1102.25/292.07	WORST_CASE(Omega(n^1), ?)
1102.25/292.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1102.25/292.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.25/292.08	
1102.25/292.08	(0) CpxTRS
1102.25/292.08	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1102.25/292.08	(2) CpxTRS
1102.25/292.08	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1102.25/292.08	(4) typed CpxTrs
1102.25/292.08	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1102.25/292.08	(6) typed CpxTrs
1102.25/292.08	(7) RewriteLemmaProof [LOWER BOUND(ID), 523 ms]
1102.25/292.08	(8) BEST
1102.25/292.08	    (9) proven lower bound
1102.25/292.08	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1102.25/292.08	        (11) BOUNDS(n^1, INF)
1102.25/292.08	    (12) typed CpxTrs
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(0)
1102.25/292.08	Obligation:
1102.25/292.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The TRS R consists of the following rules:
1102.25/292.08	
1102.25/292.08	   minus(minus(x)) -> x
1102.25/292.08	   minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	   minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	   f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	S is empty.
1102.25/292.08	Rewrite Strategy: FULL
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1102.25/292.08	Renamed function symbols to avoid clashes with predefined symbol.
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(2)
1102.25/292.08	Obligation:
1102.25/292.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The TRS R consists of the following rules:
1102.25/292.08	
1102.25/292.08	   minus(minus(x)) -> x
1102.25/292.08	   minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	   minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	   f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	S is empty.
1102.25/292.08	Rewrite Strategy: FULL
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1102.25/292.08	Infered types.
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(4)
1102.25/292.08	Obligation:
1102.25/292.08	TRS:
1102.25/292.08	Rules:
1102.25/292.08	minus(minus(x)) -> x
1102.25/292.08	minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	Types:
1102.25/292.08	minus :: +':*' -> +':*'
1102.25/292.08	+' :: +':*' -> +':*' -> +':*'
1102.25/292.08	*' :: +':*' -> +':*' -> +':*'
1102.25/292.08	f :: +':*' -> +':*'
1102.25/292.08	hole_+':*'1_0 :: +':*'
1102.25/292.08	gen_+':*'2_0 :: Nat -> +':*'
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(5) OrderProof (LOWER BOUND(ID))
1102.25/292.08	Heuristically decided to analyse the following defined symbols:
1102.25/292.08	minus, f
1102.25/292.08	
1102.25/292.08	They will be analysed ascendingly in the following order:
1102.25/292.08	minus < f
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(6)
1102.25/292.08	Obligation:
1102.25/292.08	TRS:
1102.25/292.08	Rules:
1102.25/292.08	minus(minus(x)) -> x
1102.25/292.08	minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	Types:
1102.25/292.08	minus :: +':*' -> +':*'
1102.25/292.08	+' :: +':*' -> +':*' -> +':*'
1102.25/292.08	*' :: +':*' -> +':*' -> +':*'
1102.25/292.08	f :: +':*' -> +':*'
1102.25/292.08	hole_+':*'1_0 :: +':*'
1102.25/292.08	gen_+':*'2_0 :: Nat -> +':*'
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	Generator Equations:
1102.25/292.08	gen_+':*'2_0(0) <=> hole_+':*'1_0
1102.25/292.08	gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The following defined symbols remain to be analysed:
1102.25/292.08	minus, f
1102.25/292.08	
1102.25/292.08	They will be analysed ascendingly in the following order:
1102.25/292.08	minus < f
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(7) RewriteLemmaProof (LOWER BOUND(ID))
1102.25/292.08	Proved the following rewrite lemma:
1102.25/292.08	minus(gen_+':*'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
1102.25/292.08	
1102.25/292.08	Induction Base:
1102.25/292.08	minus(gen_+':*'2_0(0))
1102.25/292.08	
1102.25/292.08	Induction Step:
1102.25/292.08	minus(gen_+':*'2_0(+(n4_0, 1))) ->_R^Omega(1)
1102.25/292.08	*'(minus(minus(minus(hole_+':*'1_0))), minus(minus(minus(gen_+':*'2_0(n4_0))))) ->_R^Omega(1)
1102.25/292.08	*'(minus(hole_+':*'1_0), minus(minus(minus(gen_+':*'2_0(n4_0))))) ->_IH
1102.25/292.08	*'(minus(hole_+':*'1_0), minus(minus(*3_0)))
1102.25/292.08	
1102.25/292.08	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(8)
1102.25/292.08	Complex Obligation (BEST)
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(9)
1102.25/292.08	Obligation:
1102.25/292.08	Proved the lower bound n^1 for the following obligation:
1102.25/292.08	
1102.25/292.08	TRS:
1102.25/292.08	Rules:
1102.25/292.08	minus(minus(x)) -> x
1102.25/292.08	minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	Types:
1102.25/292.08	minus :: +':*' -> +':*'
1102.25/292.08	+' :: +':*' -> +':*' -> +':*'
1102.25/292.08	*' :: +':*' -> +':*' -> +':*'
1102.25/292.08	f :: +':*' -> +':*'
1102.25/292.08	hole_+':*'1_0 :: +':*'
1102.25/292.08	gen_+':*'2_0 :: Nat -> +':*'
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	Generator Equations:
1102.25/292.08	gen_+':*'2_0(0) <=> hole_+':*'1_0
1102.25/292.08	gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The following defined symbols remain to be analysed:
1102.25/292.08	minus, f
1102.25/292.08	
1102.25/292.08	They will be analysed ascendingly in the following order:
1102.25/292.08	minus < f
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(10) LowerBoundPropagationProof (FINISHED)
1102.25/292.08	Propagated lower bound.
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(11)
1102.25/292.08	BOUNDS(n^1, INF)
1102.25/292.08	
1102.25/292.08	----------------------------------------
1102.25/292.08	
1102.25/292.08	(12)
1102.25/292.08	Obligation:
1102.25/292.08	TRS:
1102.25/292.08	Rules:
1102.25/292.08	minus(minus(x)) -> x
1102.25/292.08	minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
1102.25/292.08	f(minus(x)) -> minus(minus(minus(f(x))))
1102.25/292.08	
1102.25/292.08	Types:
1102.25/292.08	minus :: +':*' -> +':*'
1102.25/292.08	+' :: +':*' -> +':*' -> +':*'
1102.25/292.08	*' :: +':*' -> +':*' -> +':*'
1102.25/292.08	f :: +':*' -> +':*'
1102.25/292.08	hole_+':*'1_0 :: +':*'
1102.25/292.08	gen_+':*'2_0 :: Nat -> +':*'
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	Lemmas:
1102.25/292.08	minus(gen_+':*'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	Generator Equations:
1102.25/292.08	gen_+':*'2_0(0) <=> hole_+':*'1_0
1102.25/292.08	gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))
1102.25/292.08	
1102.25/292.08	
1102.25/292.08	The following defined symbols remain to be analysed:
1102.25/292.08	f
1102.87/292.25	EOF