3.39/1.66	YES
3.39/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.39/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.39/1.67	
3.39/1.67	
3.39/1.67	Quasi decreasingness of the given CTRS could be proven:
3.39/1.67	
3.39/1.67	(0) CTRS
3.39/1.67	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.39/1.67	(2) QTRS
3.39/1.67	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.39/1.67	(4) QTRS
3.39/1.67	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.39/1.67	(6) YES
3.39/1.67	
3.39/1.67	
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(0)
3.39/1.67	Obligation:
3.39/1.67	Conditional term rewrite system:
3.39/1.67	The TRS R consists of the following rules:
3.39/1.67	
3.39/1.67	   sub(z, 0) -> z
3.39/1.67	
3.39/1.67	The conditional TRS C consists of the following conditional rules:
3.39/1.67	
3.39/1.67	   sub(s(z), s(y)) -> x <= sub(z, y) -> x
3.39/1.67	
3.39/1.67	
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(1) CTRSToQTRSProof (SOUND)
3.39/1.67	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(2)
3.39/1.67	Obligation:
3.39/1.67	Q restricted rewrite system:
3.39/1.67	The TRS R consists of the following rules:
3.39/1.67	
3.39/1.67	   sub(s(z), s(y)) -> U1(sub(z, y))
3.39/1.67	   U1(x) -> x
3.39/1.67	   sub(z, 0) -> z
3.39/1.67	
3.39/1.67	Q is empty.
3.39/1.67	
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(3) QTRSRRRProof (EQUIVALENT)
3.39/1.67	Used ordering:
3.39/1.67	Knuth-Bendix order [KBO] with precedence:0 > sub_2 > U1_1 > s_1
3.39/1.67	
3.39/1.67	and weight map:
3.39/1.67	
3.39/1.67	   0=1
3.39/1.67	   s_1=1
3.39/1.67	   U1_1=2
3.39/1.67	   sub_2=0
3.39/1.67	
3.39/1.67	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.39/1.67	
3.39/1.67	   sub(s(z), s(y)) -> U1(sub(z, y))
3.39/1.67	   U1(x) -> x
3.39/1.67	   sub(z, 0) -> z
3.39/1.67	
3.39/1.67	
3.39/1.67	
3.39/1.67	
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(4)
3.39/1.67	Obligation:
3.39/1.67	Q restricted rewrite system:
3.39/1.67	R is empty.
3.39/1.67	Q is empty.
3.39/1.67	
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(5) RisEmptyProof (EQUIVALENT)
3.39/1.67	The TRS R is empty. Hence, termination is trivially proven.
3.39/1.67	----------------------------------------
3.39/1.67	
3.39/1.67	(6)
3.39/1.67	YES
3.39/1.69	EOF