21.01/6.26	YES
21.01/6.30	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.01/6.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.01/6.30	
21.01/6.30	
21.01/6.30	Termination w.r.t. Q of the given QTRS could be proven:
21.01/6.30	
21.01/6.30	(0) QTRS
21.01/6.30	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.01/6.30	(2) QTRS
21.01/6.30	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
21.01/6.30	(4) QDP
21.01/6.30	(5) QDPOrderProof [EQUIVALENT, 65 ms]
21.01/6.30	(6) QDP
21.01/6.30	(7) QDPOrderProof [EQUIVALENT, 84 ms]
21.01/6.30	(8) QDP
21.01/6.30	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
21.01/6.30	(10) TRUE
21.01/6.30	
21.01/6.30	
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(0)
21.01/6.30	Obligation:
21.01/6.30	Q restricted rewrite system:
21.01/6.30	The TRS R consists of the following rules:
21.01/6.30	
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(b(c(a(x1))))
21.01/6.30	
21.01/6.30	Q is empty.
21.01/6.30	
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(1) QTRS Reverse (EQUIVALENT)
21.01/6.30	We applied the QTRS Reverse Processor [REVERSE].
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(2)
21.01/6.30	Obligation:
21.01/6.30	Q restricted rewrite system:
21.01/6.30	The TRS R consists of the following rules:
21.01/6.30	
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	
21.01/6.30	Q is empty.
21.01/6.30	
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(3) DependencyPairsProof (EQUIVALENT)
21.01/6.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(4)
21.01/6.30	Obligation:
21.01/6.30	Q DP problem:
21.01/6.30	The TRS P consists of the following rules:
21.01/6.30	
21.01/6.30	   A(x1) -> B(x1)
21.01/6.30	   B(b(x1)) -> C(x1)
21.01/6.30	   C(a(c(x1))) -> A(c(b(a(x1))))
21.01/6.30	   C(a(c(x1))) -> C(b(a(x1)))
21.01/6.30	   C(a(c(x1))) -> B(a(x1))
21.01/6.30	   C(a(c(x1))) -> A(x1)
21.01/6.30	
21.01/6.30	The TRS R consists of the following rules:
21.01/6.30	
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	
21.01/6.30	Q is empty.
21.01/6.30	We have to consider all minimal (P,Q,R)-chains.
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(5) QDPOrderProof (EQUIVALENT)
21.01/6.30	We use the reduction pair processor [LPAR04,JAR06].
21.01/6.30	
21.01/6.30	
21.01/6.30	The following pairs can be oriented strictly and are deleted.
21.01/6.30	
21.01/6.30	   C(a(c(x1))) -> C(b(a(x1)))
21.01/6.30	   C(a(c(x1))) -> B(a(x1))
21.01/6.30	   C(a(c(x1))) -> A(x1)
21.01/6.30	The remaining pairs can at least be oriented weakly.
21.01/6.30	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
21.01/6.30	
21.01/6.30	POL( A_1(x_1) ) = 2x_1
21.01/6.30	POL( B_1(x_1) ) = 2x_1
21.01/6.30	POL( C_1(x_1) ) = 2x_1 + 2
21.01/6.30	POL( b_1(x_1) ) = x_1 + 1
21.01/6.30	POL( c_1(x_1) ) = x_1 + 2
21.01/6.30	POL( a_1(x_1) ) = x_1 + 1
21.01/6.30	
21.01/6.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.01/6.30	
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	
21.01/6.30	
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(6)
21.01/6.30	Obligation:
21.01/6.30	Q DP problem:
21.01/6.30	The TRS P consists of the following rules:
21.01/6.30	
21.01/6.30	   A(x1) -> B(x1)
21.01/6.30	   B(b(x1)) -> C(x1)
21.01/6.30	   C(a(c(x1))) -> A(c(b(a(x1))))
21.01/6.30	
21.01/6.30	The TRS R consists of the following rules:
21.01/6.30	
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	
21.01/6.30	Q is empty.
21.01/6.30	We have to consider all minimal (P,Q,R)-chains.
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(7) QDPOrderProof (EQUIVALENT)
21.01/6.30	We use the reduction pair processor [LPAR04,JAR06].
21.01/6.30	
21.01/6.30	
21.01/6.30	The following pairs can be oriented strictly and are deleted.
21.01/6.30	
21.01/6.30	   B(b(x1)) -> C(x1)
21.01/6.30	The remaining pairs can at least be oriented weakly.
21.01/6.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	   <<<
21.01/6.30	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [0A, 1A, 0A], [-I, 1A, -I]] 	* 	x_1
21.01/6.30	>>>
21.01/6.30	
21.01/6.30	
21.01/6.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.01/6.30	
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	
21.01/6.30	
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(8)
21.01/6.30	Obligation:
21.01/6.30	Q DP problem:
21.01/6.30	The TRS P consists of the following rules:
21.01/6.30	
21.01/6.30	   A(x1) -> B(x1)
21.01/6.30	   C(a(c(x1))) -> A(c(b(a(x1))))
21.01/6.30	
21.01/6.30	The TRS R consists of the following rules:
21.01/6.30	
21.01/6.30	   a(x1) -> b(x1)
21.01/6.30	   b(b(x1)) -> c(x1)
21.01/6.30	   c(a(c(x1))) -> a(c(b(a(x1))))
21.01/6.30	
21.01/6.30	Q is empty.
21.01/6.30	We have to consider all minimal (P,Q,R)-chains.
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(9) DependencyGraphProof (EQUIVALENT)
21.01/6.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
21.01/6.30	----------------------------------------
21.01/6.30	
21.01/6.30	(10)
21.01/6.30	TRUE
21.30/6.38	EOF