40.60/11.40	YES
40.85/11.44	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
40.85/11.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.85/11.44	
40.85/11.44	
40.85/11.44	Termination w.r.t. Q of the given QTRS could be proven:
40.85/11.44	
40.85/11.44	(0) QTRS
40.85/11.44	(1) DependencyPairsProof [EQUIVALENT, 14 ms]
40.85/11.44	(2) QDP
40.85/11.44	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
40.85/11.44	(4) AND
40.85/11.44	    (5) QDP
40.85/11.44	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
40.85/11.44	        (7) QDP
40.85/11.44	        (8) QDPOrderProof [EQUIVALENT, 185 ms]
40.85/11.44	        (9) QDP
40.85/11.44	        (10) QDPOrderProof [EQUIVALENT, 58 ms]
40.85/11.44	        (11) QDP
40.85/11.44	        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
40.85/11.44	        (13) QDP
40.85/11.44	        (14) QDPOrderProof [EQUIVALENT, 27 ms]
40.85/11.44	        (15) QDP
40.85/11.44	        (16) QDPOrderProof [EQUIVALENT, 17 ms]
40.85/11.44	        (17) QDP
40.85/11.44	        (18) PisEmptyProof [EQUIVALENT, 0 ms]
40.85/11.44	        (19) YES
40.85/11.44	    (20) QDP
40.85/11.44	        (21) UsableRulesProof [EQUIVALENT, 0 ms]
40.85/11.44	        (22) QDP
40.85/11.44	        (23) QDPOrderProof [EQUIVALENT, 65 ms]
40.85/11.44	        (24) QDP
40.85/11.44	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
40.85/11.44	        (26) YES
40.85/11.44	    (27) QDP
40.85/11.44	        (28) QDPOrderProof [EQUIVALENT, 122 ms]
40.85/11.44	        (29) QDP
40.85/11.44	        (30) QDPOrderProof [EQUIVALENT, 0 ms]
40.85/11.44	        (31) QDP
40.85/11.44	        (32) PisEmptyProof [EQUIVALENT, 0 ms]
40.85/11.44	        (33) YES
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(0)
40.85/11.44	Obligation:
40.85/11.44	Q restricted rewrite system:
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(1) DependencyPairsProof (EQUIVALENT)
40.85/11.44	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(2)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   A(c(x1)) -> C(b(c(c(a(x1)))))
40.85/11.44	   A(c(x1)) -> B(c(c(a(x1))))
40.85/11.44	   A(c(x1)) -> C(c(a(x1)))
40.85/11.44	   A(c(x1)) -> C(a(x1))
40.85/11.44	   A(c(x1)) -> A(x1)
40.85/11.44	   B(b(b(x1))) -> C(b(x1))
40.85/11.44	   D(d(x1)) -> D(b(d(b(d(x1)))))
40.85/11.44	   D(d(x1)) -> B(d(b(d(x1))))
40.85/11.44	   D(d(x1)) -> D(b(d(x1)))
40.85/11.44	   D(d(x1)) -> B(d(x1))
40.85/11.44	   A(a(x1)) -> A(d(a(x1)))
40.85/11.44	   A(a(x1)) -> D(a(x1))
40.85/11.44	   A(b(x1)) -> C(c(a(x1)))
40.85/11.44	   A(b(x1)) -> C(a(x1))
40.85/11.44	   A(b(x1)) -> A(x1)
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	   C(c(x1)) -> B(c(b(c(x1))))
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   C(c(x1)) -> B(c(x1))
40.85/11.44	   C(c(c(x1))) -> C(b(b(x1)))
40.85/11.44	   C(c(c(x1))) -> B(b(x1))
40.85/11.44	   C(c(c(x1))) -> B(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(3) DependencyGraphProof (EQUIVALENT)
40.85/11.44	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 11 less nodes.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(4)
40.85/11.44	Complex Obligation (AND)
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(5)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   C(c(c(x1))) -> C(b(b(x1)))
40.85/11.44	   C(c(c(x1))) -> B(b(x1))
40.85/11.44	   B(b(b(x1))) -> C(b(x1))
40.85/11.44	   C(c(c(x1))) -> B(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(6) UsableRulesProof (EQUIVALENT)
40.85/11.44	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.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(7)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   C(c(c(x1))) -> C(b(b(x1)))
40.85/11.44	   C(c(c(x1))) -> B(b(x1))
40.85/11.44	   B(b(b(x1))) -> C(b(x1))
40.85/11.44	   C(c(c(x1))) -> B(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(8) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   C(c(c(x1))) -> C(b(b(x1)))
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [0A, 1A, 1A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(9)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   C(c(c(x1))) -> B(b(x1))
40.85/11.44	   B(b(b(x1))) -> C(b(x1))
40.85/11.44	   C(c(c(x1))) -> B(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(10) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   C(c(c(x1))) -> B(b(x1))
40.85/11.44	   C(c(c(x1))) -> B(x1)
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(11)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   B(b(b(x1))) -> C(b(x1))
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(12) DependencyGraphProof (EQUIVALENT)
40.85/11.44	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(13)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(14) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(x1)))
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(15)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(16) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   C(c(x1)) -> C(b(c(b(c(x1)))))
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, -I]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	   <<<
40.85/11.44	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
40.85/11.44	>>>
40.85/11.44	
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(17)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	P is empty.
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(18) PisEmptyProof (EQUIVALENT)
40.85/11.44	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(19)
40.85/11.44	YES
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(20)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   D(d(x1)) -> D(b(d(x1)))
40.85/11.44	   D(d(x1)) -> D(b(d(b(d(x1)))))
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(21) UsableRulesProof (EQUIVALENT)
40.85/11.44	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.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(22)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   D(d(x1)) -> D(b(d(x1)))
40.85/11.44	   D(d(x1)) -> D(b(d(b(d(x1)))))
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(23) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   D(d(x1)) -> D(b(d(x1)))
40.85/11.44	   D(d(x1)) -> D(b(d(b(d(x1)))))
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
40.85/11.44	
40.85/11.44	POL( D_1(x_1) ) = x_1
40.85/11.44	POL( b_1(x_1) ) = 1
40.85/11.44	POL( d_1(x_1) ) = 2x_1 + 2
40.85/11.44	POL( c_1(x_1) ) = max{0, -2}
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(24)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	P is empty.
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(25) PisEmptyProof (EQUIVALENT)
40.85/11.44	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(26)
40.85/11.44	YES
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(27)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   A(a(x1)) -> A(d(a(x1)))
40.85/11.44	   A(c(x1)) -> A(x1)
40.85/11.44	   A(b(x1)) -> A(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(28) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   A(b(x1)) -> A(x1)
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Polynomial interpretation [POLO]:
40.85/11.44	
40.85/11.44	   POL(A(x_1)) = 2*x_1
40.85/11.44	   POL(a(x_1)) = 0
40.85/11.44	   POL(b(x_1)) = 1 + x_1
40.85/11.44	   POL(c(x_1)) = 4*x_1
40.85/11.44	   POL(d(x_1)) = 0
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(29)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	The TRS P consists of the following rules:
40.85/11.44	
40.85/11.44	   A(a(x1)) -> A(d(a(x1)))
40.85/11.44	   A(c(x1)) -> A(x1)
40.85/11.44	
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(30) QDPOrderProof (EQUIVALENT)
40.85/11.44	We use the reduction pair processor [LPAR04,JAR06].
40.85/11.44	
40.85/11.44	
40.85/11.44	The following pairs can be oriented strictly and are deleted.
40.85/11.44	
40.85/11.44	   A(a(x1)) -> A(d(a(x1)))
40.85/11.44	   A(c(x1)) -> A(x1)
40.85/11.44	The remaining pairs can at least be oriented weakly.
40.85/11.44	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
40.85/11.44	
40.85/11.44	POL( A_1(x_1) ) = x_1 + 1
40.85/11.44	POL( d_1(x_1) ) = 0
40.85/11.44	POL( a_1(x_1) ) = 2
40.85/11.44	POL( c_1(x_1) ) = 2x_1 + 1
40.85/11.44	POL( b_1(x_1) ) = 0
40.85/11.44	
40.85/11.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.85/11.44	
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	
40.85/11.44	
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(31)
40.85/11.44	Obligation:
40.85/11.44	Q DP problem:
40.85/11.44	P is empty.
40.85/11.44	The TRS R consists of the following rules:
40.85/11.44	
40.85/11.44	   a(c(x1)) -> c(b(c(c(a(x1)))))
40.85/11.44	   b(b(b(x1))) -> c(b(x1))
40.85/11.44	   d(d(x1)) -> d(b(d(b(d(x1)))))
40.85/11.44	   a(a(x1)) -> a(d(a(x1)))
40.85/11.44	   a(b(x1)) -> c(c(a(x1)))
40.85/11.44	   c(c(x1)) -> c(b(c(b(c(x1)))))
40.85/11.44	   c(c(c(x1))) -> c(b(b(x1)))
40.85/11.44	
40.85/11.44	Q is empty.
40.85/11.44	We have to consider all minimal (P,Q,R)-chains.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(32) PisEmptyProof (EQUIVALENT)
40.85/11.44	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.85/11.44	----------------------------------------
40.85/11.44	
40.85/11.44	(33)
40.85/11.44	YES
41.01/11.61	EOF