35.91/10.10	YES
36.57/10.27	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
36.57/10.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.57/10.27	
36.57/10.27	
36.57/10.27	Termination w.r.t. Q of the given QTRS could be proven:
36.57/10.27	
36.57/10.27	(0) QTRS
36.57/10.27	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
36.57/10.27	(2) QDP
36.57/10.27	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
36.57/10.27	(4) AND
36.57/10.27	    (5) QDP
36.57/10.27	        (6) QDPOrderProof [EQUIVALENT, 23 ms]
36.57/10.27	        (7) QDP
36.57/10.27	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
36.57/10.27	        (9) YES
36.57/10.27	    (10) QDP
36.57/10.27	        (11) QDPOrderProof [EQUIVALENT, 70 ms]
36.57/10.27	        (12) QDP
36.57/10.27	        (13) QDPOrderProof [EQUIVALENT, 209 ms]
36.57/10.27	        (14) QDP
36.57/10.27	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
36.57/10.27	        (16) YES
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(0)
36.57/10.27	Obligation:
36.57/10.27	Q restricted rewrite system:
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(1) DependencyPairsProof (EQUIVALENT)
36.57/10.27	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(2)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   A(b(b(x1))) -> A(b(x1))
36.57/10.27	   A(P(x1)) -> A(x(x1))
36.57/10.27	   A(x(x1)) -> A(x1)
36.57/10.27	   B(P(x1)) -> B(Q(x1))
36.57/10.27	   B(P(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(x(x1)) -> A(Q(x1))
36.57/10.27	   Q^1(x(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(a(x1)) -> B(b(a(x1)))
36.57/10.27	   Q^1(a(x1)) -> B(a(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(3) DependencyGraphProof (EQUIVALENT)
36.57/10.27	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(4)
36.57/10.27	Complex Obligation (AND)
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(5)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   A(P(x1)) -> A(x(x1))
36.57/10.27	   A(x(x1)) -> A(x1)
36.57/10.27	   A(b(b(x1))) -> A(b(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(6) QDPOrderProof (EQUIVALENT)
36.57/10.27	We use the reduction pair processor [LPAR04,JAR06].
36.57/10.27	
36.57/10.27	
36.57/10.27	The following pairs can be oriented strictly and are deleted.
36.57/10.27	
36.57/10.27	   A(P(x1)) -> A(x(x1))
36.57/10.27	The remaining pairs can at least be oriented weakly.
36.57/10.27	Used ordering:  Polynomial interpretation [POLO]:
36.57/10.27	
36.57/10.27	   POL(A(x_1)) = x_1
36.57/10.27	   POL(P(x_1)) = 1 + x_1
36.57/10.27	   POL(Q(x_1)) = 0
36.57/10.27	   POL(a(x_1)) = 0
36.57/10.27	   POL(b(x_1)) = 0
36.57/10.27	   POL(x(x_1)) = x_1
36.57/10.27	
36.57/10.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.57/10.27	
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(7)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   A(x(x1)) -> A(x1)
36.57/10.27	   A(b(b(x1))) -> A(b(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(8) QDPSizeChangeProof (EQUIVALENT)
36.57/10.27	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
36.57/10.27	
36.57/10.27	From the DPs we obtained the following set of size-change graphs:
36.57/10.27	*A(x(x1)) -> A(x1)
36.57/10.27	The graph contains the following edges 1 > 1
36.57/10.27	
36.57/10.27	
36.57/10.27	*A(b(b(x1))) -> A(b(x1))
36.57/10.27	The graph contains the following edges 1 > 1
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(9)
36.57/10.27	YES
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(10)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   Q^1(x(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(a(x1)) -> B(b(a(x1)))
36.57/10.27	   B(P(x1)) -> B(Q(x1))
36.57/10.27	   B(P(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(a(x1)) -> B(a(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(11) QDPOrderProof (EQUIVALENT)
36.57/10.27	We use the reduction pair processor [LPAR04,JAR06].
36.57/10.27	
36.57/10.27	
36.57/10.27	The following pairs can be oriented strictly and are deleted.
36.57/10.27	
36.57/10.27	   Q^1(a(x1)) -> B(b(a(x1)))
36.57/10.27	The remaining pairs can at least be oriented weakly.
36.57/10.27	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
36.57/10.27	
36.57/10.27	POL( B_1(x_1) ) = max{0, 2x_1 - 1}
36.57/10.27	POL( a_1(x_1) ) = 1
36.57/10.27	POL( b_1(x_1) ) = max{0, -2}
36.57/10.27	POL( P_1(x_1) ) = 1
36.57/10.27	POL( x_1(x_1) ) = max{0, x_1 - 2}
36.57/10.27	POL( Q_1(x_1) ) = 1
36.57/10.27	POL( Q^1_1(x_1) ) = 1
36.57/10.27	
36.57/10.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(12)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   Q^1(x(x1)) -> Q^1(x1)
36.57/10.27	   B(P(x1)) -> B(Q(x1))
36.57/10.27	   B(P(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(a(x1)) -> B(a(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(13) QDPOrderProof (EQUIVALENT)
36.57/10.27	We use the reduction pair processor [LPAR04,JAR06].
36.57/10.27	
36.57/10.27	
36.57/10.27	The following pairs can be oriented strictly and are deleted.
36.57/10.27	
36.57/10.27	   B(P(x1)) -> B(Q(x1))
36.57/10.27	The remaining pairs can at least be oriented weakly.
36.57/10.27	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(Q^1(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(x(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [-I, 1A, -I], [-I, -I, 0A]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(P(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [0A, 1A, 0A]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(Q(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	   <<<
36.57/10.27	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
36.57/10.27	>>>
36.57/10.27	
36.57/10.27	
36.57/10.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.57/10.27	
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(14)
36.57/10.27	Obligation:
36.57/10.27	Q DP problem:
36.57/10.27	The TRS P consists of the following rules:
36.57/10.27	
36.57/10.27	   Q^1(x(x1)) -> Q^1(x1)
36.57/10.27	   B(P(x1)) -> Q^1(x1)
36.57/10.27	   Q^1(a(x1)) -> B(a(x1))
36.57/10.27	
36.57/10.27	The TRS R consists of the following rules:
36.57/10.27	
36.57/10.27	   a(b(b(x1))) -> P(a(b(x1)))
36.57/10.27	   a(P(x1)) -> P(a(x(x1)))
36.57/10.27	   a(x(x1)) -> x(a(x1))
36.57/10.27	   b(P(x1)) -> b(Q(x1))
36.57/10.27	   Q(x(x1)) -> a(Q(x1))
36.57/10.27	   Q(a(x1)) -> b(b(a(x1)))
36.57/10.27	
36.57/10.27	Q is empty.
36.57/10.27	We have to consider all minimal (P,Q,R)-chains.
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(15) QDPSizeChangeProof (EQUIVALENT)
36.57/10.27	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
36.57/10.27	
36.57/10.27	From the DPs we obtained the following set of size-change graphs:
36.57/10.27	*Q^1(x(x1)) -> Q^1(x1)
36.57/10.27	The graph contains the following edges 1 > 1
36.57/10.27	
36.57/10.27	
36.57/10.27	*Q^1(a(x1)) -> B(a(x1))
36.57/10.27	The graph contains the following edges 1 >= 1
36.57/10.27	
36.57/10.27	
36.57/10.27	*B(P(x1)) -> Q^1(x1)
36.57/10.27	The graph contains the following edges 1 > 1
36.57/10.27	
36.57/10.27	
36.57/10.27	----------------------------------------
36.57/10.27	
36.57/10.27	(16)
36.57/10.27	YES
36.88/10.43	EOF