23.13/6.91	YES
23.54/6.99	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
23.54/6.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.54/6.99	
23.54/6.99	
23.54/6.99	Termination w.r.t. Q of the given QTRS could be proven:
23.54/6.99	
23.54/6.99	(0) QTRS
23.54/6.99	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.54/6.99	(2) QTRS
23.54/6.99	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
23.54/6.99	(4) QDP
23.54/6.99	(5) DependencyGraphProof [EQUIVALENT, 2 ms]
23.54/6.99	(6) QDP
23.54/6.99	(7) UsableRulesProof [EQUIVALENT, 0 ms]
23.54/6.99	(8) QDP
23.54/6.99	(9) MNOCProof [EQUIVALENT, 0 ms]
23.54/6.99	(10) QDP
23.54/6.99	(11) QDPOrderProof [EQUIVALENT, 167 ms]
23.54/6.99	(12) QDP
23.54/6.99	(13) QDPOrderProof [EQUIVALENT, 0 ms]
23.54/6.99	(14) QDP
23.54/6.99	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
23.54/6.99	(16) TRUE
23.54/6.99	
23.54/6.99	
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(0)
23.54/6.99	Obligation:
23.54/6.99	Q restricted rewrite system:
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   a(a(x1)) -> b(b(x1))
23.54/6.99	   c(c(b(x1))) -> d(c(a(x1)))
23.54/6.99	   a(x1) -> d(c(c(x1)))
23.54/6.99	   c(d(x1)) -> b(c(x1))
23.54/6.99	
23.54/6.99	Q is empty.
23.54/6.99	
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(1) QTRS Reverse (EQUIVALENT)
23.54/6.99	We applied the QTRS Reverse Processor [REVERSE].
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(2)
23.54/6.99	Obligation:
23.54/6.99	Q restricted rewrite system:
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   a(a(x1)) -> b(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	
23.54/6.99	Q is empty.
23.54/6.99	
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(3) DependencyPairsProof (EQUIVALENT)
23.54/6.99	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(4)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   A(a(x1)) -> B(b(x1))
23.54/6.99	   A(a(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	   B(c(c(x1))) -> D(x1)
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   a(a(x1)) -> b(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	
23.54/6.99	Q is empty.
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(5) DependencyGraphProof (EQUIVALENT)
23.54/6.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(6)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	   B(c(c(x1))) -> D(x1)
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   a(a(x1)) -> b(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	
23.54/6.99	Q is empty.
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(7) UsableRulesProof (EQUIVALENT)
23.54/6.99	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.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(8)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	   B(c(c(x1))) -> D(x1)
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	Q is empty.
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(9) MNOCProof (EQUIVALENT)
23.54/6.99	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(10)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	   B(c(c(x1))) -> D(x1)
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	The set Q consists of the following terms:
23.54/6.99	
23.54/6.99	   d(c(x0))
23.54/6.99	   b(c(c(x0)))
23.54/6.99	   a(x0)
23.54/6.99	
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(11) QDPOrderProof (EQUIVALENT)
23.54/6.99	We use the reduction pair processor [LPAR04,JAR06].
23.54/6.99	
23.54/6.99	
23.54/6.99	The following pairs can be oriented strictly and are deleted.
23.54/6.99	
23.54/6.99	   B(c(c(x1))) -> D(x1)
23.54/6.99	The remaining pairs can at least be oriented weakly.
23.54/6.99	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(D(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(d(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(a(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	
23.54/6.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(12)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	The set Q consists of the following terms:
23.54/6.99	
23.54/6.99	   d(c(x0))
23.54/6.99	   b(c(c(x0)))
23.54/6.99	   a(x0)
23.54/6.99	
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(13) QDPOrderProof (EQUIVALENT)
23.54/6.99	We use the reduction pair processor [LPAR04,JAR06].
23.54/6.99	
23.54/6.99	
23.54/6.99	The following pairs can be oriented strictly and are deleted.
23.54/6.99	
23.54/6.99	   A(x1) -> D(x1)
23.54/6.99	The remaining pairs can at least be oriented weakly.
23.54/6.99	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(D(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(d(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	   <<<
23.54/6.99	 POL(a(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 1A], [-I, 0A, 1A]] 	* 	x_1
23.54/6.99	>>>
23.54/6.99	
23.54/6.99	
23.54/6.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(14)
23.54/6.99	Obligation:
23.54/6.99	Q DP problem:
23.54/6.99	The TRS P consists of the following rules:
23.54/6.99	
23.54/6.99	   D(c(x1)) -> B(x1)
23.54/6.99	   B(c(c(x1))) -> A(c(d(x1)))
23.54/6.99	
23.54/6.99	The TRS R consists of the following rules:
23.54/6.99	
23.54/6.99	   d(c(x1)) -> c(b(x1))
23.54/6.99	   b(c(c(x1))) -> a(c(d(x1)))
23.54/6.99	   a(x1) -> c(c(d(x1)))
23.54/6.99	
23.54/6.99	The set Q consists of the following terms:
23.54/6.99	
23.54/6.99	   d(c(x0))
23.54/6.99	   b(c(c(x0)))
23.54/6.99	   a(x0)
23.54/6.99	
23.54/6.99	We have to consider all minimal (P,Q,R)-chains.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(15) DependencyGraphProof (EQUIVALENT)
23.54/6.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
23.54/6.99	----------------------------------------
23.54/6.99	
23.54/6.99	(16)
23.54/6.99	TRUE
24.02/7.19	EOF