6.01/2.54	YES
6.15/2.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.15/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.15/2.55	
6.15/2.55	
6.15/2.55	Termination w.r.t. Q of the given QTRS could be proven:
6.15/2.55	
6.15/2.55	(0) QTRS
6.15/2.55	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
6.15/2.55	(2) QDP
6.15/2.55	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
6.15/2.55	(4) AND
6.15/2.55	    (5) QDP
6.15/2.55	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
6.15/2.55	        (7) QDP
6.15/2.55	        (8) QReductionProof [EQUIVALENT, 0 ms]
6.15/2.55	        (9) QDP
6.15/2.55	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.15/2.55	        (11) YES
6.15/2.55	    (12) QDP
6.15/2.55	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
6.15/2.55	        (14) QDP
6.15/2.55	        (15) QReductionProof [EQUIVALENT, 0 ms]
6.15/2.55	        (16) QDP
6.15/2.55	        (17) TransformationProof [EQUIVALENT, 0 ms]
6.15/2.55	        (18) QDP
6.15/2.55	        (19) TransformationProof [EQUIVALENT, 0 ms]
6.15/2.55	        (20) QDP
6.15/2.55	        (21) UsableRulesProof [EQUIVALENT, 0 ms]
6.15/2.55	        (22) QDP
6.15/2.55	        (23) QReductionProof [EQUIVALENT, 0 ms]
6.15/2.55	        (24) QDP
6.15/2.55	        (25) TransformationProof [EQUIVALENT, 0 ms]
6.15/2.55	        (26) QDP
6.15/2.55	        (27) TransformationProof [EQUIVALENT, 0 ms]
6.15/2.55	        (28) QDP
6.15/2.55	        (29) TransformationProof [EQUIVALENT, 0 ms]
6.15/2.55	        (30) QDP
6.15/2.55	        (31) UsableRulesProof [EQUIVALENT, 0 ms]
6.15/2.55	        (32) QDP
6.15/2.55	        (33) QReductionProof [EQUIVALENT, 0 ms]
6.15/2.55	        (34) QDP
6.15/2.55	        (35) QDPOrderProof [EQUIVALENT, 41 ms]
6.15/2.55	        (36) QDP
6.15/2.55	        (37) QDPOrderProof [EQUIVALENT, 8 ms]
6.15/2.55	        (38) QDP
6.15/2.55	        (39) PisEmptyProof [EQUIVALENT, 0 ms]
6.15/2.55	        (40) YES
6.15/2.55	    (41) QDP
6.15/2.55	        (42) UsableRulesProof [EQUIVALENT, 0 ms]
6.15/2.55	        (43) QDP
6.15/2.55	        (44) QReductionProof [EQUIVALENT, 0 ms]
6.15/2.55	        (45) QDP
6.15/2.55	        (46) QDPOrderProof [EQUIVALENT, 28 ms]
6.15/2.55	        (47) QDP
6.15/2.55	        (48) QDPOrderProof [EQUIVALENT, 0 ms]
6.15/2.55	        (49) QDP
6.15/2.55	        (50) PisEmptyProof [EQUIVALENT, 0 ms]
6.15/2.55	        (51) YES
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(0)
6.15/2.55	Obligation:
6.15/2.55	Q restricted rewrite system:
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   quot(0, s(y), s(z)) -> 0
6.15/2.55	   quot(s(x), s(y), z) -> quot(x, y, z)
6.15/2.55	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.55	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   plus(zero, y) -> y
6.15/2.55	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
6.15/2.55	   id(x) -> x
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(1) DependencyPairsProof (EQUIVALENT)
6.15/2.55	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(2)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))
6.15/2.55	   PLUS(s(x), s(y)) -> IF(gt(x, y), x, y)
6.15/2.55	   PLUS(s(x), s(y)) -> GT(x, y)
6.15/2.55	   PLUS(s(x), s(y)) -> IF(not(gt(x, y)), id(x), id(y))
6.15/2.55	   PLUS(s(x), s(y)) -> NOT(gt(x, y))
6.15/2.55	   PLUS(s(x), s(y)) -> ID(x)
6.15/2.55	   PLUS(s(x), s(y)) -> ID(y)
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   PLUS(s(x), x) -> IF(gt(x, x), id(x), id(x))
6.15/2.55	   PLUS(s(x), x) -> GT(x, x)
6.15/2.55	   PLUS(s(x), x) -> ID(x)
6.15/2.55	   PLUS(id(x), s(y)) -> PLUS(x, if(gt(s(y), y), y, s(y)))
6.15/2.55	   PLUS(id(x), s(y)) -> IF(gt(s(y), y), y, s(y))
6.15/2.55	   PLUS(id(x), s(y)) -> GT(s(y), y)
6.15/2.55	   NOT(x) -> IF(x, false, true)
6.15/2.55	   GT(s(x), s(y)) -> GT(x, y)
6.15/2.55	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.55	   QUOT(x, 0, s(z)) -> PLUS(z, s(0))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   quot(0, s(y), s(z)) -> 0
6.15/2.55	   quot(s(x), s(y), z) -> quot(x, y, z)
6.15/2.55	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.55	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   plus(zero, y) -> y
6.15/2.55	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
6.15/2.55	   id(x) -> x
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(3) DependencyGraphProof (EQUIVALENT)
6.15/2.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 14 less nodes.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(4)
6.15/2.55	Complex Obligation (AND)
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(5)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   GT(s(x), s(y)) -> GT(x, y)
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   quot(0, s(y), s(z)) -> 0
6.15/2.55	   quot(s(x), s(y), z) -> quot(x, y, z)
6.15/2.55	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.55	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   plus(zero, y) -> y
6.15/2.55	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
6.15/2.55	   id(x) -> x
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(6) UsableRulesProof (EQUIVALENT)
6.15/2.55	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.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(7)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   GT(s(x), s(y)) -> GT(x, y)
6.15/2.55	
6.15/2.55	R is empty.
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(8) QReductionProof (EQUIVALENT)
6.15/2.55	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(9)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   GT(s(x), s(y)) -> GT(x, y)
6.15/2.55	
6.15/2.55	R is empty.
6.15/2.55	Q is empty.
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(10) QDPSizeChangeProof (EQUIVALENT)
6.15/2.55	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. 
6.15/2.55	
6.15/2.55	From the DPs we obtained the following set of size-change graphs:
6.15/2.55	*GT(s(x), s(y)) -> GT(x, y)
6.15/2.55	The graph contains the following edges 1 > 1, 2 > 2
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(11)
6.15/2.55	YES
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(12)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   quot(0, s(y), s(z)) -> 0
6.15/2.55	   quot(s(x), s(y), z) -> quot(x, y, z)
6.15/2.55	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.55	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   plus(zero, y) -> y
6.15/2.55	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
6.15/2.55	   id(x) -> x
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(13) UsableRulesProof (EQUIVALENT)
6.15/2.55	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.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(14)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(15) QReductionProof (EQUIVALENT)
6.15/2.55	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.15/2.55	
6.15/2.55	   quot(0, s(x0), s(x1))
6.15/2.55	   quot(s(x0), s(x1), x2)
6.15/2.55	   plus(s(x0), s(x1))
6.15/2.55	   plus(s(x0), x0)
6.15/2.55	   plus(zero, x0)
6.15/2.55	   quot(x0, 0, s(x1))
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(16)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(17) TransformationProof (EQUIVALENT)
6.15/2.55	By rewriting [LPAR04] the rule PLUS(s(x), x) -> PLUS(if(gt(x, x), id(x), id(x)), s(x)) at position [0,1] we obtained the following new rules [LPAR04]:
6.15/2.55	
6.15/2.55	   (PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x)),PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x)))
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(18)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(19) TransformationProof (EQUIVALENT)
6.15/2.55	By rewriting [LPAR04] the rule PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) at position [1,0] we obtained the following new rules [LPAR04]:
6.15/2.55	
6.15/2.55	   (PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y))),PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y))))
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(20)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   not(x) -> if(x, false, true)
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(21) UsableRulesProof (EQUIVALENT)
6.15/2.55	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.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(22)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   not(x0)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(23) QReductionProof (EQUIVALENT)
6.15/2.55	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.15/2.55	
6.15/2.55	   not(x0)
6.15/2.55	
6.15/2.55	
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(24)
6.15/2.55	Obligation:
6.15/2.55	Q DP problem:
6.15/2.55	The TRS P consists of the following rules:
6.15/2.55	
6.15/2.55	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x))
6.15/2.55	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y)))
6.15/2.55	
6.15/2.55	The TRS R consists of the following rules:
6.15/2.55	
6.15/2.55	   gt(s(x), zero) -> true
6.15/2.55	   gt(zero, y) -> false
6.15/2.55	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.55	   if(true, x, y) -> x
6.15/2.55	   if(false, x, y) -> y
6.15/2.55	   id(x) -> x
6.15/2.55	
6.15/2.55	The set Q consists of the following terms:
6.15/2.55	
6.15/2.55	   id(x0)
6.15/2.55	   if(true, x0, x1)
6.15/2.55	   if(false, x0, x1)
6.15/2.55	   gt(s(x0), zero)
6.15/2.55	   gt(zero, x0)
6.15/2.55	   gt(s(x0), s(x1))
6.15/2.55	
6.15/2.55	We have to consider all minimal (P,Q,R)-chains.
6.15/2.55	----------------------------------------
6.15/2.55	
6.15/2.55	(25) TransformationProof (EQUIVALENT)
6.15/2.55	By rewriting [LPAR04] the rule PLUS(s(x), x) -> PLUS(if(gt(x, x), x, id(x)), s(x)) at position [0,2] we obtained the following new rules [LPAR04]:
6.15/2.55	
6.15/2.55	   (PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x)),PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x)))
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(26)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y)))
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   id(x) -> x
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(27) TransformationProof (EQUIVALENT)
6.15/2.56	By rewriting [LPAR04] the rule PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), id(x), id(y))) at position [1,1] we obtained the following new rules [LPAR04]:
6.15/2.56	
6.15/2.56	   (PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, id(y))),PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, id(y))))
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(28)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, id(y)))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   id(x) -> x
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(29) TransformationProof (EQUIVALENT)
6.15/2.56	By rewriting [LPAR04] the rule PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, id(y))) at position [1,2] we obtained the following new rules [LPAR04]:
6.15/2.56	
6.15/2.56	   (PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y)),PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y)))
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(30)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   id(x) -> x
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(31) UsableRulesProof (EQUIVALENT)
6.15/2.56	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.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(32)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(33) QReductionProof (EQUIVALENT)
6.15/2.56	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.15/2.56	
6.15/2.56	   id(x0)
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(34)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(35) QDPOrderProof (EQUIVALENT)
6.15/2.56	We use the reduction pair processor [LPAR04,JAR06].
6.15/2.56	
6.15/2.56	
6.15/2.56	The following pairs can be oriented strictly and are deleted.
6.15/2.56	
6.15/2.56	   PLUS(s(x), x) -> PLUS(if(gt(x, x), x, x), s(x))
6.15/2.56	The remaining pairs can at least be oriented weakly.
6.15/2.56	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(PLUS(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(s(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(if(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(gt(x_1, x_2)) =  	[[-I]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(false) =  	[[0A]]
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(true) =  	[[5A]]
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	   <<<
6.15/2.56	 POL(zero) =  	[[2A]]
6.15/2.56	>>>
6.15/2.56	
6.15/2.56	
6.15/2.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.15/2.56	
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(36)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(37) QDPOrderProof (EQUIVALENT)
6.15/2.56	We use the reduction pair processor [LPAR04,JAR06].
6.15/2.56	
6.15/2.56	
6.15/2.56	The following pairs can be oriented strictly and are deleted.
6.15/2.56	
6.15/2.56	   PLUS(s(x), s(y)) -> PLUS(if(gt(x, y), x, y), if(if(gt(x, y), false, true), x, y))
6.15/2.56	The remaining pairs can at least be oriented weakly.
6.15/2.56	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
6.15/2.56	
6.15/2.56	POL( PLUS_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
6.15/2.56	POL( if_3(x_1, ..., x_3) ) = x_2 + x_3
6.15/2.56	POL( gt_2(x_1, x_2) ) = 0
6.15/2.56	POL( s_1(x_1) ) = 2x_1 + 1
6.15/2.56	POL( zero ) = 0
6.15/2.56	POL( true ) = 0
6.15/2.56	POL( false ) = 0
6.15/2.56	
6.15/2.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.15/2.56	
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(38)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	P is empty.
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(39) PisEmptyProof (EQUIVALENT)
6.15/2.56	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(40)
6.15/2.56	YES
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(41)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.56	   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   quot(0, s(y), s(z)) -> 0
6.15/2.56	   quot(s(x), s(y), z) -> quot(x, y, z)
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
6.15/2.56	   id(x) -> x
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   not(x) -> if(x, false, true)
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   quot(0, s(x0), s(x1))
6.15/2.56	   quot(s(x0), s(x1), x2)
6.15/2.56	   plus(s(x0), s(x1))
6.15/2.56	   plus(s(x0), x0)
6.15/2.56	   plus(zero, x0)
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   not(x0)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	   quot(x0, 0, s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(42) UsableRulesProof (EQUIVALENT)
6.15/2.56	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.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(43)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.56	   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   id(x) -> x
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   not(x) -> if(x, false, true)
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   quot(0, s(x0), s(x1))
6.15/2.56	   quot(s(x0), s(x1), x2)
6.15/2.56	   plus(s(x0), s(x1))
6.15/2.56	   plus(s(x0), x0)
6.15/2.56	   plus(zero, x0)
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   not(x0)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	   quot(x0, 0, s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(44) QReductionProof (EQUIVALENT)
6.15/2.56	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.15/2.56	
6.15/2.56	   quot(0, s(x0), s(x1))
6.15/2.56	   quot(s(x0), s(x1), x2)
6.15/2.56	   quot(x0, 0, s(x1))
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(45)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.56	   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   id(x) -> x
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   not(x) -> if(x, false, true)
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   plus(s(x0), s(x1))
6.15/2.56	   plus(s(x0), x0)
6.15/2.56	   plus(zero, x0)
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   not(x0)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(46) QDPOrderProof (EQUIVALENT)
6.15/2.56	We use the reduction pair processor [LPAR04,JAR06].
6.15/2.56	
6.15/2.56	
6.15/2.56	The following pairs can be oriented strictly and are deleted.
6.15/2.56	
6.15/2.56	   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
6.15/2.56	The remaining pairs can at least be oriented weakly.
6.15/2.56	Used ordering:  Combined order from the following AFS and order.
6.15/2.56	QUOT(x1, x2, x3)  =  x1
6.15/2.56	
6.15/2.56	s(x1)  =  s(x1)
6.15/2.56	
6.15/2.56	
6.15/2.56	Knuth-Bendix order [KBO] with precedence:trivial
6.15/2.56	
6.15/2.56	and weight map:
6.15/2.56	
6.15/2.56	   s_1=1
6.15/2.56	   dummyConstant=1
6.15/2.56	
6.15/2.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.15/2.56	none
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(47)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	The TRS P consists of the following rules:
6.15/2.56	
6.15/2.56	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.56	
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   id(x) -> x
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   not(x) -> if(x, false, true)
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   plus(s(x0), s(x1))
6.15/2.56	   plus(s(x0), x0)
6.15/2.56	   plus(zero, x0)
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   not(x0)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(48) QDPOrderProof (EQUIVALENT)
6.15/2.56	We use the reduction pair processor [LPAR04,JAR06].
6.15/2.56	
6.15/2.56	
6.15/2.56	The following pairs can be oriented strictly and are deleted.
6.15/2.56	
6.15/2.56	   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
6.15/2.56	The remaining pairs can at least be oriented weakly.
6.15/2.56	Used ordering:  Combined order from the following AFS and order.
6.15/2.56	QUOT(x1, x2, x3)  =  x2
6.15/2.56	
6.15/2.56	0  =  0
6.15/2.56	
6.15/2.56	plus(x1, x2)  =  x2
6.15/2.56	
6.15/2.56	s(x1)  =  s
6.15/2.56	
6.15/2.56	
6.15/2.56	Knuth-Bendix order [KBO] with precedence:trivial
6.15/2.56	
6.15/2.56	and weight map:
6.15/2.56	
6.15/2.56	   s=1
6.15/2.56	   0=2
6.15/2.56	
6.15/2.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.15/2.56	
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	
6.15/2.56	
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(49)
6.15/2.56	Obligation:
6.15/2.56	Q DP problem:
6.15/2.56	P is empty.
6.15/2.56	The TRS R consists of the following rules:
6.15/2.56	
6.15/2.56	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
6.15/2.56	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
6.15/2.56	   plus(zero, y) -> y
6.15/2.56	   gt(zero, y) -> false
6.15/2.56	   gt(s(x), s(y)) -> gt(x, y)
6.15/2.56	   id(x) -> x
6.15/2.56	   if(true, x, y) -> x
6.15/2.56	   if(false, x, y) -> y
6.15/2.56	   gt(s(x), zero) -> true
6.15/2.56	   not(x) -> if(x, false, true)
6.15/2.56	
6.15/2.56	The set Q consists of the following terms:
6.15/2.56	
6.15/2.56	   plus(s(x0), s(x1))
6.15/2.56	   plus(s(x0), x0)
6.15/2.56	   plus(zero, x0)
6.15/2.56	   id(x0)
6.15/2.56	   if(true, x0, x1)
6.15/2.56	   if(false, x0, x1)
6.15/2.56	   not(x0)
6.15/2.56	   gt(s(x0), zero)
6.15/2.56	   gt(zero, x0)
6.15/2.56	   gt(s(x0), s(x1))
6.15/2.56	
6.15/2.56	We have to consider all minimal (P,Q,R)-chains.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(50) PisEmptyProof (EQUIVALENT)
6.15/2.56	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.15/2.56	----------------------------------------
6.15/2.56	
6.15/2.56	(51)
6.15/2.56	YES
6.15/2.61	EOF