8.18/3.37	YES
8.18/3.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.18/3.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.18/3.38	
8.18/3.38	
8.18/3.38	Termination w.r.t. Q of the given QTRS could be proven:
8.18/3.38	
8.18/3.38	(0) QTRS
8.18/3.38	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
8.18/3.38	(2) QDP
8.18/3.38	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
8.18/3.38	(4) AND
8.18/3.38	    (5) QDP
8.18/3.38	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
8.18/3.38	        (7) QDP
8.18/3.38	        (8) QReductionProof [EQUIVALENT, 0 ms]
8.18/3.38	        (9) QDP
8.18/3.38	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.18/3.38	        (11) YES
8.18/3.38	    (12) QDP
8.18/3.38	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
8.18/3.38	        (14) QDP
8.18/3.38	        (15) QReductionProof [EQUIVALENT, 0 ms]
8.18/3.38	        (16) QDP
8.18/3.38	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.18/3.38	        (18) YES
8.18/3.38	    (19) QDP
8.18/3.38	        (20) UsableRulesProof [EQUIVALENT, 0 ms]
8.18/3.38	        (21) QDP
8.18/3.38	        (22) QReductionProof [EQUIVALENT, 0 ms]
8.18/3.38	        (23) QDP
8.18/3.38	        (24) TransformationProof [EQUIVALENT, 0 ms]
8.18/3.38	        (25) QDP
8.18/3.38	        (26) DependencyGraphProof [EQUIVALENT, 0 ms]
8.18/3.38	        (27) QDP
8.18/3.38	        (28) TransformationProof [EQUIVALENT, 0 ms]
8.18/3.38	        (29) QDP
8.18/3.38	        (30) DependencyGraphProof [EQUIVALENT, 0 ms]
8.18/3.38	        (31) QDP
8.18/3.38	        (32) QDPOrderProof [EQUIVALENT, 0 ms]
8.18/3.38	        (33) QDP
8.18/3.38	        (34) DependencyGraphProof [EQUIVALENT, 0 ms]
8.18/3.38	        (35) TRUE
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(0)
8.18/3.38	Obligation:
8.18/3.38	Q restricted rewrite system:
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
8.18/3.38	   if_quot(true, x, y) -> s(quot(minus(x, y), y))
8.18/3.38	   if_quot(false, x, y) -> 0
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(1) DependencyPairsProof (EQUIVALENT)
8.18/3.38	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(2)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   MINUS(s(x), s(y)) -> MINUS(x, y)
8.18/3.38	   LE(s(x), s(y)) -> LE(x, y)
8.18/3.38	   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
8.18/3.38	   QUOT(x, s(y)) -> LE(s(y), x)
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	   IF_QUOT(true, x, y) -> MINUS(x, y)
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
8.18/3.38	   if_quot(true, x, y) -> s(quot(minus(x, y), y))
8.18/3.38	   if_quot(false, x, y) -> 0
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(3) DependencyGraphProof (EQUIVALENT)
8.18/3.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(4)
8.18/3.38	Complex Obligation (AND)
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(5)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   LE(s(x), s(y)) -> LE(x, y)
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
8.18/3.38	   if_quot(true, x, y) -> s(quot(minus(x, y), y))
8.18/3.38	   if_quot(false, x, y) -> 0
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(6) UsableRulesProof (EQUIVALENT)
8.18/3.38	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(7)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   LE(s(x), s(y)) -> LE(x, y)
8.18/3.38	
8.18/3.38	R is empty.
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(8) QReductionProof (EQUIVALENT)
8.18/3.38	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(9)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   LE(s(x), s(y)) -> LE(x, y)
8.18/3.38	
8.18/3.38	R is empty.
8.18/3.38	Q is empty.
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(10) QDPSizeChangeProof (EQUIVALENT)
8.18/3.38	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. 
8.18/3.38	
8.18/3.38	From the DPs we obtained the following set of size-change graphs:
8.18/3.38	*LE(s(x), s(y)) -> LE(x, y)
8.18/3.38	The graph contains the following edges 1 > 1, 2 > 2
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(11)
8.18/3.38	YES
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(12)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   MINUS(s(x), s(y)) -> MINUS(x, y)
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
8.18/3.38	   if_quot(true, x, y) -> s(quot(minus(x, y), y))
8.18/3.38	   if_quot(false, x, y) -> 0
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(13) UsableRulesProof (EQUIVALENT)
8.18/3.38	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(14)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   MINUS(s(x), s(y)) -> MINUS(x, y)
8.18/3.38	
8.18/3.38	R is empty.
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(15) QReductionProof (EQUIVALENT)
8.18/3.38	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(16)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   MINUS(s(x), s(y)) -> MINUS(x, y)
8.18/3.38	
8.18/3.38	R is empty.
8.18/3.38	Q is empty.
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(17) QDPSizeChangeProof (EQUIVALENT)
8.18/3.38	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. 
8.18/3.38	
8.18/3.38	From the DPs we obtained the following set of size-change graphs:
8.18/3.38	*MINUS(s(x), s(y)) -> MINUS(x, y)
8.18/3.38	The graph contains the following edges 1 > 1, 2 > 2
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(18)
8.18/3.38	YES
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(19)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
8.18/3.38	   if_quot(true, x, y) -> s(quot(minus(x, y), y))
8.18/3.38	   if_quot(false, x, y) -> 0
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(20) UsableRulesProof (EQUIVALENT)
8.18/3.38	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(21)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(22) QReductionProof (EQUIVALENT)
8.18/3.38	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
8.18/3.38	
8.18/3.38	   quot(x0, s(x1))
8.18/3.38	   if_quot(true, x0, x1)
8.18/3.38	   if_quot(false, x0, x1)
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(23)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	   QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(24) TransformationProof (EQUIVALENT)
8.18/3.38	By narrowing [LPAR04] the rule QUOT(x, s(y)) -> IF_QUOT(le(s(y), x), x, s(y)) at position [0] we obtained the following new rules [LPAR04]:
8.18/3.38	
8.18/3.38	   (QUOT(0, s(x0)) -> IF_QUOT(false, 0, s(x0)),QUOT(0, s(x0)) -> IF_QUOT(false, 0, s(x0)))
8.18/3.38	   (QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0)),QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0)))
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(25)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	   QUOT(0, s(x0)) -> IF_QUOT(false, 0, s(x0))
8.18/3.38	   QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(26) DependencyGraphProof (EQUIVALENT)
8.18/3.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(27)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0))
8.18/3.38	   IF_QUOT(true, x, y) -> QUOT(minus(x, y), y)
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(28) TransformationProof (EQUIVALENT)
8.18/3.38	By narrowing [LPAR04] the rule IF_QUOT(true, x, y) -> QUOT(minus(x, y), y) at position [0] we obtained the following new rules [LPAR04]:
8.18/3.38	
8.18/3.38	   (IF_QUOT(true, x0, 0) -> QUOT(x0, 0),IF_QUOT(true, x0, 0) -> QUOT(x0, 0))
8.18/3.38	   (IF_QUOT(true, s(x0), s(x1)) -> QUOT(minus(x0, x1), s(x1)),IF_QUOT(true, s(x0), s(x1)) -> QUOT(minus(x0, x1), s(x1)))
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(29)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0))
8.18/3.38	   IF_QUOT(true, x0, 0) -> QUOT(x0, 0)
8.18/3.38	   IF_QUOT(true, s(x0), s(x1)) -> QUOT(minus(x0, x1), s(x1))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(30) DependencyGraphProof (EQUIVALENT)
8.18/3.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(31)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   IF_QUOT(true, s(x0), s(x1)) -> QUOT(minus(x0, x1), s(x1))
8.18/3.38	   QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(32) QDPOrderProof (EQUIVALENT)
8.18/3.38	We use the reduction pair processor [LPAR04,JAR06].
8.18/3.38	
8.18/3.38	
8.18/3.38	The following pairs can be oriented strictly and are deleted.
8.18/3.38	
8.18/3.38	   IF_QUOT(true, s(x0), s(x1)) -> QUOT(minus(x0, x1), s(x1))
8.18/3.38	The remaining pairs can at least be oriented weakly.
8.18/3.38	Used ordering:  Combined order from the following AFS and order.
8.18/3.38	IF_QUOT(x1, x2, x3)  =  x2
8.18/3.38	
8.18/3.38	s(x1)  =  s(x1)
8.18/3.38	
8.18/3.38	QUOT(x1, x2)  =  x1
8.18/3.38	
8.18/3.38	minus(x1, x2)  =  x1
8.18/3.38	
8.18/3.38	
8.18/3.38	Knuth-Bendix order [KBO] with precedence:trivial
8.18/3.38	
8.18/3.38	and weight map:
8.18/3.38	
8.18/3.38	   s_1=1
8.18/3.38	   dummyConstant=1
8.18/3.38	
8.18/3.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
8.18/3.38	
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(33)
8.18/3.38	Obligation:
8.18/3.38	Q DP problem:
8.18/3.38	The TRS P consists of the following rules:
8.18/3.38	
8.18/3.38	   QUOT(s(x1), s(x0)) -> IF_QUOT(le(x0, x1), s(x1), s(x0))
8.18/3.38	
8.18/3.38	The TRS R consists of the following rules:
8.18/3.38	
8.18/3.38	   le(s(x), 0) -> false
8.18/3.38	   le(s(x), s(y)) -> le(x, y)
8.18/3.38	   le(0, y) -> true
8.18/3.38	   minus(x, 0) -> x
8.18/3.38	   minus(s(x), s(y)) -> minus(x, y)
8.18/3.38	
8.18/3.38	The set Q consists of the following terms:
8.18/3.38	
8.18/3.38	   minus(x0, 0)
8.18/3.38	   minus(s(x0), s(x1))
8.18/3.38	   le(0, x0)
8.18/3.38	   le(s(x0), 0)
8.18/3.38	   le(s(x0), s(x1))
8.18/3.38	
8.18/3.38	We have to consider all minimal (P,Q,R)-chains.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(34) DependencyGraphProof (EQUIVALENT)
8.18/3.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
8.18/3.38	----------------------------------------
8.18/3.38	
8.18/3.38	(35)
8.18/3.38	TRUE
8.18/3.41	EOF