[-I], [-I], [-1A]] 	 +  	[[0A, -I, 0A], [-I, 0A, -I], [-I, 2A, 0A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(b(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[1A, 1A, 1A], [-1A, -I, -1A], [-1A, 1A, -1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	   <<<
88.34/23.22	 POL(d(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-1A, -I, -1A], [-1A, -I, -I], [-I, -I, -1A]] 	* 	x_1
88.34/23.22	>>>
88.34/23.22	
88.34/23.22	
88.34/23.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
88.34/23.22	
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(10)
88.34/23.22	Obligation:
88.34/23.22	Q DP problem:
88.34/23.22	The TRS P consists of the following rules:
88.34/23.22	
88.34/23.22	   B(x1) -> C(x1)
88.34/23.22	   C(a(a(x1))) -> B(x1)
88.34/23.22	
88.34/23.22	The TRS R consists of the following rules:
88.34/23.22	
88.34/23.22	   c(a(a(x1))) -> a(b(c(b(x1))))
88.34/23.22	   b(x1) -> a(d(x1))
88.34/23.22	   d(c(a(b(x1)))) -> a(a(a(x1)))
88.34/23.22	   c(x1) -> x1
88.34/23.22	   b(x1) -> d(c(x1))
88.34/23.22	
88.34/23.22	Q is empty.
88.34/23.22	We have to consider all minimal (P,Q,R)-chains.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(11) UsableRulesProof (EQUIVALENT)
88.34/23.22	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.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(12)
88.34/23.22	Obligation:
88.34/23.22	Q DP problem:
88.34/23.22	The TRS P consists of the following rules:
88.34/23.22	
88.34/23.22	   B(x1) -> C(x1)
88.34/23.22	   C(a(a(x1))) -> B(x1)
88.34/23.22	
88.34/23.22	R is empty.
88.34/23.22	Q is empty.
88.34/23.22	We have to consider all minimal (P,Q,R)-chains.
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(13) QDPSizeChangeProof (EQUIVALENT)
88.34/23.22	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. 
88.34/23.22	
88.34/23.22	From the DPs we obtained the following set of size-change graphs:
88.34/23.22	*C(a(a(x1))) -> B(x1)
88.34/23.22	The graph contains the following edges 1 > 1
88.34/23.22	
88.34/23.22	
88.34/23.22	*B(x1) -> C(x1)
88.34/23.22	The graph contains the following edges 1 >= 1
88.34/23.22	
88.34/23.22	
88.34/23.22	----------------------------------------
88.34/23.22	
88.34/23.22	(14)
88.34/23.22	YES
88.56/23.37	EOF