23.18/6.83	YES
23.50/6.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.50/6.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.50/6.89	
23.50/6.89	
23.50/6.89	Termination w.r.t. Q of the given QTRS could be proven:
23.50/6.89	
23.50/6.89	(0) QTRS
23.50/6.89	(1) DependencyPairsProof [EQUIVALENT, 7 ms]
23.50/6.89	(2) QDP
23.50/6.89	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
23.50/6.89	(4) AND
23.50/6.89	    (5) QDP
23.50/6.89	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
23.50/6.89	        (7) QDP
23.50/6.89	        (8) QDPOrderProof [EQUIVALENT, 5 ms]
23.50/6.89	        (9) QDP
23.50/6.89	        (10) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (11) YES
23.50/6.89	    (12) QDP
23.50/6.89	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
23.50/6.89	        (14) QDP
23.50/6.89	        (15) QDPOrderProof [EQUIVALENT, 6 ms]
23.50/6.89	        (16) QDP
23.50/6.89	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (18) YES
23.50/6.89	    (19) QDP
23.50/6.89	        (20) UsableRulesProof [EQUIVALENT, 0 ms]
23.50/6.89	        (21) QDP
23.50/6.89	        (22) QDPOrderProof [EQUIVALENT, 2 ms]
23.50/6.89	        (23) QDP
23.50/6.89	        (24) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (25) YES
23.50/6.89	    (26) QDP
23.50/6.89	        (27) UsableRulesProof [EQUIVALENT, 3 ms]
23.50/6.89	        (28) QDP
23.50/6.89	        (29) QDPOrderProof [EQUIVALENT, 5 ms]
23.50/6.89	        (30) QDP
23.50/6.89	        (31) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (32) YES
23.50/6.89	    (33) QDP
23.50/6.89	        (34) UsableRulesProof [EQUIVALENT, 0 ms]
23.50/6.89	        (35) QDP
23.50/6.89	        (36) QDPOrderProof [EQUIVALENT, 5 ms]
23.50/6.89	        (37) QDP
23.50/6.89	        (38) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (39) YES
23.50/6.89	    (40) QDP
23.50/6.89	        (41) QDPOrderProof [EQUIVALENT, 86 ms]
23.50/6.89	        (42) QDP
23.50/6.89	        (43) PisEmptyProof [EQUIVALENT, 0 ms]
23.50/6.89	        (44) YES
23.50/6.89	
23.50/6.89	
23.50/6.89	----------------------------------------
23.50/6.89	
23.50/6.89	(0)
23.50/6.89	Obligation:
23.50/6.89	Q restricted rewrite system:
23.50/6.89	The TRS R consists of the following rules:
23.50/6.89	
23.50/6.89	   r0(0(x1)) -> 0(r0(x1))
23.50/6.89	   r0(1(x1)) -> 1(r0(x1))
23.50/6.89	   r0(m(x1)) -> m(r0(x1))
23.50/6.89	   r1(0(x1)) -> 0(r1(x1))
23.50/6.89	   r1(1(x1)) -> 1(r1(x1))
23.50/6.89	   r1(m(x1)) -> m(r1(x1))
23.50/6.89	   r0(b(x1)) -> qr(0(b(x1)))
23.50/6.89	   r1(b(x1)) -> qr(1(b(x1)))
23.50/6.89	   0(qr(x1)) -> qr(0(x1))
23.50/6.89	   1(qr(x1)) -> qr(1(x1))
23.50/6.89	   m(qr(x1)) -> ql(m(x1))
23.50/6.89	   0(ql(x1)) -> ql(0(x1))
23.50/6.89	   1(ql(x1)) -> ql(1(x1))
23.50/6.89	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.50/6.89	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.50/6.89	
23.50/6.89	Q is empty.
23.50/6.89	
23.50/6.89	----------------------------------------
23.50/6.89	
23.50/6.89	(1) DependencyPairsProof (EQUIVALENT)
23.50/6.89	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.50/6.89	----------------------------------------
23.50/6.89	
23.50/6.89	(2)
23.50/6.89	Obligation:
23.50/6.89	Q DP problem:
23.50/6.89	The TRS P consists of the following rules:
23.50/6.89	
23.50/6.89	   R0(0(x1)) -> 0^1(r0(x1))
23.50/6.89	   R0(0(x1)) -> R0(x1)
23.50/6.89	   R0(1(x1)) -> 1^1(r0(x1))
23.50/6.89	   R0(1(x1)) -> R0(x1)
23.50/6.89	   R0(m(x1)) -> M(r0(x1))
23.50/6.89	   R0(m(x1)) -> R0(x1)
23.50/6.89	   R1(0(x1)) -> 0^1(r1(x1))
23.50/6.89	   R1(0(x1)) -> R1(x1)
23.50/6.89	   R1(1(x1)) -> 1^1(r1(x1))
23.50/6.89	   R1(1(x1)) -> R1(x1)
23.50/6.89	   R1(m(x1)) -> M(r1(x1))
23.50/6.89	   R1(m(x1)) -> R1(x1)
23.50/6.89	   R0(b(x1)) -> 0^1(b(x1))
23.50/6.89	   R1(b(x1)) -> 1^1(b(x1))
23.50/6.89	   0^1(qr(x1)) -> 0^1(x1)
23.50/6.89	   1^1(qr(x1)) -> 1^1(x1)
23.50/6.89	   M(qr(x1)) -> M(x1)
23.50/6.89	   0^1(ql(x1)) -> 0^1(x1)
23.50/6.89	   1^1(ql(x1)) -> 1^1(x1)
23.50/6.89	   B(ql(0(x1))) -> 0^1(b(r0(x1)))
23.50/6.89	   B(ql(0(x1))) -> B(r0(x1))
23.50/6.89	   B(ql(0(x1))) -> R0(x1)
23.50/6.89	   B(ql(1(x1))) -> 1^1(b(r1(x1)))
23.50/6.89	   B(ql(1(x1))) -> B(r1(x1))
23.50/6.89	   B(ql(1(x1))) -> R1(x1)
23.50/6.89	
23.50/6.89	The TRS R consists of the following rules:
23.50/6.89	
23.50/6.89	   r0(0(x1)) -> 0(r0(x1))
23.50/6.89	   r0(1(x1)) -> 1(r0(x1))
23.55/6.89	   r0(m(x1)) -> m(r0(x1))
23.55/6.89	   r1(0(x1)) -> 0(r1(x1))
23.55/6.89	   r1(1(x1)) -> 1(r1(x1))
23.55/6.89	   r1(m(x1)) -> m(r1(x1))
23.55/6.89	   r0(b(x1)) -> qr(0(b(x1)))
23.55/6.89	   r1(b(x1)) -> qr(1(b(x1)))
23.55/6.89	   0(qr(x1)) -> qr(0(x1))
23.55/6.89	   1(qr(x1)) -> qr(1(x1))
23.55/6.89	   m(qr(x1)) -> ql(m(x1))
23.55/6.89	   0(ql(x1)) -> ql(0(x1))
23.55/6.89	   1(ql(x1)) -> ql(1(x1))
23.55/6.89	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.55/6.89	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.55/6.89	
23.55/6.89	Q is empty.
23.55/6.89	We have to consider all minimal (P,Q,R)-chains.
23.55/6.89	----------------------------------------
23.55/6.89	
23.55/6.89	(3) DependencyGraphProof (EQUIVALENT)
23.55/6.89	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 12 less nodes.
23.55/6.89	----------------------------------------
23.55/6.89	
23.55/6.89	(4)
23.55/6.89	Complex Obligation (AND)
23.55/6.89	
23.55/6.89	----------------------------------------
23.55/6.89	
23.55/6.89	(5)
23.55/6.89	Obligation:
23.55/6.89	Q DP problem:
23.55/6.89	The TRS P consists of the following rules:
23.55/6.89	
23.55/6.89	   M(qr(x1)) -> M(x1)
23.55/6.89	
23.55/6.89	The TRS R consists of the following rules:
23.55/6.89	
23.55/6.89	   r0(0(x1)) -> 0(r0(x1))
23.55/6.89	   r0(1(x1)) -> 1(r0(x1))
23.55/6.89	   r0(m(x1)) -> m(r0(x1))
23.55/6.89	   r1(0(x1)) -> 0(r1(x1))
23.55/6.89	   r1(1(x1)) -> 1(r1(x1))
23.55/6.89	   r1(m(x1)) -> m(r1(x1))
23.55/6.89	   r0(b(x1)) -> qr(0(b(x1)))
23.55/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.55/6.91	   0(qr(x1)) -> qr(0(x1))
23.55/6.91	   1(qr(x1)) -> qr(1(x1))
23.55/6.91	   m(qr(x1)) -> ql(m(x1))
23.55/6.91	   0(ql(x1)) -> ql(0(x1))
23.55/6.91	   1(ql(x1)) -> ql(1(x1))
23.55/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.55/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.55/6.91	
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(6) UsableRulesProof (EQUIVALENT)
23.55/6.91	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.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(7)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   M(qr(x1)) -> M(x1)
23.55/6.91	
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(8) QDPOrderProof (EQUIVALENT)
23.55/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.55/6.91	
23.55/6.91	
23.55/6.91	The following pairs can be oriented strictly and are deleted.
23.55/6.91	
23.55/6.91	   M(qr(x1)) -> M(x1)
23.55/6.91	The remaining pairs can at least be oriented weakly.
23.55/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.55/6.91	
23.55/6.91	POL( M_1(x_1) ) = 2x_1
23.55/6.91	POL( qr_1(x_1) ) = x_1 + 2
23.55/6.91	
23.55/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.55/6.91	none
23.55/6.91	
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(9)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	P is empty.
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(10) PisEmptyProof (EQUIVALENT)
23.55/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(11)
23.55/6.91	YES
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(12)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   1^1(ql(x1)) -> 1^1(x1)
23.55/6.91	   1^1(qr(x1)) -> 1^1(x1)
23.55/6.91	
23.55/6.91	The TRS R consists of the following rules:
23.55/6.91	
23.55/6.91	   r0(0(x1)) -> 0(r0(x1))
23.55/6.91	   r0(1(x1)) -> 1(r0(x1))
23.55/6.91	   r0(m(x1)) -> m(r0(x1))
23.55/6.91	   r1(0(x1)) -> 0(r1(x1))
23.55/6.91	   r1(1(x1)) -> 1(r1(x1))
23.55/6.91	   r1(m(x1)) -> m(r1(x1))
23.55/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.55/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.55/6.91	   0(qr(x1)) -> qr(0(x1))
23.55/6.91	   1(qr(x1)) -> qr(1(x1))
23.55/6.91	   m(qr(x1)) -> ql(m(x1))
23.55/6.91	   0(ql(x1)) -> ql(0(x1))
23.55/6.91	   1(ql(x1)) -> ql(1(x1))
23.55/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.55/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.55/6.91	
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(13) UsableRulesProof (EQUIVALENT)
23.55/6.91	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.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(14)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   1^1(ql(x1)) -> 1^1(x1)
23.55/6.91	   1^1(qr(x1)) -> 1^1(x1)
23.55/6.91	
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(15) QDPOrderProof (EQUIVALENT)
23.55/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.55/6.91	
23.55/6.91	
23.55/6.91	The following pairs can be oriented strictly and are deleted.
23.55/6.91	
23.55/6.91	   1^1(ql(x1)) -> 1^1(x1)
23.55/6.91	   1^1(qr(x1)) -> 1^1(x1)
23.55/6.91	The remaining pairs can at least be oriented weakly.
23.55/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.55/6.91	
23.55/6.91	POL( 1^1_1(x_1) ) = 2x_1
23.55/6.91	POL( ql_1(x_1) ) = 2x_1 + 1
23.55/6.91	POL( qr_1(x_1) ) = x_1 + 1
23.55/6.91	
23.55/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.55/6.91	none
23.55/6.91	
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(16)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	P is empty.
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(17) PisEmptyProof (EQUIVALENT)
23.55/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(18)
23.55/6.91	YES
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(19)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   0^1(ql(x1)) -> 0^1(x1)
23.55/6.91	   0^1(qr(x1)) -> 0^1(x1)
23.55/6.91	
23.55/6.91	The TRS R consists of the following rules:
23.55/6.91	
23.55/6.91	   r0(0(x1)) -> 0(r0(x1))
23.55/6.91	   r0(1(x1)) -> 1(r0(x1))
23.55/6.91	   r0(m(x1)) -> m(r0(x1))
23.55/6.91	   r1(0(x1)) -> 0(r1(x1))
23.55/6.91	   r1(1(x1)) -> 1(r1(x1))
23.55/6.91	   r1(m(x1)) -> m(r1(x1))
23.55/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.55/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.55/6.91	   0(qr(x1)) -> qr(0(x1))
23.55/6.91	   1(qr(x1)) -> qr(1(x1))
23.55/6.91	   m(qr(x1)) -> ql(m(x1))
23.55/6.91	   0(ql(x1)) -> ql(0(x1))
23.55/6.91	   1(ql(x1)) -> ql(1(x1))
23.55/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.55/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.55/6.91	
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(20) UsableRulesProof (EQUIVALENT)
23.55/6.91	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.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(21)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   0^1(ql(x1)) -> 0^1(x1)
23.55/6.91	   0^1(qr(x1)) -> 0^1(x1)
23.55/6.91	
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(22) QDPOrderProof (EQUIVALENT)
23.55/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.55/6.91	
23.55/6.91	
23.55/6.91	The following pairs can be oriented strictly and are deleted.
23.55/6.91	
23.55/6.91	   0^1(ql(x1)) -> 0^1(x1)
23.55/6.91	   0^1(qr(x1)) -> 0^1(x1)
23.55/6.91	The remaining pairs can at least be oriented weakly.
23.55/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.55/6.91	
23.55/6.91	POL( 0^1_1(x_1) ) = 2x_1
23.55/6.91	POL( ql_1(x_1) ) = 2x_1 + 1
23.55/6.91	POL( qr_1(x_1) ) = x_1 + 1
23.55/6.91	
23.55/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.55/6.91	none
23.55/6.91	
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(23)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	P is empty.
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(24) PisEmptyProof (EQUIVALENT)
23.55/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(25)
23.55/6.91	YES
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(26)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   R1(1(x1)) -> R1(x1)
23.55/6.91	   R1(0(x1)) -> R1(x1)
23.55/6.91	   R1(m(x1)) -> R1(x1)
23.55/6.91	
23.55/6.91	The TRS R consists of the following rules:
23.55/6.91	
23.55/6.91	   r0(0(x1)) -> 0(r0(x1))
23.55/6.91	   r0(1(x1)) -> 1(r0(x1))
23.55/6.91	   r0(m(x1)) -> m(r0(x1))
23.55/6.91	   r1(0(x1)) -> 0(r1(x1))
23.55/6.91	   r1(1(x1)) -> 1(r1(x1))
23.55/6.91	   r1(m(x1)) -> m(r1(x1))
23.55/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.55/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.55/6.91	   0(qr(x1)) -> qr(0(x1))
23.55/6.91	   1(qr(x1)) -> qr(1(x1))
23.55/6.91	   m(qr(x1)) -> ql(m(x1))
23.55/6.91	   0(ql(x1)) -> ql(0(x1))
23.55/6.91	   1(ql(x1)) -> ql(1(x1))
23.55/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.55/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.55/6.91	
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(27) UsableRulesProof (EQUIVALENT)
23.55/6.91	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.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(28)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	The TRS P consists of the following rules:
23.55/6.91	
23.55/6.91	   R1(1(x1)) -> R1(x1)
23.55/6.91	   R1(0(x1)) -> R1(x1)
23.55/6.91	   R1(m(x1)) -> R1(x1)
23.55/6.91	
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(29) QDPOrderProof (EQUIVALENT)
23.55/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.55/6.91	
23.55/6.91	
23.55/6.91	The following pairs can be oriented strictly and are deleted.
23.55/6.91	
23.55/6.91	   R1(1(x1)) -> R1(x1)
23.55/6.91	   R1(0(x1)) -> R1(x1)
23.55/6.91	   R1(m(x1)) -> R1(x1)
23.55/6.91	The remaining pairs can at least be oriented weakly.
23.55/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.55/6.91	
23.55/6.91	POL( R1_1(x_1) ) = x_1
23.55/6.91	POL( 1_1(x_1) ) = 2x_1 + 2
23.55/6.91	POL( 0_1(x_1) ) = 2x_1 + 2
23.55/6.91	POL( m_1(x_1) ) = 2x_1 + 2
23.55/6.91	
23.55/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.55/6.91	none
23.55/6.91	
23.55/6.91	
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(30)
23.55/6.91	Obligation:
23.55/6.91	Q DP problem:
23.55/6.91	P is empty.
23.55/6.91	R is empty.
23.55/6.91	Q is empty.
23.55/6.91	We have to consider all minimal (P,Q,R)-chains.
23.55/6.91	----------------------------------------
23.55/6.91	
23.55/6.91	(31) PisEmptyProof (EQUIVALENT)
23.55/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.55/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(32)
23.64/6.91	YES
23.64/6.91	
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(33)
23.64/6.91	Obligation:
23.64/6.91	Q DP problem:
23.64/6.91	The TRS P consists of the following rules:
23.64/6.91	
23.64/6.91	   R0(1(x1)) -> R0(x1)
23.64/6.91	   R0(0(x1)) -> R0(x1)
23.64/6.91	   R0(m(x1)) -> R0(x1)
23.64/6.91	
23.64/6.91	The TRS R consists of the following rules:
23.64/6.91	
23.64/6.91	   r0(0(x1)) -> 0(r0(x1))
23.64/6.91	   r0(1(x1)) -> 1(r0(x1))
23.64/6.91	   r0(m(x1)) -> m(r0(x1))
23.64/6.91	   r1(0(x1)) -> 0(r1(x1))
23.64/6.91	   r1(1(x1)) -> 1(r1(x1))
23.64/6.91	   r1(m(x1)) -> m(r1(x1))
23.64/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.64/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.64/6.91	   0(qr(x1)) -> qr(0(x1))
23.64/6.91	   1(qr(x1)) -> qr(1(x1))
23.64/6.91	   m(qr(x1)) -> ql(m(x1))
23.64/6.91	   0(ql(x1)) -> ql(0(x1))
23.64/6.91	   1(ql(x1)) -> ql(1(x1))
23.64/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.64/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.64/6.91	
23.64/6.91	Q is empty.
23.64/6.91	We have to consider all minimal (P,Q,R)-chains.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(34) UsableRulesProof (EQUIVALENT)
23.64/6.91	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.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(35)
23.64/6.91	Obligation:
23.64/6.91	Q DP problem:
23.64/6.91	The TRS P consists of the following rules:
23.64/6.91	
23.64/6.91	   R0(1(x1)) -> R0(x1)
23.64/6.91	   R0(0(x1)) -> R0(x1)
23.64/6.91	   R0(m(x1)) -> R0(x1)
23.64/6.91	
23.64/6.91	R is empty.
23.64/6.91	Q is empty.
23.64/6.91	We have to consider all minimal (P,Q,R)-chains.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(36) QDPOrderProof (EQUIVALENT)
23.64/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.64/6.91	
23.64/6.91	
23.64/6.91	The following pairs can be oriented strictly and are deleted.
23.64/6.91	
23.64/6.91	   R0(1(x1)) -> R0(x1)
23.64/6.91	   R0(0(x1)) -> R0(x1)
23.64/6.91	   R0(m(x1)) -> R0(x1)
23.64/6.91	The remaining pairs can at least be oriented weakly.
23.64/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.64/6.91	
23.64/6.91	POL( R0_1(x_1) ) = x_1
23.64/6.91	POL( 1_1(x_1) ) = 2x_1 + 2
23.64/6.91	POL( 0_1(x_1) ) = 2x_1 + 2
23.64/6.91	POL( m_1(x_1) ) = 2x_1 + 2
23.64/6.91	
23.64/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.64/6.91	none
23.64/6.91	
23.64/6.91	
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(37)
23.64/6.91	Obligation:
23.64/6.91	Q DP problem:
23.64/6.91	P is empty.
23.64/6.91	R is empty.
23.64/6.91	Q is empty.
23.64/6.91	We have to consider all minimal (P,Q,R)-chains.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(38) PisEmptyProof (EQUIVALENT)
23.64/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(39)
23.64/6.91	YES
23.64/6.91	
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(40)
23.64/6.91	Obligation:
23.64/6.91	Q DP problem:
23.64/6.91	The TRS P consists of the following rules:
23.64/6.91	
23.64/6.91	   B(ql(1(x1))) -> B(r1(x1))
23.64/6.91	   B(ql(0(x1))) -> B(r0(x1))
23.64/6.91	
23.64/6.91	The TRS R consists of the following rules:
23.64/6.91	
23.64/6.91	   r0(0(x1)) -> 0(r0(x1))
23.64/6.91	   r0(1(x1)) -> 1(r0(x1))
23.64/6.91	   r0(m(x1)) -> m(r0(x1))
23.64/6.91	   r1(0(x1)) -> 0(r1(x1))
23.64/6.91	   r1(1(x1)) -> 1(r1(x1))
23.64/6.91	   r1(m(x1)) -> m(r1(x1))
23.64/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.64/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.64/6.91	   0(qr(x1)) -> qr(0(x1))
23.64/6.91	   1(qr(x1)) -> qr(1(x1))
23.64/6.91	   m(qr(x1)) -> ql(m(x1))
23.64/6.91	   0(ql(x1)) -> ql(0(x1))
23.64/6.91	   1(ql(x1)) -> ql(1(x1))
23.64/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.64/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.64/6.91	
23.64/6.91	Q is empty.
23.64/6.91	We have to consider all minimal (P,Q,R)-chains.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(41) QDPOrderProof (EQUIVALENT)
23.64/6.91	We use the reduction pair processor [LPAR04,JAR06].
23.64/6.91	
23.64/6.91	
23.64/6.91	The following pairs can be oriented strictly and are deleted.
23.64/6.91	
23.64/6.91	   B(ql(1(x1))) -> B(r1(x1))
23.64/6.91	   B(ql(0(x1))) -> B(r0(x1))
23.64/6.91	The remaining pairs can at least be oriented weakly.
23.64/6.91	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
23.64/6.91	
23.64/6.91	POL( B_1(x_1) ) = max{0, 2x_1 - 1}
23.64/6.91	POL( r1_1(x_1) ) = 2x_1
23.64/6.91	POL( 0_1(x_1) ) = 2x_1 + 1
23.64/6.91	POL( 1_1(x_1) ) = 2x_1 + 2
23.64/6.91	POL( m_1(x_1) ) = 2
23.64/6.91	POL( b_1(x_1) ) = 2
23.64/6.91	POL( qr_1(x_1) ) = max{0, -2}
23.64/6.91	POL( r0_1(x_1) ) = 2x_1
23.64/6.91	POL( ql_1(x_1) ) = x_1
23.64/6.91	
23.64/6.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.64/6.91	
23.64/6.91	   r1(0(x1)) -> 0(r1(x1))
23.64/6.91	   r1(1(x1)) -> 1(r1(x1))
23.64/6.91	   r1(m(x1)) -> m(r1(x1))
23.64/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.64/6.91	   r0(0(x1)) -> 0(r0(x1))
23.64/6.91	   r0(1(x1)) -> 1(r0(x1))
23.64/6.91	   r0(m(x1)) -> m(r0(x1))
23.64/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.64/6.91	   1(qr(x1)) -> qr(1(x1))
23.64/6.91	   1(ql(x1)) -> ql(1(x1))
23.64/6.91	   0(qr(x1)) -> qr(0(x1))
23.64/6.91	   0(ql(x1)) -> ql(0(x1))
23.64/6.91	   m(qr(x1)) -> ql(m(x1))
23.64/6.91	
23.64/6.91	
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(42)
23.64/6.91	Obligation:
23.64/6.91	Q DP problem:
23.64/6.91	P is empty.
23.64/6.91	The TRS R consists of the following rules:
23.64/6.91	
23.64/6.91	   r0(0(x1)) -> 0(r0(x1))
23.64/6.91	   r0(1(x1)) -> 1(r0(x1))
23.64/6.91	   r0(m(x1)) -> m(r0(x1))
23.64/6.91	   r1(0(x1)) -> 0(r1(x1))
23.64/6.91	   r1(1(x1)) -> 1(r1(x1))
23.64/6.91	   r1(m(x1)) -> m(r1(x1))
23.64/6.91	   r0(b(x1)) -> qr(0(b(x1)))
23.64/6.91	   r1(b(x1)) -> qr(1(b(x1)))
23.64/6.91	   0(qr(x1)) -> qr(0(x1))
23.64/6.91	   1(qr(x1)) -> qr(1(x1))
23.64/6.91	   m(qr(x1)) -> ql(m(x1))
23.64/6.91	   0(ql(x1)) -> ql(0(x1))
23.64/6.91	   1(ql(x1)) -> ql(1(x1))
23.64/6.91	   b(ql(0(x1))) -> 0(b(r0(x1)))
23.64/6.91	   b(ql(1(x1))) -> 1(b(r1(x1)))
23.64/6.91	
23.64/6.91	Q is empty.
23.64/6.91	We have to consider all minimal (P,Q,R)-chains.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(43) PisEmptyProof (EQUIVALENT)
23.64/6.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.64/6.91	----------------------------------------
23.64/6.91	
23.64/6.91	(44)
23.64/6.91	YES
23.69/6.96	EOF