42.31/11.71	YES
43.13/11.93	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
43.13/11.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.13/11.93	
43.13/11.93	
43.13/11.93	Termination w.r.t. Q of the given QTRS could be proven:
43.13/11.93	
43.13/11.93	(0) QTRS
43.13/11.93	(1) QTRS Reverse [EQUIVALENT, 0 ms]
43.13/11.93	(2) QTRS
43.13/11.93	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
43.13/11.93	(4) QDP
43.13/11.93	(5) QDPOrderProof [EQUIVALENT, 89 ms]
43.13/11.93	(6) QDP
43.13/11.93	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
43.13/11.93	(8) QDP
43.13/11.93	(9) QDPOrderProof [EQUIVALENT, 19 ms]
43.13/11.93	(10) QDP
43.13/11.93	(11) QDPOrderProof [EQUIVALENT, 1096 ms]
43.13/11.93	(12) QDP
43.13/11.93	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
43.13/11.93	(14) TRUE
43.13/11.93	
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(0)
43.13/11.93	Obligation:
43.13/11.93	Q restricted rewrite system:
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   d(a(x1)) -> b(d(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> a(d(x1))
43.13/11.93	   b(d(d(x1))) -> c(c(d(d(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(1) QTRS Reverse (EQUIVALENT)
43.13/11.93	We applied the QTRS Reverse Processor [REVERSE].
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(2)
43.13/11.93	Obligation:
43.13/11.93	Q restricted rewrite system:
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(3) DependencyPairsProof (EQUIVALENT)
43.13/11.93	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(4)
43.13/11.93	Obligation:
43.13/11.93	Q DP problem:
43.13/11.93	The TRS P consists of the following rules:
43.13/11.93	
43.13/11.93	   A(d(x1)) -> D(b(x1))
43.13/11.93	   A(d(x1)) -> B(x1)
43.13/11.93	   B(x1) -> A(a(a(x1)))
43.13/11.93	   B(x1) -> A(a(x1))
43.13/11.93	   B(x1) -> A(x1)
43.13/11.93	   C(d(c(x1))) -> D(a(x1))
43.13/11.93	   C(d(c(x1))) -> A(x1)
43.13/11.93	   D(d(b(x1))) -> C(d(d(c(c(x1)))))
43.13/11.93	   D(d(b(x1))) -> D(d(c(c(x1))))
43.13/11.93	   D(d(b(x1))) -> D(c(c(x1)))
43.13/11.93	   D(d(b(x1))) -> C(c(x1))
43.13/11.93	   D(d(b(x1))) -> C(x1)
43.13/11.93	
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	We have to consider all minimal (P,Q,R)-chains.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(5) QDPOrderProof (EQUIVALENT)
43.13/11.93	We use the reduction pair processor [LPAR04,JAR06].
43.13/11.93	
43.13/11.93	
43.13/11.93	The following pairs can be oriented strictly and are deleted.
43.13/11.93	
43.13/11.93	   A(d(x1)) -> D(b(x1))
43.13/11.93	   B(x1) -> A(a(a(x1)))
43.13/11.93	   B(x1) -> A(a(x1))
43.13/11.93	   B(x1) -> A(x1)
43.13/11.93	   C(d(c(x1))) -> A(x1)
43.13/11.93	   D(d(b(x1))) -> D(c(c(x1)))
43.13/11.93	   D(d(b(x1))) -> C(c(x1))
43.13/11.93	   D(d(b(x1))) -> C(x1)
43.13/11.93	The remaining pairs can at least be oriented weakly.
43.13/11.93	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
43.13/11.93	
43.13/11.93	POL( D_1(x_1) ) = 2x_1 + 1
43.13/11.93	POL( A_1(x_1) ) = 2x_1
43.13/11.93	POL( d_1(x_1) ) = x_1 + 1
43.13/11.93	POL( C_1(x_1) ) = max{0, 2x_1 - 1}
43.13/11.93	POL( b_1(x_1) ) = x_1
43.13/11.93	POL( a_1(x_1) ) = x_1
43.13/11.93	POL( c_1(x_1) ) = x_1
43.13/11.93	POL( B_1(x_1) ) = 2x_1 + 2
43.13/11.93	
43.13/11.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.13/11.93	
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(6)
43.13/11.93	Obligation:
43.13/11.93	Q DP problem:
43.13/11.93	The TRS P consists of the following rules:
43.13/11.93	
43.13/11.93	   A(d(x1)) -> B(x1)
43.13/11.93	   C(d(c(x1))) -> D(a(x1))
43.13/11.93	   D(d(b(x1))) -> C(d(d(c(c(x1)))))
43.13/11.93	   D(d(b(x1))) -> D(d(c(c(x1))))
43.13/11.93	
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	We have to consider all minimal (P,Q,R)-chains.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(7) DependencyGraphProof (EQUIVALENT)
43.13/11.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(8)
43.13/11.93	Obligation:
43.13/11.93	Q DP problem:
43.13/11.93	The TRS P consists of the following rules:
43.13/11.93	
43.13/11.93	   D(d(b(x1))) -> C(d(d(c(c(x1)))))
43.13/11.93	   C(d(c(x1))) -> D(a(x1))
43.13/11.93	   D(d(b(x1))) -> D(d(c(c(x1))))
43.13/11.93	
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	We have to consider all minimal (P,Q,R)-chains.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(9) QDPOrderProof (EQUIVALENT)
43.13/11.93	We use the reduction pair processor [LPAR04,JAR06].
43.13/11.93	
43.13/11.93	
43.13/11.93	The following pairs can be oriented strictly and are deleted.
43.13/11.93	
43.13/11.93	   D(d(b(x1))) -> D(d(c(c(x1))))
43.13/11.93	The remaining pairs can at least be oriented weakly.
43.13/11.93	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
43.13/11.93	
43.13/11.93	POL( C_1(x_1) ) = 1
43.13/11.93	POL( D_1(x_1) ) = max{0, 2x_1 - 1}
43.13/11.93	POL( d_1(x_1) ) = max{0, x_1 - 1}
43.13/11.93	POL( b_1(x_1) ) = 2
43.13/11.93	POL( c_1(x_1) ) = max{0, -2}
43.13/11.93	POL( a_1(x_1) ) = 1
43.13/11.93	
43.13/11.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.13/11.93	
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(10)
43.13/11.93	Obligation:
43.13/11.93	Q DP problem:
43.13/11.93	The TRS P consists of the following rules:
43.13/11.93	
43.13/11.93	   D(d(b(x1))) -> C(d(d(c(c(x1)))))
43.13/11.93	   C(d(c(x1))) -> D(a(x1))
43.13/11.93	
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	We have to consider all minimal (P,Q,R)-chains.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(11) QDPOrderProof (EQUIVALENT)
43.13/11.93	We use the reduction pair processor [LPAR04,JAR06].
43.13/11.93	
43.13/11.93	
43.13/11.93	The following pairs can be oriented strictly and are deleted.
43.13/11.93	
43.13/11.93	   C(d(c(x1))) -> D(a(x1))
43.13/11.93	The remaining pairs can at least be oriented weakly.
43.13/11.93	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(D(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(d(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [1A, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(b(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	   <<<
43.13/11.93	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
43.13/11.93	>>>
43.13/11.93	
43.13/11.93	
43.13/11.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.13/11.93	
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	
43.13/11.93	
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(12)
43.13/11.93	Obligation:
43.13/11.93	Q DP problem:
43.13/11.93	The TRS P consists of the following rules:
43.13/11.93	
43.13/11.93	   D(d(b(x1))) -> C(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	The TRS R consists of the following rules:
43.13/11.93	
43.13/11.93	   a(d(x1)) -> d(b(x1))
43.13/11.93	   b(x1) -> a(a(a(x1)))
43.13/11.93	   c(d(c(x1))) -> d(a(x1))
43.13/11.93	   d(d(b(x1))) -> c(d(d(c(c(x1)))))
43.13/11.93	
43.13/11.93	Q is empty.
43.13/11.93	We have to consider all minimal (P,Q,R)-chains.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(13) DependencyGraphProof (EQUIVALENT)
43.13/11.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
43.13/11.93	----------------------------------------
43.13/11.93	
43.13/11.93	(14)
43.13/11.93	TRUE
43.47/12.06	EOF