3.71/1.73	YES
3.71/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.71/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.73	
3.71/1.73	
3.71/1.73	Quasi decreasingness of the given CTRS could be proven:
3.71/1.73	
3.71/1.73	(0) CTRS
3.71/1.73	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.71/1.73	(2) QTRS
3.71/1.73	(3) QTRSRRRProof [EQUIVALENT, 63 ms]
3.71/1.73	(4) QTRS
3.71/1.73	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.71/1.73	(6) QTRS
3.71/1.73	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.71/1.73	(8) YES
3.71/1.73	
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(0)
3.71/1.73	Obligation:
3.71/1.73	Conditional term rewrite system:
3.71/1.73	The TRS R consists of the following rules:
3.71/1.73	
3.71/1.73	   g(d, x, x) -> A
3.71/1.73	   h(x, x) -> g(x, x, f(k))
3.71/1.73	   c -> e'
3.71/1.73	   a -> c
3.71/1.73	   a -> d
3.71/1.73	   b -> c
3.71/1.73	   b -> d
3.71/1.73	   c -> e
3.71/1.73	   c -> l
3.71/1.73	   k -> l
3.71/1.73	   k -> m
3.71/1.73	   d -> m
3.71/1.73	
3.71/1.73	The conditional TRS C consists of the following conditional rules:
3.71/1.73	
3.71/1.73	   f(x) -> x <= x -> e, x -> e'
3.71/1.73	
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(1) CTRSToQTRSProof (SOUND)
3.71/1.73	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(2)
3.71/1.73	Obligation:
3.71/1.73	Q restricted rewrite system:
3.71/1.73	The TRS R consists of the following rules:
3.71/1.73	
3.71/1.73	   f(x) -> U1(x, x)
3.71/1.73	   U1(e, x) -> U2(x, x)
3.71/1.73	   U2(e', x) -> x
3.71/1.73	   g(d, x, x) -> A
3.71/1.73	   h(x, x) -> g(x, x, f(k))
3.71/1.73	   c -> e'
3.71/1.73	   a -> c
3.71/1.73	   a -> d
3.71/1.73	   b -> c
3.71/1.73	   b -> d
3.71/1.73	   c -> e
3.71/1.73	   c -> l
3.71/1.73	   k -> l
3.71/1.73	   k -> m
3.71/1.73	   d -> m
3.71/1.73	
3.71/1.73	Q is empty.
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(3) QTRSRRRProof (EQUIVALENT)
3.71/1.73	Used ordering:
3.71/1.73	Quasi precedence:
3.71/1.73	[d, c, a, b] > e
3.71/1.73	[d, c, a, b] > e'
3.71/1.73	[d, c, a, b] > A
3.71/1.73	[d, c, a, b] > l
3.71/1.73	[d, c, a, b] > m
3.71/1.73	h_2 > f_1 > U1_2 > U2_2
3.71/1.73	h_2 > g_3 > A
3.71/1.73	h_2 > k > l
3.71/1.73	h_2 > k > m
3.71/1.73	
3.71/1.73	
3.71/1.73	Status:
3.71/1.73	f_1: [1]
3.71/1.73	U1_2: [2,1]
3.71/1.73	e: multiset status
3.71/1.73	U2_2: [2,1]
3.71/1.73	e': multiset status
3.71/1.73	g_3: multiset status
3.71/1.73	d: multiset status
3.71/1.73	A: multiset status
3.71/1.73	h_2: [1,2]
3.71/1.73	k: multiset status
3.71/1.73	c: multiset status
3.71/1.73	a: multiset status
3.71/1.73	b: multiset status
3.71/1.73	l: multiset status
3.71/1.73	m: multiset status
3.71/1.73	
3.71/1.73	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.71/1.73	
3.71/1.73	   f(x) -> U1(x, x)
3.71/1.73	   U1(e, x) -> U2(x, x)
3.71/1.73	   U2(e', x) -> x
3.71/1.73	   g(d, x, x) -> A
3.71/1.73	   h(x, x) -> g(x, x, f(k))
3.71/1.73	   c -> e'
3.71/1.73	   c -> e
3.71/1.73	   c -> l
3.71/1.73	   k -> l
3.71/1.73	   k -> m
3.71/1.73	   d -> m
3.71/1.73	
3.71/1.73	
3.71/1.73	
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(4)
3.71/1.73	Obligation:
3.71/1.73	Q restricted rewrite system:
3.71/1.73	The TRS R consists of the following rules:
3.71/1.73	
3.71/1.73	   a -> c
3.71/1.73	   a -> d
3.71/1.73	   b -> c
3.71/1.73	   b -> d
3.71/1.73	
3.71/1.73	Q is empty.
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(5) QTRSRRRProof (EQUIVALENT)
3.71/1.73	Used ordering:
3.71/1.73	Knuth-Bendix order [KBO] with precedence:a > b > c > d
3.71/1.73	
3.71/1.73	and weight map:
3.71/1.73	
3.71/1.73	   a=1
3.71/1.73	   c=1
3.71/1.73	   d=1
3.71/1.73	   b=1
3.71/1.73	
3.71/1.73	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.71/1.73	
3.71/1.73	   a -> c
3.71/1.73	   a -> d
3.71/1.73	   b -> c
3.71/1.73	   b -> d
3.71/1.73	
3.71/1.73	
3.71/1.73	
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(6)
3.71/1.73	Obligation:
3.71/1.73	Q restricted rewrite system:
3.71/1.73	R is empty.
3.71/1.73	Q is empty.
3.71/1.73	
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(7) RisEmptyProof (EQUIVALENT)
3.71/1.73	The TRS R is empty. Hence, termination is trivially proven.
3.71/1.73	----------------------------------------
3.71/1.73	
3.71/1.73	(8)
3.71/1.73	YES
3.82/1.76	EOF