21.07/6.38	YES
39.50/11.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
39.50/11.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.50/11.08	
39.50/11.08	
39.50/11.08	Termination w.r.t. Q of the given QTRS could be proven:
39.50/11.08	
39.50/11.08	(0) QTRS
39.50/11.08	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
39.50/11.08	(2) QDP
39.50/11.08	(3) QDPOrderProof [EQUIVALENT, 98 ms]
39.50/11.08	(4) QDP
39.50/11.08	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
39.50/11.08	(6) QDP
39.50/11.08	(7) QDPOrderProof [EQUIVALENT, 76 ms]
39.50/11.08	(8) QDP
39.50/11.08	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
39.50/11.08	(10) TRUE
39.50/11.08	
39.50/11.08	
39.50/11.08	----------------------------------------
39.50/11.08	
39.50/11.08	(0)
39.50/11.08	Obligation:
39.50/11.08	Q restricted rewrite system:
39.50/11.08	The TRS R consists of the following rules:
39.50/11.08	
39.50/11.08	   a(x1) -> b(b(x1))
39.50/11.08	   c(b(x1)) -> d(x1)
39.50/11.08	   e(b(x1)) -> c(c(x1))
39.50/11.08	   d(b(x1)) -> b(f(x1))
39.50/11.08	   f(x1) -> a(e(x1))
39.50/11.08	   c(x1) -> x1
39.50/11.08	   a(a(x1)) -> f(x1)
39.50/11.08	
39.50/11.08	Q is empty.
39.50/11.08	
39.50/11.08	----------------------------------------
39.50/11.08	
39.50/11.08	(1) DependencyPairsProof (EQUIVALENT)
39.50/11.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.50/11.08	----------------------------------------
39.50/11.08	
39.50/11.08	(2)
39.50/11.08	Obligation:
39.50/11.08	Q DP problem:
39.50/11.08	The TRS P consists of the following rules:
39.50/11.08	
39.50/11.08	   C(b(x1)) -> D(x1)
39.50/11.08	   E(b(x1)) -> C(c(x1))
39.50/11.08	   E(b(x1)) -> C(x1)
39.50/11.08	   D(b(x1)) -> F(x1)
39.50/11.08	   F(x1) -> A(e(x1))
39.50/11.08	   F(x1) -> E(x1)
39.50/11.08	   A(a(x1)) -> F(x1)
39.50/11.08	
39.50/11.08	The TRS R consists of the following rules:
39.50/11.08	
39.50/11.08	   a(x1) -> b(b(x1))
39.50/11.08	   c(b(x1)) -> d(x1)
39.50/11.08	   e(b(x1)) -> c(c(x1))
39.50/11.08	   d(b(x1)) -> b(f(x1))
39.50/11.08	   f(x1) -> a(e(x1))
39.50/11.08	   c(x1) -> x1
39.50/11.08	   a(a(x1)) -> f(x1)
39.50/11.08	
39.50/11.08	Q is empty.
39.50/11.08	We have to consider all minimal (P,Q,R)-chains.
39.50/11.08	----------------------------------------
39.50/11.08	
39.50/11.08	(3) QDPOrderProof (EQUIVALENT)
39.50/11.08	We use the reduction pair processor [LPAR04,JAR06].
39.50/11.08	
39.50/11.08	
39.50/11.08	The following pairs can be oriented strictly and are deleted.
39.50/11.08	
39.50/11.08	   C(b(x1)) -> D(x1)
39.50/11.08	The remaining pairs can at least be oriented weakly.
39.50/11.08	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.50/11.08	
39.50/11.08	   <<<
39.50/11.08	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
39.50/11.08	>>>
39.50/11.08	
39.50/11.08	   <<<
39.50/11.08	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
39.50/11.08	>>>
39.50/11.08	
39.50/11.08	   <<<
39.50/11.08	 POL(D(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
39.50/11.08	>>>
39.50/11.08	
39.50/11.08	   <<<
39.50/11.08	 POL(E(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(F(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(e(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, 1A], [1A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(d(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, -I, 1A], [0A, -I, 1A], [-I, -I, 1A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(f(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[1A, -I, 1A], [1A, -I, 1A], [0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	
39.50/11.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.50/11.09	
39.50/11.09	   c(b(x1)) -> d(x1)
39.50/11.09	   c(x1) -> x1
39.50/11.09	   e(b(x1)) -> c(c(x1))
39.50/11.09	   d(b(x1)) -> b(f(x1))
39.50/11.09	   a(a(x1)) -> f(x1)
39.50/11.09	   f(x1) -> a(e(x1))
39.50/11.09	   a(x1) -> b(b(x1))
39.50/11.09	
39.50/11.09	
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(4)
39.50/11.09	Obligation:
39.50/11.09	Q DP problem:
39.50/11.09	The TRS P consists of the following rules:
39.50/11.09	
39.50/11.09	   E(b(x1)) -> C(c(x1))
39.50/11.09	   E(b(x1)) -> C(x1)
39.50/11.09	   D(b(x1)) -> F(x1)
39.50/11.09	   F(x1) -> A(e(x1))
39.50/11.09	   F(x1) -> E(x1)
39.50/11.09	   A(a(x1)) -> F(x1)
39.50/11.09	
39.50/11.09	The TRS R consists of the following rules:
39.50/11.09	
39.50/11.09	   a(x1) -> b(b(x1))
39.50/11.09	   c(b(x1)) -> d(x1)
39.50/11.09	   e(b(x1)) -> c(c(x1))
39.50/11.09	   d(b(x1)) -> b(f(x1))
39.50/11.09	   f(x1) -> a(e(x1))
39.50/11.09	   c(x1) -> x1
39.50/11.09	   a(a(x1)) -> f(x1)
39.50/11.09	
39.50/11.09	Q is empty.
39.50/11.09	We have to consider all minimal (P,Q,R)-chains.
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(5) DependencyGraphProof (EQUIVALENT)
39.50/11.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(6)
39.50/11.09	Obligation:
39.50/11.09	Q DP problem:
39.50/11.09	The TRS P consists of the following rules:
39.50/11.09	
39.50/11.09	   F(x1) -> A(e(x1))
39.50/11.09	   A(a(x1)) -> F(x1)
39.50/11.09	
39.50/11.09	The TRS R consists of the following rules:
39.50/11.09	
39.50/11.09	   a(x1) -> b(b(x1))
39.50/11.09	   c(b(x1)) -> d(x1)
39.50/11.09	   e(b(x1)) -> c(c(x1))
39.50/11.09	   d(b(x1)) -> b(f(x1))
39.50/11.09	   f(x1) -> a(e(x1))
39.50/11.09	   c(x1) -> x1
39.50/11.09	   a(a(x1)) -> f(x1)
39.50/11.09	
39.50/11.09	Q is empty.
39.50/11.09	We have to consider all minimal (P,Q,R)-chains.
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(7) QDPOrderProof (EQUIVALENT)
39.50/11.09	We use the reduction pair processor [LPAR04,JAR06].
39.50/11.09	
39.50/11.09	
39.50/11.09	The following pairs can be oriented strictly and are deleted.
39.50/11.09	
39.50/11.09	   F(x1) -> A(e(x1))
39.50/11.09	The remaining pairs can at least be oriented weakly.
39.50/11.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(F(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(e(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(d(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	   <<<
39.50/11.09	 POL(f(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [-I, 0A, 0A], [0A, 1A, 1A]] 	* 	x_1
39.50/11.09	>>>
39.50/11.09	
39.50/11.09	
39.50/11.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.50/11.09	
39.50/11.09	   e(b(x1)) -> c(c(x1))
39.50/11.09	   c(b(x1)) -> d(x1)
39.50/11.09	   d(b(x1)) -> b(f(x1))
39.50/11.09	   a(a(x1)) -> f(x1)
39.50/11.09	   f(x1) -> a(e(x1))
39.50/11.09	   c(x1) -> x1
39.50/11.09	   a(x1) -> b(b(x1))
39.50/11.09	
39.50/11.09	
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(8)
39.50/11.09	Obligation:
39.50/11.09	Q DP problem:
39.50/11.09	The TRS P consists of the following rules:
39.50/11.09	
39.50/11.09	   A(a(x1)) -> F(x1)
39.50/11.09	
39.50/11.09	The TRS R consists of the following rules:
39.50/11.09	
39.50/11.09	   a(x1) -> b(b(x1))
39.50/11.09	   c(b(x1)) -> d(x1)
39.50/11.09	   e(b(x1)) -> c(c(x1))
39.50/11.09	   d(b(x1)) -> b(f(x1))
39.50/11.09	   f(x1) -> a(e(x1))
39.50/11.09	   c(x1) -> x1
39.50/11.09	   a(a(x1)) -> f(x1)
39.50/11.09	
39.50/11.09	Q is empty.
39.50/11.09	We have to consider all minimal (P,Q,R)-chains.
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(9) DependencyGraphProof (EQUIVALENT)
39.50/11.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
39.50/11.09	----------------------------------------
39.50/11.09	
39.50/11.09	(10)
39.50/11.09	TRUE
39.62/11.15	EOF