3.72/1.82	YES
3.83/1.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.83/1.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.83/1.83	
3.83/1.83	
3.83/1.83	Quasi decreasingness of the given CTRS could be proven:
3.83/1.83	
3.83/1.83	(0) CTRS
3.83/1.83	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.83/1.83	(2) QTRS
3.83/1.83	(3) QTRSRRRProof [EQUIVALENT, 109 ms]
3.83/1.83	(4) QTRS
3.83/1.83	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.83/1.83	(6) QTRS
3.83/1.83	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.83/1.83	(8) YES
3.83/1.83	
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(0)
3.83/1.83	Obligation:
3.83/1.83	Conditional term rewrite system:
3.83/1.83	The TRS R consists of the following rules:
3.83/1.83	
3.83/1.83	   lte(0, n) -> true
3.83/1.83	   lte(s(m), 0) -> false
3.83/1.83	   lte(s(m), s(n)) -> lte(m, n)
3.83/1.83	   insert(nil, m) -> cons(m, nil)
3.83/1.83	   ordered(nil) -> true
3.83/1.83	   ordered(cons(m, nil)) -> true
3.83/1.83	
3.83/1.83	The conditional TRS C consists of the following conditional rules:
3.83/1.83	
3.83/1.83	   insert(cons(n, l), m) -> cons(m, cons(n, l)) <= lte(m, n) -> true
3.83/1.83	   insert(cons(n, l), m) -> cons(n, insert(l, m)) <= lte(m, n) -> false
3.83/1.83	   ordered(cons(m, cons(n, l))) -> ordered(cons(n, l)) <= lte(m, n) -> true
3.83/1.83	   ordered(cons(m, cons(n, l))) -> false <= lte(m, n) -> false
3.83/1.83	
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(1) CTRSToQTRSProof (SOUND)
3.83/1.83	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(2)
3.83/1.83	Obligation:
3.83/1.83	Q restricted rewrite system:
3.83/1.83	The TRS R consists of the following rules:
3.83/1.83	
3.83/1.83	   insert(cons(n, l), m) -> U1(lte(m, n), n, l, m)
3.83/1.83	   U1(false, n, l, m) -> cons(n, insert(l, m))
3.83/1.83	   U1(true, n, l, m) -> cons(m, cons(n, l))
3.83/1.83	   ordered(cons(m, cons(n, l))) -> U2(lte(m, n), n, l)
3.83/1.83	   U2(true, n, l) -> ordered(cons(n, l))
3.83/1.83	   U2(false, n, l) -> false
3.83/1.83	   lte(0, n) -> true
3.83/1.83	   lte(s(m), 0) -> false
3.83/1.83	   lte(s(m), s(n)) -> lte(m, n)
3.83/1.83	   insert(nil, m) -> cons(m, nil)
3.83/1.83	   ordered(nil) -> true
3.83/1.83	   ordered(cons(m, nil)) -> true
3.83/1.83	
3.83/1.83	Q is empty.
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(3) QTRSRRRProof (EQUIVALENT)
3.83/1.83	Used ordering:
3.83/1.83	insert/2(YES,YES)
3.83/1.83	cons/2(YES,YES)
3.83/1.83	U1/4(YES,YES,YES,YES)
3.83/1.83	lte/2(YES,YES)
3.83/1.83	false/0)
3.83/1.83	true/0)
3.83/1.83	ordered/1)YES(
3.83/1.83	U2/3(YES,YES,YES)
3.83/1.83	0/0)
3.83/1.83	s/1)YES(
3.83/1.83	nil/0)
3.83/1.83	
3.83/1.83	Quasi precedence:
3.83/1.83	[insert_2, U1_4, nil] > [true, 0] > [cons_2, lte_2, U2_3] > false
3.83/1.83	
3.83/1.83	
3.83/1.83	Status:
3.83/1.83	insert_2: [1,2]
3.83/1.83	cons_2: [2,1]
3.83/1.83	U1_4: [3,4,1,2]
3.83/1.83	lte_2: [2,1]
3.83/1.83	false: multiset status
3.83/1.83	true: multiset status
3.83/1.83	U2_3: [3,2,1]
3.83/1.83	0: multiset status
3.83/1.83	nil: multiset status
3.83/1.83	
3.83/1.83	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.83/1.83	
3.83/1.83	   insert(cons(n, l), m) -> U1(lte(m, n), n, l, m)
3.83/1.83	   U1(false, n, l, m) -> cons(n, insert(l, m))
3.83/1.83	   U1(true, n, l, m) -> cons(m, cons(n, l))
3.83/1.83	   ordered(cons(m, cons(n, l))) -> U2(lte(m, n), n, l)
3.83/1.83	   U2(true, n, l) -> ordered(cons(n, l))
3.83/1.83	   U2(false, n, l) -> false
3.83/1.83	   lte(0, n) -> true
3.83/1.83	   lte(s(m), 0) -> false
3.83/1.83	   insert(nil, m) -> cons(m, nil)
3.83/1.83	   ordered(nil) -> true
3.83/1.83	   ordered(cons(m, nil)) -> true
3.83/1.83	
3.83/1.83	
3.83/1.83	
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(4)
3.83/1.83	Obligation:
3.83/1.83	Q restricted rewrite system:
3.83/1.83	The TRS R consists of the following rules:
3.83/1.83	
3.83/1.83	   lte(s(m), s(n)) -> lte(m, n)
3.83/1.83	
3.83/1.83	Q is empty.
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(5) QTRSRRRProof (EQUIVALENT)
3.83/1.83	Used ordering:
3.83/1.83	Knuth-Bendix order [KBO] with precedence:s_1 > lte_2
3.83/1.83	
3.83/1.83	and weight map:
3.83/1.83	
3.83/1.83	   s_1=0
3.83/1.83	   lte_2=0
3.83/1.83	
3.83/1.83	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.83/1.83	
3.83/1.83	   lte(s(m), s(n)) -> lte(m, n)
3.83/1.83	
3.83/1.83	
3.83/1.83	
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(6)
3.83/1.83	Obligation:
3.83/1.83	Q restricted rewrite system:
3.83/1.83	R is empty.
3.83/1.83	Q is empty.
3.83/1.83	
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(7) RisEmptyProof (EQUIVALENT)
3.83/1.83	The TRS R is empty. Hence, termination is trivially proven.
3.83/1.83	----------------------------------------
3.83/1.83	
3.83/1.83	(8)
3.83/1.83	YES
3.84/1.85	EOF