6.64/3.04	YES
6.64/3.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.64/3.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.64/3.06	
6.64/3.06	
6.64/3.06	Termination w.r.t. Q of the given QTRS could be proven:
6.64/3.06	
6.64/3.06	(0) QTRS
6.64/3.06	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
6.64/3.06	(2) QDP
6.64/3.06	(3) DependencyGraphProof [EQUIVALENT, 3 ms]
6.64/3.06	(4) AND
6.64/3.06	    (5) QDP
6.64/3.06	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
6.64/3.06	        (7) QDP
6.64/3.06	        (8) QReductionProof [EQUIVALENT, 0 ms]
6.64/3.06	        (9) QDP
6.64/3.06	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.64/3.06	        (11) YES
6.64/3.06	    (12) QDP
6.64/3.06	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
6.64/3.06	        (14) QDP
6.64/3.06	        (15) QReductionProof [EQUIVALENT, 0 ms]
6.64/3.06	        (16) QDP
6.64/3.06	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.64/3.06	        (18) YES
6.64/3.06	    (19) QDP
6.64/3.06	        (20) UsableRulesProof [EQUIVALENT, 0 ms]
6.64/3.06	        (21) QDP
6.64/3.06	        (22) QReductionProof [EQUIVALENT, 0 ms]
6.64/3.06	        (23) QDP
6.64/3.06	        (24) TransformationProof [EQUIVALENT, 0 ms]
6.64/3.06	        (25) QDP
6.64/3.06	        (26) Induction-Processor [SOUND, 30 ms]
6.64/3.06	        (27) AND
6.64/3.06	            (28) QDP
6.64/3.06	                (29) DependencyGraphProof [EQUIVALENT, 0 ms]
6.64/3.06	                (30) TRUE
6.64/3.06	            (31) QTRS
6.64/3.06	                (32) QTRSRRRProof [EQUIVALENT, 0 ms]
6.64/3.06	                (33) QTRS
6.64/3.06	                (34) QTRSRRRProof [EQUIVALENT, 0 ms]
6.64/3.06	                (35) QTRS
6.64/3.06	                (36) RisEmptyProof [EQUIVALENT, 0 ms]
6.64/3.06	                (37) YES
6.64/3.06	
6.64/3.06	
6.64/3.06	----------------------------------------
6.64/3.06	
6.64/3.06	(0)
6.64/3.06	Obligation:
6.64/3.06	Q restricted rewrite system:
6.64/3.06	The TRS R consists of the following rules:
6.64/3.06	
6.64/3.06	   le(0, y) -> true
6.64/3.06	   le(s(x), 0) -> false
6.64/3.06	   le(s(x), s(y)) -> le(x, y)
6.64/3.06	   minus(x, 0) -> x
6.64/3.06	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.06	   mod(0, y) -> 0
6.64/3.06	   mod(s(x), 0) -> 0
6.64/3.06	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
6.64/3.06	   if_mod(true, x, y) -> mod(minus(x, y), y)
6.64/3.06	   if_mod(false, s(x), s(y)) -> s(x)
6.64/3.06	
6.64/3.06	The set Q consists of the following terms:
6.64/3.06	
6.64/3.06	   le(0, x0)
6.64/3.06	   le(s(x0), 0)
6.64/3.06	   le(s(x0), s(x1))
6.64/3.06	   minus(x0, 0)
6.64/3.06	   minus(s(x0), s(x1))
6.64/3.06	   mod(0, x0)
6.64/3.06	   mod(s(x0), 0)
6.64/3.06	   mod(s(x0), s(x1))
6.64/3.06	   if_mod(true, x0, x1)
6.64/3.06	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(1) DependencyPairsProof (EQUIVALENT)
6.64/3.07	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(2)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   LE(s(x), s(y)) -> LE(x, y)
6.64/3.07	   MINUS(s(x), s(y)) -> MINUS(x, y)
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	   MOD(s(x), s(y)) -> LE(y, x)
6.64/3.07	   IF_MOD(true, x, y) -> MOD(minus(x, y), y)
6.64/3.07	   IF_MOD(true, x, y) -> MINUS(x, y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   mod(0, y) -> 0
6.64/3.07	   mod(s(x), 0) -> 0
6.64/3.07	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
6.64/3.07	   if_mod(true, x, y) -> mod(minus(x, y), y)
6.64/3.07	   if_mod(false, s(x), s(y)) -> s(x)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(3) DependencyGraphProof (EQUIVALENT)
6.64/3.07	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(4)
6.64/3.07	Complex Obligation (AND)
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(5)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MINUS(s(x), s(y)) -> MINUS(x, y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   mod(0, y) -> 0
6.64/3.07	   mod(s(x), 0) -> 0
6.64/3.07	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
6.64/3.07	   if_mod(true, x, y) -> mod(minus(x, y), y)
6.64/3.07	   if_mod(false, s(x), s(y)) -> s(x)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(6) UsableRulesProof (EQUIVALENT)
6.64/3.07	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.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(7)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MINUS(s(x), s(y)) -> MINUS(x, y)
6.64/3.07	
6.64/3.07	R is empty.
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(8) QReductionProof (EQUIVALENT)
6.64/3.07	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(9)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MINUS(s(x), s(y)) -> MINUS(x, y)
6.64/3.07	
6.64/3.07	R is empty.
6.64/3.07	Q is empty.
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(10) QDPSizeChangeProof (EQUIVALENT)
6.64/3.07	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. 
6.64/3.07	
6.64/3.07	From the DPs we obtained the following set of size-change graphs:
6.64/3.07	*MINUS(s(x), s(y)) -> MINUS(x, y)
6.64/3.07	The graph contains the following edges 1 > 1, 2 > 2
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(11)
6.64/3.07	YES
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(12)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   LE(s(x), s(y)) -> LE(x, y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   mod(0, y) -> 0
6.64/3.07	   mod(s(x), 0) -> 0
6.64/3.07	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
6.64/3.07	   if_mod(true, x, y) -> mod(minus(x, y), y)
6.64/3.07	   if_mod(false, s(x), s(y)) -> s(x)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(13) UsableRulesProof (EQUIVALENT)
6.64/3.07	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.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(14)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   LE(s(x), s(y)) -> LE(x, y)
6.64/3.07	
6.64/3.07	R is empty.
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(15) QReductionProof (EQUIVALENT)
6.64/3.07	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(16)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   LE(s(x), s(y)) -> LE(x, y)
6.64/3.07	
6.64/3.07	R is empty.
6.64/3.07	Q is empty.
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(17) QDPSizeChangeProof (EQUIVALENT)
6.64/3.07	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. 
6.64/3.07	
6.64/3.07	From the DPs we obtained the following set of size-change graphs:
6.64/3.07	*LE(s(x), s(y)) -> LE(x, y)
6.64/3.07	The graph contains the following edges 1 > 1, 2 > 2
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(18)
6.64/3.07	YES
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(19)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	   IF_MOD(true, x, y) -> MOD(minus(x, y), y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   mod(0, y) -> 0
6.64/3.07	   mod(s(x), 0) -> 0
6.64/3.07	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
6.64/3.07	   if_mod(true, x, y) -> mod(minus(x, y), y)
6.64/3.07	   if_mod(false, s(x), s(y)) -> s(x)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(20) UsableRulesProof (EQUIVALENT)
6.64/3.07	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.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(21)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	   IF_MOD(true, x, y) -> MOD(minus(x, y), y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(22) QReductionProof (EQUIVALENT)
6.64/3.07	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.64/3.07	
6.64/3.07	   mod(0, x0)
6.64/3.07	   mod(s(x0), 0)
6.64/3.07	   mod(s(x0), s(x1))
6.64/3.07	   if_mod(true, x0, x1)
6.64/3.07	   if_mod(false, s(x0), s(x1))
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(23)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	   IF_MOD(true, x, y) -> MOD(minus(x, y), y)
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(24) TransformationProof (EQUIVALENT)
6.64/3.07	By instantiating [LPAR04] the rule IF_MOD(true, x, y) -> MOD(minus(x, y), y) we obtained the following new rules [LPAR04]:
6.64/3.07	
6.64/3.07	   (IF_MOD(true, s(z0), s(z1)) -> MOD(minus(s(z0), s(z1)), s(z1)),IF_MOD(true, s(z0), s(z1)) -> MOD(minus(s(z0), s(z1)), s(z1)))
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(25)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	   IF_MOD(true, s(z0), s(z1)) -> MOD(minus(s(z0), s(z1)), s(z1))
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(26) Induction-Processor (SOUND)
6.64/3.07	
6.64/3.07	This DP could be deleted by the Induction-Processor:
6.64/3.07	IF_MOD(true_renamed, s(z0), s(z1)) -> MOD(minus(s(z0), s(z1)), s(z1))
6.64/3.07	
6.64/3.07	
6.64/3.07	This order was computed:
6.64/3.07	Polynomial interpretation [POLO]:
6.64/3.07	
6.64/3.07	   POL(0) = 1
6.64/3.07	   POL(IF_MOD(x_1, x_2, x_3)) = x_2
6.64/3.07	   POL(MOD(x_1, x_2)) = x_1
6.64/3.07	   POL(false_renamed) = 1
6.64/3.07	   POL(le(x_1, x_2)) = x_2
6.64/3.07	   POL(minus(x_1, x_2)) = x_1
6.64/3.07	   POL(s(x_1)) = 1 + x_1
6.64/3.07	   POL(true_renamed) = 0
6.64/3.07	
6.64/3.07	At least one of these decreasing rules is always used after the deleted DP:
6.64/3.07	minus(s(x''), s(y')) -> minus(x'', y')
6.64/3.07	
6.64/3.07	
6.64/3.07	The following formula is valid:
6.64/3.07	z2:sort[a0],z3:sort[a0].((~(z2=0)/\~(z3=0))->minus'(z2, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	The transformed set:
6.64/3.07	minus'(x', 0) -> false
6.64/3.07	minus'(s(x''), s(y')) -> true
6.64/3.07	minus'(0, s(v11)) -> false
6.64/3.07	minus(x', 0) -> x'
6.64/3.07	minus(s(x''), s(y')) -> minus(x'', y')
6.64/3.07	le(0, y'') -> true_renamed
6.64/3.07	le(s(x1), 0) -> false_renamed
6.64/3.07	le(s(x2), s(y1)) -> le(x2, y1)
6.64/3.07	minus(0, s(v11)) -> 0
6.64/3.07	equal_bool(true, false) -> false
6.64/3.07	equal_bool(false, true) -> false
6.64/3.07	equal_bool(true, true) -> true
6.64/3.07	equal_bool(false, false) -> true
6.64/3.07	and(true, x) -> x
6.64/3.07	and(false, x) -> false
6.64/3.07	or(true, x) -> true
6.64/3.07	or(false, x) -> x
6.64/3.07	not(false) -> true
6.64/3.07	not(true) -> false
6.64/3.07	isa_true(true) -> true
6.64/3.07	isa_true(false) -> false
6.64/3.07	isa_false(true) -> false
6.64/3.07	isa_false(false) -> true
6.64/3.07	equal_sort[a0](0, 0) -> true
6.64/3.07	equal_sort[a0](0, s(v18)) -> false
6.64/3.07	equal_sort[a0](s(v19), 0) -> false
6.64/3.07	equal_sort[a0](s(v19), s(v20)) -> equal_sort[a0](v19, v20)
6.64/3.07	equal_sort[a19](true_renamed, true_renamed) -> true
6.64/3.07	equal_sort[a19](true_renamed, false_renamed) -> false
6.64/3.07	equal_sort[a19](false_renamed, true_renamed) -> false
6.64/3.07	equal_sort[a19](false_renamed, false_renamed) -> true
6.64/3.07	equal_sort[a24](witness_sort[a24], witness_sort[a24]) -> true
6.64/3.07	
6.64/3.07	
6.64/3.07	The proof given by the theorem prover:
6.64/3.07	The following output was given by the internal theorem prover:proof of internal
6.64/3.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.64/3.07	
6.64/3.07	
6.64/3.07	Partial correctness of the following Program
6.64/3.07	
6.64/3.07	   [x, v18, v19, v20, x', x'', y', v11, y'', x1, x2, y1]
6.64/3.07	   equal_bool(true, false) -> false
6.64/3.07	   equal_bool(false, true) -> false
6.64/3.07	   equal_bool(true, true) -> true
6.64/3.07	   equal_bool(false, false) -> true
6.64/3.07	   true and x -> x
6.64/3.07	   false and x -> false
6.64/3.07	   true or x -> true
6.64/3.07	   false or x -> x
6.64/3.07	   not(false) -> true
6.64/3.07	   not(true) -> false
6.64/3.07	   isa_true(true) -> true
6.64/3.07	   isa_true(false) -> false
6.64/3.07	   isa_false(true) -> false
6.64/3.07	   isa_false(false) -> true
6.64/3.07	   equal_sort[a0](0, 0) -> true
6.64/3.07	   equal_sort[a0](0, s(v18)) -> false
6.64/3.07	   equal_sort[a0](s(v19), 0) -> false
6.64/3.07	   equal_sort[a0](s(v19), s(v20)) -> equal_sort[a0](v19, v20)
6.64/3.07	   equal_sort[a19](true_renamed, true_renamed) -> true
6.64/3.07	   equal_sort[a19](true_renamed, false_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, true_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, false_renamed) -> true
6.64/3.07	   equal_sort[a24](witness_sort[a24], witness_sort[a24]) -> true
6.64/3.07	   minus'(x', 0) -> false
6.64/3.07	   minus'(s(x''), s(y')) -> true
6.64/3.07	   minus'(0, s(v11)) -> false
6.64/3.07	   minus(x', 0) -> x'
6.64/3.07	   minus(s(x''), s(y')) -> minus(x'', y')
6.64/3.07	   minus(0, s(v11)) -> 0
6.64/3.07	   le(0, y'') -> true_renamed
6.64/3.07	   le(s(x1), 0) -> false_renamed
6.64/3.07	   le(s(x2), s(y1)) -> le(x2, y1)
6.64/3.07	
6.64/3.07	using the following formula:
6.64/3.07	z2:sort[a0],z3:sort[a0].((~(z2=0)/\~(z3=0))->minus'(z2, z3)=true)
6.64/3.07	
6.64/3.07	could be successfully shown:
6.64/3.07	(0) Formula
6.64/3.07	(1) Induction by data structure [EQUIVALENT, 0 ms]
6.64/3.07	(2) AND
6.64/3.07	    (3) Formula
6.64/3.07	        (4) Symbolic evaluation [EQUIVALENT, 0 ms]
6.64/3.07	        (5) YES
6.64/3.07	    (6) Formula
6.64/3.07	        (7) Symbolic evaluation [EQUIVALENT, 0 ms]
6.64/3.07	        (8) Formula
6.64/3.07	        (9) Case Analysis [EQUIVALENT, 0 ms]
6.64/3.07	        (10) AND
6.64/3.07	            (11) Formula
6.64/3.07	                (12) Symbolic evaluation [EQUIVALENT, 0 ms]
6.64/3.07	                (13) YES
6.64/3.07	            (14) Formula
6.64/3.07	                (15) Symbolic evaluation [EQUIVALENT, 0 ms]
6.64/3.07	                (16) YES
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(0)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	z2:sort[a0],z3:sort[a0].((~(z2=0)/\~(z3=0))->minus'(z2, z3)=true)
6.64/3.07	
6.64/3.07	There are no hypotheses.
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(1) Induction by data structure (EQUIVALENT)
6.64/3.07	Induction by data structure sort[a0] generates the following cases:
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	1. Base Case:
6.64/3.07	Formula:
6.64/3.07	z3:sort[a0].((~(0=0)/\~(z3=0))->minus'(0, z3)=true)
6.64/3.07	
6.64/3.07	There are no hypotheses.
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	1. Step Case:
6.64/3.07	Formula:
6.64/3.07	n:sort[a0],z3:sort[a0].((~(s(n)=0)/\~(z3=0))->minus'(s(n), z3)=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(2)
6.64/3.07	Complex Obligation (AND)
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(3)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	z3:sort[a0].((~(0=0)/\~(z3=0))->minus'(0, z3)=true)
6.64/3.07	
6.64/3.07	There are no hypotheses.
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(4) Symbolic evaluation (EQUIVALENT)
6.64/3.07	Could be reduced to the following new obligation by simple symbolic evaluation:
6.64/3.07	True
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(5)
6.64/3.07	YES
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(6)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	n:sort[a0],z3:sort[a0].((~(s(n)=0)/\~(z3=0))->minus'(s(n), z3)=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(7) Symbolic evaluation (EQUIVALENT)
6.64/3.07	Could be shown by simple symbolic evaluation.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(8)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	z3:sort[a0],n:sort[a0].(~(z3=0)->minus'(s(n), z3)=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(9) Case Analysis (EQUIVALENT)
6.64/3.07	Case analysis leads to the following new obligations:
6.64/3.07	
6.64/3.07	Formula:
6.64/3.07	n:sort[a0].(~(0=0)->minus'(s(n), 0)=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	Formula:
6.64/3.07	x_1:sort[a0],n:sort[a0].(~(s(x_1)=0)->minus'(s(n), s(x_1))=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(10)
6.64/3.07	Complex Obligation (AND)
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(11)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	n:sort[a0].(~(0=0)->minus'(s(n), 0)=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(12) Symbolic evaluation (EQUIVALENT)
6.64/3.07	Could be reduced to the following new obligation by simple symbolic evaluation:
6.64/3.07	True
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(13)
6.64/3.07	YES
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(14)
6.64/3.07	Obligation:
6.64/3.07	Formula:
6.64/3.07	x_1:sort[a0],n:sort[a0].(~(s(x_1)=0)->minus'(s(n), s(x_1))=true)
6.64/3.07	
6.64/3.07	Hypotheses:
6.64/3.07	n:sort[a0],!z3:sort[a0].((~(n=0)/\~(z3=0))->minus'(n, z3)=true)
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(15) Symbolic evaluation (EQUIVALENT)
6.64/3.07	Could be reduced to the following new obligation by simple symbolic evaluation:
6.64/3.07	True
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(16)
6.64/3.07	YES
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(27)
6.64/3.07	Complex Obligation (AND)
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(28)
6.64/3.07	Obligation:
6.64/3.07	Q DP problem:
6.64/3.07	The TRS P consists of the following rules:
6.64/3.07	
6.64/3.07	   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
6.64/3.07	
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus(x, 0) -> x
6.64/3.07	   minus(s(x), s(y)) -> minus(x, y)
6.64/3.07	   le(0, y) -> true
6.64/3.07	   le(s(x), 0) -> false
6.64/3.07	   le(s(x), s(y)) -> le(x, y)
6.64/3.07	
6.64/3.07	The set Q consists of the following terms:
6.64/3.07	
6.64/3.07	   le(0, x0)
6.64/3.07	   le(s(x0), 0)
6.64/3.07	   le(s(x0), s(x1))
6.64/3.07	   minus(x0, 0)
6.64/3.07	   minus(s(x0), s(x1))
6.64/3.07	
6.64/3.07	We have to consider all minimal (P,Q,R)-chains.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(29) DependencyGraphProof (EQUIVALENT)
6.64/3.07	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(30)
6.64/3.07	TRUE
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(31)
6.64/3.07	Obligation:
6.64/3.07	Q restricted rewrite system:
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus'(x', 0) -> false
6.64/3.07	   minus'(s(x''), s(y')) -> true
6.64/3.07	   minus'(0, s(v11)) -> false
6.64/3.07	   minus(x', 0) -> x'
6.64/3.07	   minus(s(x''), s(y')) -> minus(x'', y')
6.64/3.07	   le(0, y'') -> true_renamed
6.64/3.07	   le(s(x1), 0) -> false_renamed
6.64/3.07	   le(s(x2), s(y1)) -> le(x2, y1)
6.64/3.07	   minus(0, s(v11)) -> 0
6.64/3.07	   equal_bool(true, false) -> false
6.64/3.07	   equal_bool(false, true) -> false
6.64/3.07	   equal_bool(true, true) -> true
6.64/3.07	   equal_bool(false, false) -> true
6.64/3.07	   and(true, x) -> x
6.64/3.07	   and(false, x) -> false
6.64/3.07	   or(true, x) -> true
6.64/3.07	   or(false, x) -> x
6.64/3.07	   not(false) -> true
6.64/3.07	   not(true) -> false
6.64/3.07	   isa_true(true) -> true
6.64/3.07	   isa_true(false) -> false
6.64/3.07	   isa_false(true) -> false
6.64/3.07	   isa_false(false) -> true
6.64/3.07	   equal_sort[a0](0, 0) -> true
6.64/3.07	   equal_sort[a0](0, s(v18)) -> false
6.64/3.07	   equal_sort[a0](s(v19), 0) -> false
6.64/3.07	   equal_sort[a0](s(v19), s(v20)) -> equal_sort[a0](v19, v20)
6.64/3.07	   equal_sort[a19](true_renamed, true_renamed) -> true
6.64/3.07	   equal_sort[a19](true_renamed, false_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, true_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, false_renamed) -> true
6.64/3.07	   equal_sort[a24](witness_sort[a24], witness_sort[a24]) -> true
6.64/3.07	
6.64/3.07	Q is empty.
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(32) QTRSRRRProof (EQUIVALENT)
6.64/3.07	Used ordering:
6.64/3.07	Polynomial interpretation [POLO]:
6.64/3.07	
6.64/3.07	   POL(0) = 0
6.64/3.07	   POL(and(x_1, x_2)) = 2*x_1 + x_2
6.64/3.07	   POL(equal_bool(x_1, x_2)) = 2*x_1 + 2*x_2
6.64/3.07	   POL(equal_sort[a0](x_1, x_2)) = 2 + x_1 + x_2
6.64/3.07	   POL(equal_sort[a19](x_1, x_2)) = 2 + x_1 + x_2
6.64/3.07	   POL(equal_sort[a24](x_1, x_2)) = 2 + 2*x_1 + 2*x_2
6.64/3.07	   POL(false) = 2
6.64/3.07	   POL(false_renamed) = 0
6.64/3.07	   POL(isa_false(x_1)) = 2 + x_1
6.64/3.07	   POL(isa_true(x_1)) = 2*x_1
6.64/3.07	   POL(le(x_1, x_2)) = 2 + x_1 + x_2
6.64/3.07	   POL(minus(x_1, x_2)) = 2*x_1 + 2*x_2
6.64/3.07	   POL(minus'(x_1, x_2)) = 2 + x_1 + x_2
6.64/3.07	   POL(not(x_1)) = 2*x_1
6.64/3.07	   POL(or(x_1, x_2)) = 2*x_1 + x_2
6.64/3.07	   POL(s(x_1)) = 1 + 2*x_1
6.64/3.07	   POL(true) = 1
6.64/3.07	   POL(true_renamed) = 1
6.64/3.07	   POL(witness_sort[a24]) = 1
6.64/3.07	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.64/3.07	
6.64/3.07	   minus'(s(x''), s(y')) -> true
6.64/3.07	   minus'(0, s(v11)) -> false
6.64/3.07	   minus(s(x''), s(y')) -> minus(x'', y')
6.64/3.07	   le(0, y'') -> true_renamed
6.64/3.07	   le(s(x1), 0) -> false_renamed
6.64/3.07	   le(s(x2), s(y1)) -> le(x2, y1)
6.64/3.07	   minus(0, s(v11)) -> 0
6.64/3.07	   equal_bool(true, false) -> false
6.64/3.07	   equal_bool(false, true) -> false
6.64/3.07	   equal_bool(true, true) -> true
6.64/3.07	   equal_bool(false, false) -> true
6.64/3.07	   and(true, x) -> x
6.64/3.07	   and(false, x) -> false
6.64/3.07	   or(true, x) -> true
6.64/3.07	   or(false, x) -> x
6.64/3.07	   not(false) -> true
6.64/3.07	   isa_true(true) -> true
6.64/3.07	   isa_true(false) -> false
6.64/3.07	   isa_false(true) -> false
6.64/3.07	   isa_false(false) -> true
6.64/3.07	   equal_sort[a0](0, 0) -> true
6.64/3.07	   equal_sort[a0](0, s(v18)) -> false
6.64/3.07	   equal_sort[a0](s(v19), 0) -> false
6.64/3.07	   equal_sort[a0](s(v19), s(v20)) -> equal_sort[a0](v19, v20)
6.64/3.07	   equal_sort[a19](true_renamed, true_renamed) -> true
6.64/3.07	   equal_sort[a19](true_renamed, false_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, true_renamed) -> false
6.64/3.07	   equal_sort[a19](false_renamed, false_renamed) -> true
6.64/3.07	   equal_sort[a24](witness_sort[a24], witness_sort[a24]) -> true
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(33)
6.64/3.07	Obligation:
6.64/3.07	Q restricted rewrite system:
6.64/3.07	The TRS R consists of the following rules:
6.64/3.07	
6.64/3.07	   minus'(x', 0) -> false
6.64/3.07	   minus(x', 0) -> x'
6.64/3.07	   not(true) -> false
6.64/3.07	
6.64/3.07	Q is empty.
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(34) QTRSRRRProof (EQUIVALENT)
6.64/3.07	Used ordering:
6.64/3.07	Knuth-Bendix order [KBO] with precedence:not_1 > true > minus'_2 > false > minus_2 > 0
6.64/3.07	
6.64/3.07	and weight map:
6.64/3.07	
6.64/3.07	   0=1
6.64/3.07	   false=2
6.64/3.07	   true=1
6.64/3.07	   not_1=1
6.64/3.07	   minus'_2=0
6.64/3.07	   minus_2=0
6.64/3.07	
6.64/3.07	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.64/3.07	
6.64/3.07	   minus'(x', 0) -> false
6.64/3.07	   minus(x', 0) -> x'
6.64/3.07	   not(true) -> false
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(35)
6.64/3.07	Obligation:
6.64/3.07	Q restricted rewrite system:
6.64/3.07	R is empty.
6.64/3.07	Q is empty.
6.64/3.07	
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(36) RisEmptyProof (EQUIVALENT)
6.64/3.07	The TRS R is empty. Hence, termination is trivially proven.
6.64/3.07	----------------------------------------
6.64/3.07	
6.64/3.07	(37)
6.64/3.07	YES
6.64/3.10	EOF