28.52/8.23	YES
29.05/8.33	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
29.05/8.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.05/8.33	
29.05/8.33	
29.05/8.33	Termination w.r.t. Q of the given QTRS could be proven:
29.05/8.33	
29.05/8.33	(0) QTRS
29.05/8.33	(1) QTRS Reverse [EQUIVALENT, 0 ms]
29.05/8.33	(2) QTRS
29.05/8.33	(3) DependencyPairsProof [EQUIVALENT, 26 ms]
29.05/8.33	(4) QDP
29.05/8.33	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
29.05/8.33	(6) AND
29.05/8.33	    (7) QDP
29.05/8.33	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
29.05/8.33	        (9) QDP
29.05/8.33	        (10) QDPOrderProof [EQUIVALENT, 109 ms]
29.05/8.33	        (11) QDP
29.05/8.33	        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
29.05/8.33	        (13) AND
29.05/8.33	            (14) QDP
29.05/8.33	                (15) QDPOrderProof [EQUIVALENT, 0 ms]
29.05/8.33	                (16) QDP
29.05/8.33	                (17) PisEmptyProof [EQUIVALENT, 0 ms]
29.05/8.33	                (18) YES
29.05/8.33	            (19) QDP
29.05/8.33	                (20) QDPOrderProof [EQUIVALENT, 25 ms]
29.05/8.33	                (21) QDP
29.05/8.33	                (22) DependencyGraphProof [EQUIVALENT, 0 ms]
29.05/8.33	                (23) TRUE
29.05/8.33	    (24) QDP
29.05/8.33	        (25) UsableRulesProof [EQUIVALENT, 4 ms]
29.05/8.33	        (26) QDP
29.05/8.33	        (27) QDPOrderProof [EQUIVALENT, 12 ms]
29.05/8.33	        (28) QDP
29.05/8.33	        (29) MRRProof [EQUIVALENT, 3 ms]
29.05/8.33	        (30) QDP
29.05/8.33	        (31) PisEmptyProof [EQUIVALENT, 0 ms]
29.05/8.33	        (32) YES
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(0)
29.05/8.33	Obligation:
29.05/8.33	Q restricted rewrite system:
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(s(x1)) -> s(a(x1))
29.05/8.33	   b(a(b(s(x1)))) -> a(b(s(a(x1))))
29.05/8.33	   b(a(b(b(x1)))) -> c(s(x1))
29.05/8.33	   c(s(x1)) -> a(b(a(b(x1))))
29.05/8.33	   a(b(a(a(x1)))) -> b(a(b(a(x1))))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(1) QTRS Reverse (EQUIVALENT)
29.05/8.33	We applied the QTRS Reverse Processor [REVERSE].
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(2)
29.05/8.33	Obligation:
29.05/8.33	Q restricted rewrite system:
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   s(a(x1)) -> a(s(x1))
29.05/8.33	   s(b(a(b(x1)))) -> a(s(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(3) DependencyPairsProof (EQUIVALENT)
29.05/8.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(4)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(a(x1)) -> A(s(x1))
29.05/8.33	   S(a(x1)) -> S(x1)
29.05/8.33	   S(b(a(b(x1)))) -> A(s(b(a(x1))))
29.05/8.33	   S(b(a(b(x1)))) -> S(b(a(x1)))
29.05/8.33	   S(b(a(b(x1)))) -> B(a(x1))
29.05/8.33	   S(b(a(b(x1)))) -> A(x1)
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	   S(c(x1)) -> A(b(a(x1)))
29.05/8.33	   S(c(x1)) -> B(a(x1))
29.05/8.33	   S(c(x1)) -> A(x1)
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	   A(a(b(a(x1)))) -> B(a(b(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(x1))
29.05/8.33	   A(a(b(a(x1)))) -> B(x1)
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   s(a(x1)) -> a(s(x1))
29.05/8.33	   s(b(a(b(x1)))) -> a(s(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(5) DependencyGraphProof (EQUIVALENT)
29.05/8.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(6)
29.05/8.33	Complex Obligation (AND)
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(7)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	   S(c(x1)) -> A(b(a(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	   A(a(b(a(x1)))) -> B(a(b(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(x1))
29.05/8.33	   A(a(b(a(x1)))) -> B(x1)
29.05/8.33	   S(c(x1)) -> B(a(x1))
29.05/8.33	   S(c(x1)) -> A(x1)
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   s(a(x1)) -> a(s(x1))
29.05/8.33	   s(b(a(b(x1)))) -> a(s(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(8) UsableRulesProof (EQUIVALENT)
29.05/8.33	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.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(9)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	   S(c(x1)) -> A(b(a(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	   A(a(b(a(x1)))) -> B(a(b(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(x1))
29.05/8.33	   A(a(b(a(x1)))) -> B(x1)
29.05/8.33	   S(c(x1)) -> B(a(x1))
29.05/8.33	   S(c(x1)) -> A(x1)
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(10) QDPOrderProof (EQUIVALENT)
29.05/8.33	We use the reduction pair processor [LPAR04,JAR06].
29.05/8.33	
29.05/8.33	
29.05/8.33	The following pairs can be oriented strictly and are deleted.
29.05/8.33	
29.05/8.33	   S(c(x1)) -> A(b(a(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> B(a(b(x1)))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(x1))
29.05/8.33	   A(a(b(a(x1)))) -> B(x1)
29.05/8.33	   S(c(x1)) -> B(a(x1))
29.05/8.33	   S(c(x1)) -> A(x1)
29.05/8.33	The remaining pairs can at least be oriented weakly.
29.05/8.33	Used ordering:  Polynomial interpretation [POLO]:
29.05/8.33	
29.05/8.33	   POL(A(x_1)) = 2*x_1
29.05/8.33	   POL(B(x_1)) = 2*x_1
29.05/8.33	   POL(S(x_1)) = 4 + 2*x_1
29.05/8.33	   POL(a(x_1)) = 1 + x_1
29.05/8.33	   POL(b(x_1)) = 1 + x_1
29.05/8.33	   POL(c(x_1)) = 1 + x_1
29.05/8.33	   POL(s(x_1)) = 3 + x_1
29.05/8.33	
29.05/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(11)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(12) DependencyGraphProof (EQUIVALENT)
29.05/8.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(13)
29.05/8.33	Complex Obligation (AND)
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(14)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(15) QDPOrderProof (EQUIVALENT)
29.05/8.33	We use the reduction pair processor [LPAR04,JAR06].
29.05/8.33	
29.05/8.33	
29.05/8.33	The following pairs can be oriented strictly and are deleted.
29.05/8.33	
29.05/8.33	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
29.05/8.33	The remaining pairs can at least be oriented weakly.
29.05/8.33	Used ordering:  Polynomial interpretation [POLO]:
29.05/8.33	
29.05/8.33	   POL(A(x_1)) = x_1
29.05/8.33	   POL(a(x_1)) = 1
29.05/8.33	   POL(b(x_1)) = 0
29.05/8.33	   POL(c(x_1)) = 0
29.05/8.33	   POL(s(x_1)) = x_1
29.05/8.33	
29.05/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.05/8.33	
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(16)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	P is empty.
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(17) PisEmptyProof (EQUIVALENT)
29.05/8.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(18)
29.05/8.33	YES
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(19)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(20) QDPOrderProof (EQUIVALENT)
29.05/8.33	We use the reduction pair processor [LPAR04,JAR06].
29.05/8.33	
29.05/8.33	
29.05/8.33	The following pairs can be oriented strictly and are deleted.
29.05/8.33	
29.05/8.33	   S(c(x1)) -> B(a(b(a(x1))))
29.05/8.33	The remaining pairs can at least be oriented weakly.
29.05/8.33	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
29.05/8.33	
29.05/8.33	POL( B_1(x_1) ) = x_1 + 2
29.05/8.33	POL( a_1(x_1) ) = 0
29.05/8.33	POL( b_1(x_1) ) = 1
29.05/8.33	POL( s_1(x_1) ) = 2x_1
29.05/8.33	POL( c_1(x_1) ) = 2
29.05/8.33	POL( S_1(x_1) ) = max{0, 2x_1 - 1}
29.05/8.33	
29.05/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(21)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   B(b(a(b(x1)))) -> S(c(x1))
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(22) DependencyGraphProof (EQUIVALENT)
29.05/8.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(23)
29.05/8.33	TRUE
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(24)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(b(a(b(x1)))) -> S(b(a(x1)))
29.05/8.33	   S(a(x1)) -> S(x1)
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   s(a(x1)) -> a(s(x1))
29.05/8.33	   s(b(a(b(x1)))) -> a(s(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(25) UsableRulesProof (EQUIVALENT)
29.05/8.33	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.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(26)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(b(a(b(x1)))) -> S(b(a(x1)))
29.05/8.33	   S(a(x1)) -> S(x1)
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(27) QDPOrderProof (EQUIVALENT)
29.05/8.33	We use the reduction pair processor [LPAR04,JAR06].
29.05/8.33	
29.05/8.33	
29.05/8.33	The following pairs can be oriented strictly and are deleted.
29.05/8.33	
29.05/8.33	   S(a(x1)) -> S(x1)
29.05/8.33	The remaining pairs can at least be oriented weakly.
29.05/8.33	Used ordering:  Polynomial interpretation [POLO]:
29.05/8.33	
29.05/8.33	   POL(S(x_1)) = x_1
29.05/8.33	   POL(a(x_1)) = 1 + x_1
29.05/8.33	   POL(b(x_1)) = 1
29.05/8.33	   POL(c(x_1)) = 1
29.05/8.33	   POL(s(x_1)) = x_1
29.05/8.33	
29.05/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.05/8.33	
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(28)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	The TRS P consists of the following rules:
29.05/8.33	
29.05/8.33	   S(b(a(b(x1)))) -> S(b(a(x1)))
29.05/8.33	
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(29) MRRProof (EQUIVALENT)
29.05/8.33	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
29.05/8.33	
29.05/8.33	Strictly oriented dependency pairs:
29.05/8.33	
29.05/8.33	   S(b(a(b(x1)))) -> S(b(a(x1)))
29.05/8.33	
29.05/8.33	
29.05/8.33	Used ordering: Polynomial interpretation [POLO]:
29.05/8.33	
29.05/8.33	   POL(S(x_1)) = 2*x_1
29.05/8.33	   POL(a(x_1)) = 1 + x_1
29.05/8.33	   POL(b(x_1)) = 1 + x_1
29.05/8.33	   POL(c(x_1)) = 1 + x_1
29.05/8.33	   POL(s(x_1)) = 3 + x_1
29.05/8.33	
29.05/8.33	
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(30)
29.05/8.33	Obligation:
29.05/8.33	Q DP problem:
29.05/8.33	P is empty.
29.05/8.33	The TRS R consists of the following rules:
29.05/8.33	
29.05/8.33	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
29.05/8.33	   s(c(x1)) -> b(a(b(a(x1))))
29.05/8.33	   b(b(a(b(x1)))) -> s(c(x1))
29.05/8.33	
29.05/8.33	Q is empty.
29.05/8.33	We have to consider all minimal (P,Q,R)-chains.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(31) PisEmptyProof (EQUIVALENT)
29.05/8.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.05/8.33	----------------------------------------
29.05/8.33	
29.05/8.33	(32)
29.05/8.33	YES
29.19/8.42	EOF