34.21/9.84	YES
34.21/9.85	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
34.21/9.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.21/9.85	
34.21/9.85	
34.21/9.85	Termination w.r.t. Q of the given QTRS could be proven:
34.21/9.85	
34.21/9.85	(0) QTRS
34.21/9.85	(1) DependencyPairsProof [EQUIVALENT, 17 ms]
34.21/9.85	(2) QDP
34.21/9.85	(3) QDPOrderProof [EQUIVALENT, 133 ms]
34.21/9.85	(4) QDP
34.21/9.85	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
34.21/9.85	(6) QDP
34.21/9.85	(7) QDPOrderProof [EQUIVALENT, 13 ms]
34.21/9.85	(8) QDP
34.21/9.85	(9) QDPOrderProof [EQUIVALENT, 0 ms]
34.21/9.85	(10) QDP
34.21/9.85	(11) PisEmptyProof [EQUIVALENT, 0 ms]
34.21/9.85	(12) YES
34.21/9.85	
34.21/9.85	
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(0)
34.21/9.85	Obligation:
34.21/9.85	Q restricted rewrite system:
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(1) DependencyPairsProof (EQUIVALENT)
34.21/9.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(2)
34.21/9.85	Obligation:
34.21/9.85	Q DP problem:
34.21/9.85	The TRS P consists of the following rules:
34.21/9.85	
34.21/9.85	   A(x1) -> C(x1)
34.21/9.85	   C(c(a(x1))) -> A(a(c(c(x1))))
34.21/9.85	   C(c(a(x1))) -> A(c(c(x1)))
34.21/9.85	   C(c(a(x1))) -> C(c(x1))
34.21/9.85	   C(c(a(x1))) -> C(x1)
34.21/9.85	
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	We have to consider all minimal (P,Q,R)-chains.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(3) QDPOrderProof (EQUIVALENT)
34.21/9.85	We use the reduction pair processor [LPAR04,JAR06].
34.21/9.85	
34.21/9.85	
34.21/9.85	The following pairs can be oriented strictly and are deleted.
34.21/9.85	
34.21/9.85	   A(x1) -> C(x1)
34.21/9.85	The remaining pairs can at least be oriented weakly.
34.21/9.85	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.21/9.85	
34.21/9.85	   <<<
34.21/9.85	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 1A]] 	* 	x_1
34.21/9.85	>>>
34.21/9.85	
34.21/9.85	   <<<
34.21/9.85	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
34.21/9.85	>>>
34.21/9.85	
34.21/9.85	   <<<
34.21/9.85	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
34.21/9.85	>>>
34.21/9.85	
34.21/9.85	   <<<
34.21/9.85	 POL(a(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [-I, -I, 0A]] 	* 	x_1
34.21/9.85	>>>
34.21/9.85	
34.21/9.85	   <<<
34.21/9.85	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
34.21/9.85	>>>
34.21/9.85	
34.21/9.85	
34.21/9.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.21/9.85	
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	
34.21/9.85	
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(4)
34.21/9.85	Obligation:
34.21/9.85	Q DP problem:
34.21/9.85	The TRS P consists of the following rules:
34.21/9.85	
34.21/9.85	   C(c(a(x1))) -> A(a(c(c(x1))))
34.21/9.85	   C(c(a(x1))) -> A(c(c(x1)))
34.21/9.85	   C(c(a(x1))) -> C(c(x1))
34.21/9.85	   C(c(a(x1))) -> C(x1)
34.21/9.85	
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	We have to consider all minimal (P,Q,R)-chains.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(5) DependencyGraphProof (EQUIVALENT)
34.21/9.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(6)
34.21/9.85	Obligation:
34.21/9.85	Q DP problem:
34.21/9.85	The TRS P consists of the following rules:
34.21/9.85	
34.21/9.85	   C(c(a(x1))) -> C(x1)
34.21/9.85	   C(c(a(x1))) -> C(c(x1))
34.21/9.85	
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	We have to consider all minimal (P,Q,R)-chains.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(7) QDPOrderProof (EQUIVALENT)
34.21/9.85	We use the reduction pair processor [LPAR04,JAR06].
34.21/9.85	
34.21/9.85	
34.21/9.85	The following pairs can be oriented strictly and are deleted.
34.21/9.85	
34.21/9.85	   C(c(a(x1))) -> C(x1)
34.21/9.85	The remaining pairs can at least be oriented weakly.
34.21/9.85	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
34.21/9.85	
34.21/9.85	POL( C_1(x_1) ) = max{0, x_1 - 1}
34.21/9.85	POL( c_1(x_1) ) = x_1 + 2
34.21/9.85	POL( a_1(x_1) ) = x_1
34.21/9.85	POL( b_1(x_1) ) = max{0, -2}
34.21/9.85	
34.21/9.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.21/9.85	
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   a(x1) -> x1
34.21/9.85	
34.21/9.85	
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(8)
34.21/9.85	Obligation:
34.21/9.85	Q DP problem:
34.21/9.85	The TRS P consists of the following rules:
34.21/9.85	
34.21/9.85	   C(c(a(x1))) -> C(c(x1))
34.21/9.85	
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	We have to consider all minimal (P,Q,R)-chains.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(9) QDPOrderProof (EQUIVALENT)
34.21/9.85	We use the reduction pair processor [LPAR04,JAR06].
34.21/9.85	
34.21/9.85	
34.21/9.85	The following pairs can be oriented strictly and are deleted.
34.21/9.85	
34.21/9.85	   C(c(a(x1))) -> C(c(x1))
34.21/9.85	The remaining pairs can at least be oriented weakly.
34.21/9.85	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
34.21/9.85	
34.21/9.85	POL( C_1(x_1) ) = max{0, 2x_1 - 2}
34.21/9.85	POL( c_1(x_1) ) = 2x_1
34.21/9.85	POL( a_1(x_1) ) = x_1 + 2
34.21/9.85	POL( b_1(x_1) ) = 2
34.21/9.85	
34.21/9.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.21/9.85	
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   a(x1) -> x1
34.21/9.85	
34.21/9.85	
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(10)
34.21/9.85	Obligation:
34.21/9.85	Q DP problem:
34.21/9.85	P is empty.
34.21/9.85	The TRS R consists of the following rules:
34.21/9.85	
34.21/9.85	   a(x1) -> x1
34.21/9.85	   a(x1) -> b(b(c(x1)))
34.21/9.85	   c(c(a(x1))) -> a(a(c(c(x1))))
34.21/9.85	
34.21/9.85	Q is empty.
34.21/9.85	We have to consider all minimal (P,Q,R)-chains.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(11) PisEmptyProof (EQUIVALENT)
34.21/9.85	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.21/9.85	----------------------------------------
34.21/9.85	
34.21/9.85	(12)
34.21/9.85	YES
34.21/9.91	EOF