3.32/1.64	YES
3.32/1.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.32/1.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.32/1.65	
3.32/1.65	
3.32/1.65	Quasi decreasingness of the given CTRS could be proven:
3.32/1.65	
3.32/1.65	(0) CTRS
3.32/1.65	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.32/1.65	(2) QTRS
3.32/1.65	(3) QTRSRRRProof [EQUIVALENT, 43 ms]
3.32/1.65	(4) QTRS
3.32/1.65	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.32/1.65	(6) QTRS
3.32/1.65	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.32/1.65	(8) YES
3.32/1.65	
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(0)
3.32/1.65	Obligation:
3.32/1.65	Conditional term rewrite system:
3.32/1.65	The TRS R consists of the following rules:
3.32/1.65	
3.32/1.65	   add(x, 0) -> x
3.32/1.65	   add(x, s(y)) -> s(add(x, y))
3.32/1.65	
3.32/1.65	The conditional TRS C consists of the following conditional rules:
3.32/1.65	
3.32/1.65	   quad(x) -> z <= add(x, x) -> y, add(y, y) -> z
3.32/1.65	
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(1) CTRSToQTRSProof (SOUND)
3.32/1.65	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(2)
3.32/1.65	Obligation:
3.32/1.65	Q restricted rewrite system:
3.32/1.65	The TRS R consists of the following rules:
3.32/1.65	
3.32/1.65	   quad(x) -> U1(add(x, x))
3.32/1.65	   U1(y) -> U2(add(y, y))
3.32/1.65	   U2(z) -> z
3.32/1.65	   add(x, 0) -> x
3.32/1.65	   add(x, s(y)) -> s(add(x, y))
3.32/1.65	
3.32/1.65	Q is empty.
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(3) QTRSRRRProof (EQUIVALENT)
3.32/1.65	Used ordering:
3.32/1.65	quad/1(YES)
3.32/1.65	U1/1(YES)
3.32/1.65	add/2(YES,YES)
3.32/1.65	U2/1)YES(
3.32/1.65	0/0)
3.32/1.65	s/1)YES(
3.32/1.65	
3.32/1.65	Quasi precedence:
3.32/1.65	quad_1 > U1_1 > add_2
3.32/1.65	0 > add_2
3.32/1.65	
3.32/1.65	
3.32/1.65	Status:
3.32/1.65	quad_1: multiset status
3.32/1.65	U1_1: multiset status
3.32/1.65	add_2: multiset status
3.32/1.65	0: multiset status
3.32/1.65	
3.32/1.65	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.32/1.65	
3.32/1.65	   quad(x) -> U1(add(x, x))
3.32/1.65	   U1(y) -> U2(add(y, y))
3.32/1.65	   add(x, 0) -> x
3.32/1.65	
3.32/1.65	
3.32/1.65	
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(4)
3.32/1.65	Obligation:
3.32/1.65	Q restricted rewrite system:
3.32/1.65	The TRS R consists of the following rules:
3.32/1.65	
3.32/1.65	   U2(z) -> z
3.32/1.65	   add(x, s(y)) -> s(add(x, y))
3.32/1.65	
3.32/1.65	Q is empty.
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(5) QTRSRRRProof (EQUIVALENT)
3.32/1.65	Used ordering:
3.32/1.65	Knuth-Bendix order [KBO] with precedence:add_2 > s_1 > U2_1
3.32/1.65	
3.32/1.65	and weight map:
3.32/1.65	
3.32/1.65	   U2_1=1
3.32/1.65	   s_1=1
3.32/1.65	   add_2=0
3.32/1.65	
3.32/1.65	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.32/1.65	
3.32/1.65	   U2(z) -> z
3.32/1.65	   add(x, s(y)) -> s(add(x, y))
3.32/1.65	
3.32/1.65	
3.32/1.65	
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(6)
3.32/1.65	Obligation:
3.32/1.65	Q restricted rewrite system:
3.32/1.65	R is empty.
3.32/1.65	Q is empty.
3.32/1.65	
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(7) RisEmptyProof (EQUIVALENT)
3.32/1.65	The TRS R is empty. Hence, termination is trivially proven.
3.32/1.65	----------------------------------------
3.32/1.65	
3.32/1.65	(8)
3.32/1.65	YES
3.53/1.67	EOF