6.06/2.41	YES
6.06/2.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.06/2.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.06/2.42	
6.06/2.42	
6.06/2.42	Termination w.r.t. Q of the given QTRS could be proven:
6.06/2.42	
6.06/2.42	(0) QTRS
6.06/2.42	(1) DependencyPairsProof [EQUIVALENT, 42 ms]
6.06/2.42	(2) QDP
6.06/2.42	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.42	(4) AND
6.06/2.42	    (5) QDP
6.06/2.42	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.42	        (7) QDP
6.06/2.42	        (8) QReductionProof [EQUIVALENT, 0 ms]
6.06/2.42	        (9) QDP
6.06/2.42	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.06/2.42	        (11) YES
6.06/2.42	    (12) QDP
6.06/2.42	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.42	        (14) QDP
6.06/2.42	        (15) QReductionProof [EQUIVALENT, 0 ms]
6.06/2.42	        (16) QDP
6.06/2.42	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.06/2.42	        (18) YES
6.06/2.42	    (19) QDP
6.06/2.42	        (20) QDPOrderProof [EQUIVALENT, 59 ms]
6.06/2.42	        (21) QDP
6.06/2.42	        (22) DependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.42	        (23) QDP
6.06/2.42	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.06/2.42	        (25) YES
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(0)
6.06/2.42	Obligation:
6.06/2.42	Q restricted rewrite system:
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(1) DependencyPairsProof (EQUIVALENT)
6.06/2.42	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(2)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__MINUS(s(X), s(Y)) -> A__MINUS(X, Y)
6.06/2.42	   A__GEQ(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	   A__DIV(s(X), s(Y)) -> A__IF(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   A__DIV(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	   A__IF(true, X, Y) -> MARK(X)
6.06/2.42	   A__IF(false, X, Y) -> MARK(Y)
6.06/2.42	   MARK(minus(X1, X2)) -> A__MINUS(X1, X2)
6.06/2.42	   MARK(geq(X1, X2)) -> A__GEQ(X1, X2)
6.06/2.42	   MARK(div(X1, X2)) -> A__DIV(mark(X1), X2)
6.06/2.42	   MARK(div(X1, X2)) -> MARK(X1)
6.06/2.42	   MARK(if(X1, X2, X3)) -> A__IF(mark(X1), X2, X3)
6.06/2.42	   MARK(if(X1, X2, X3)) -> MARK(X1)
6.06/2.42	   MARK(s(X)) -> MARK(X)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(3) DependencyGraphProof (EQUIVALENT)
6.06/2.42	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(4)
6.06/2.42	Complex Obligation (AND)
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(5)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__GEQ(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(6) UsableRulesProof (EQUIVALENT)
6.06/2.42	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.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(7)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__GEQ(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	
6.06/2.42	R is empty.
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(8) QReductionProof (EQUIVALENT)
6.06/2.42	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(9)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__GEQ(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	
6.06/2.42	R is empty.
6.06/2.42	Q is empty.
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(10) QDPSizeChangeProof (EQUIVALENT)
6.06/2.42	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. 
6.06/2.42	
6.06/2.42	From the DPs we obtained the following set of size-change graphs:
6.06/2.42	*A__GEQ(s(X), s(Y)) -> A__GEQ(X, Y)
6.06/2.42	The graph contains the following edges 1 > 1, 2 > 2
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(11)
6.06/2.42	YES
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(12)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__MINUS(s(X), s(Y)) -> A__MINUS(X, Y)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(13) UsableRulesProof (EQUIVALENT)
6.06/2.42	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.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(14)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__MINUS(s(X), s(Y)) -> A__MINUS(X, Y)
6.06/2.42	
6.06/2.42	R is empty.
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(15) QReductionProof (EQUIVALENT)
6.06/2.42	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(16)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   A__MINUS(s(X), s(Y)) -> A__MINUS(X, Y)
6.06/2.42	
6.06/2.42	R is empty.
6.06/2.42	Q is empty.
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(17) QDPSizeChangeProof (EQUIVALENT)
6.06/2.42	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. 
6.06/2.42	
6.06/2.42	From the DPs we obtained the following set of size-change graphs:
6.06/2.42	*A__MINUS(s(X), s(Y)) -> A__MINUS(X, Y)
6.06/2.42	The graph contains the following edges 1 > 1, 2 > 2
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(18)
6.06/2.42	YES
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(19)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   MARK(div(X1, X2)) -> A__DIV(mark(X1), X2)
6.06/2.42	   A__DIV(s(X), s(Y)) -> A__IF(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   A__IF(true, X, Y) -> MARK(X)
6.06/2.42	   MARK(div(X1, X2)) -> MARK(X1)
6.06/2.42	   MARK(if(X1, X2, X3)) -> A__IF(mark(X1), X2, X3)
6.06/2.42	   A__IF(false, X, Y) -> MARK(Y)
6.06/2.42	   MARK(if(X1, X2, X3)) -> MARK(X1)
6.06/2.42	   MARK(s(X)) -> MARK(X)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(20) QDPOrderProof (EQUIVALENT)
6.06/2.42	We use the reduction pair processor [LPAR04,JAR06].
6.06/2.42	
6.06/2.42	
6.06/2.42	The following pairs can be oriented strictly and are deleted.
6.06/2.42	
6.06/2.42	   A__DIV(s(X), s(Y)) -> A__IF(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   MARK(s(X)) -> MARK(X)
6.06/2.42	The remaining pairs can at least be oriented weakly.
6.06/2.42	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
6.06/2.42	
6.06/2.42	POL( A__DIV_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
6.06/2.42	POL( A__IF_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
6.06/2.42	POL( mark_1(x_1) ) = x_1
6.06/2.42	POL( minus_2(x_1, x_2) ) = 0
6.06/2.42	POL( a__minus_2(x_1, x_2) ) = 0
6.06/2.42	POL( geq_2(x_1, x_2) ) = 0
6.06/2.42	POL( a__geq_2(x_1, x_2) ) = 0
6.06/2.42	POL( div_2(x_1, x_2) ) = 2x_1 + 2x_2
6.06/2.42	POL( a__div_2(x_1, x_2) ) = 2x_1 + 2x_2
6.06/2.42	POL( s_1(x_1) ) = x_1 + 1
6.06/2.42	POL( a__if_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3
6.06/2.42	POL( 0 ) = 0
6.06/2.42	POL( true ) = 0
6.06/2.42	POL( if_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3
6.06/2.42	POL( false ) = 0
6.06/2.42	POL( MARK_1(x_1) ) = x_1 + 2
6.06/2.42	
6.06/2.42	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.06/2.42	
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(21)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   MARK(div(X1, X2)) -> A__DIV(mark(X1), X2)
6.06/2.42	   A__IF(true, X, Y) -> MARK(X)
6.06/2.42	   MARK(div(X1, X2)) -> MARK(X1)
6.06/2.42	   MARK(if(X1, X2, X3)) -> A__IF(mark(X1), X2, X3)
6.06/2.42	   A__IF(false, X, Y) -> MARK(Y)
6.06/2.42	   MARK(if(X1, X2, X3)) -> MARK(X1)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(22) DependencyGraphProof (EQUIVALENT)
6.06/2.42	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(23)
6.06/2.42	Obligation:
6.06/2.42	Q DP problem:
6.06/2.42	The TRS P consists of the following rules:
6.06/2.42	
6.06/2.42	   MARK(div(X1, X2)) -> MARK(X1)
6.06/2.42	   MARK(if(X1, X2, X3)) -> A__IF(mark(X1), X2, X3)
6.06/2.42	   A__IF(true, X, Y) -> MARK(X)
6.06/2.42	   MARK(if(X1, X2, X3)) -> MARK(X1)
6.06/2.42	   A__IF(false, X, Y) -> MARK(Y)
6.06/2.42	
6.06/2.42	The TRS R consists of the following rules:
6.06/2.42	
6.06/2.42	   a__minus(0, Y) -> 0
6.06/2.42	   a__minus(s(X), s(Y)) -> a__minus(X, Y)
6.06/2.42	   a__geq(X, 0) -> true
6.06/2.42	   a__geq(0, s(Y)) -> false
6.06/2.42	   a__geq(s(X), s(Y)) -> a__geq(X, Y)
6.06/2.42	   a__div(0, s(Y)) -> 0
6.06/2.42	   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
6.06/2.42	   a__if(true, X, Y) -> mark(X)
6.06/2.42	   a__if(false, X, Y) -> mark(Y)
6.06/2.42	   mark(minus(X1, X2)) -> a__minus(X1, X2)
6.06/2.42	   mark(geq(X1, X2)) -> a__geq(X1, X2)
6.06/2.42	   mark(div(X1, X2)) -> a__div(mark(X1), X2)
6.06/2.42	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
6.06/2.42	   mark(0) -> 0
6.06/2.42	   mark(s(X)) -> s(mark(X))
6.06/2.42	   mark(true) -> true
6.06/2.42	   mark(false) -> false
6.06/2.42	   a__minus(X1, X2) -> minus(X1, X2)
6.06/2.42	   a__geq(X1, X2) -> geq(X1, X2)
6.06/2.42	   a__div(X1, X2) -> div(X1, X2)
6.06/2.42	   a__if(X1, X2, X3) -> if(X1, X2, X3)
6.06/2.42	
6.06/2.42	The set Q consists of the following terms:
6.06/2.42	
6.06/2.42	   mark(minus(x0, x1))
6.06/2.42	   mark(geq(x0, x1))
6.06/2.42	   mark(div(x0, x1))
6.06/2.42	   mark(if(x0, x1, x2))
6.06/2.42	   mark(0)
6.06/2.42	   mark(s(x0))
6.06/2.42	   mark(true)
6.06/2.42	   mark(false)
6.06/2.42	   a__minus(x0, x1)
6.06/2.42	   a__geq(x0, x1)
6.06/2.42	   a__div(x0, x1)
6.06/2.42	   a__if(x0, x1, x2)
6.06/2.42	
6.06/2.42	We have to consider all minimal (P,Q,R)-chains.
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(24) QDPSizeChangeProof (EQUIVALENT)
6.06/2.42	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. 
6.06/2.42	
6.06/2.42	From the DPs we obtained the following set of size-change graphs:
6.06/2.42	*MARK(if(X1, X2, X3)) -> A__IF(mark(X1), X2, X3)
6.06/2.42	The graph contains the following edges 1 > 2, 1 > 3
6.06/2.42	
6.06/2.42	
6.06/2.42	*MARK(div(X1, X2)) -> MARK(X1)
6.06/2.42	The graph contains the following edges 1 > 1
6.06/2.42	
6.06/2.42	
6.06/2.42	*MARK(if(X1, X2, X3)) -> MARK(X1)
6.06/2.42	The graph contains the following edges 1 > 1
6.06/2.42	
6.06/2.42	
6.06/2.42	*A__IF(true, X, Y) -> MARK(X)
6.06/2.42	The graph contains the following edges 2 >= 1
6.06/2.42	
6.06/2.42	
6.06/2.42	*A__IF(false, X, Y) -> MARK(Y)
6.06/2.42	The graph contains the following edges 3 >= 1
6.06/2.42	
6.06/2.42	
6.06/2.42	----------------------------------------
6.06/2.42	
6.06/2.42	(25)
6.06/2.42	YES
6.31/2.47	EOF