644.63/291.51	WORST_CASE(Omega(n^1), ?)
644.63/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
644.63/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
644.63/291.52	
644.63/291.52	
644.63/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
644.63/291.52	
644.63/291.52	(0) CpxTRS
644.63/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
644.63/291.52	(2) CpxTRS
644.63/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
644.63/291.52	(4) typed CpxTrs
644.63/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
644.63/291.52	(6) typed CpxTrs
644.63/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 426 ms]
644.63/291.52	(8) proven lower bound
644.63/291.52	(9) LowerBoundPropagationProof [FINISHED, 0 ms]
644.63/291.52	(10) BOUNDS(n^1, INF)
644.63/291.52	
644.63/291.52	
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(0)
644.63/291.52	Obligation:
644.63/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
644.63/291.52	
644.63/291.52	
644.63/291.52	The TRS R consists of the following rules:
644.63/291.52	
644.63/291.52	   f(0, 1, x) -> f(g(x), g(x), x)
644.63/291.52	   f(g(x), y, z) -> g(f(x, y, z))
644.63/291.52	   f(x, g(y), z) -> g(f(x, y, z))
644.63/291.52	   f(x, y, g(z)) -> g(f(x, y, z))
644.63/291.52	
644.63/291.52	S is empty.
644.63/291.52	Rewrite Strategy: INNERMOST
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
644.63/291.52	Renamed function symbols to avoid clashes with predefined symbol.
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(2)
644.63/291.52	Obligation:
644.63/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
644.63/291.52	
644.63/291.52	
644.63/291.52	The TRS R consists of the following rules:
644.63/291.52	
644.63/291.52	   f(0', 1', x) -> f(g(x), g(x), x)
644.63/291.52	   f(g(x), y, z) -> g(f(x, y, z))
644.63/291.52	   f(x, g(y), z) -> g(f(x, y, z))
644.63/291.52	   f(x, y, g(z)) -> g(f(x, y, z))
644.63/291.52	
644.63/291.52	S is empty.
644.63/291.52	Rewrite Strategy: INNERMOST
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
644.63/291.52	Infered types.
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(4)
644.63/291.52	Obligation:
644.63/291.52	Innermost TRS:
644.63/291.52	Rules:
644.63/291.52	f(0', 1', x) -> f(g(x), g(x), x)
644.63/291.52	f(g(x), y, z) -> g(f(x, y, z))
644.63/291.52	f(x, g(y), z) -> g(f(x, y, z))
644.63/291.52	f(x, y, g(z)) -> g(f(x, y, z))
644.63/291.52	
644.63/291.52	Types:
644.63/291.52	f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
644.63/291.52	0' :: 0':1':g
644.63/291.52	1' :: 0':1':g
644.63/291.52	g :: 0':1':g -> 0':1':g
644.63/291.52	hole_0':1':g1_0 :: 0':1':g
644.63/291.52	gen_0':1':g2_0 :: Nat -> 0':1':g
644.63/291.52	
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(5) OrderProof (LOWER BOUND(ID))
644.63/291.52	Heuristically decided to analyse the following defined symbols:
644.63/291.52	f
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(6)
644.63/291.52	Obligation:
644.63/291.52	Innermost TRS:
644.63/291.52	Rules:
644.63/291.52	f(0', 1', x) -> f(g(x), g(x), x)
644.63/291.52	f(g(x), y, z) -> g(f(x, y, z))
644.63/291.52	f(x, g(y), z) -> g(f(x, y, z))
644.63/291.52	f(x, y, g(z)) -> g(f(x, y, z))
644.63/291.52	
644.63/291.52	Types:
644.63/291.52	f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
644.63/291.52	0' :: 0':1':g
644.63/291.52	1' :: 0':1':g
644.63/291.52	g :: 0':1':g -> 0':1':g
644.63/291.52	hole_0':1':g1_0 :: 0':1':g
644.63/291.52	gen_0':1':g2_0 :: Nat -> 0':1':g
644.63/291.52	
644.63/291.52	
644.63/291.52	Generator Equations:
644.63/291.52	gen_0':1':g2_0(0) <=> 1'
644.63/291.52	gen_0':1':g2_0(+(x, 1)) <=> g(gen_0':1':g2_0(x))
644.63/291.52	
644.63/291.52	
644.63/291.52	The following defined symbols remain to be analysed:
644.63/291.52	f
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
644.63/291.52	Proved the following rewrite lemma:
644.63/291.52	f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) -> *3_0, rt in Omega(n4_0)
644.63/291.52	
644.63/291.52	Induction Base:
644.63/291.52	f(gen_0':1':g2_0(+(1, 0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))
644.63/291.52	
644.63/291.52	Induction Step:
644.63/291.52	f(gen_0':1':g2_0(+(1, +(n4_0, 1))), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) ->_R^Omega(1)
644.63/291.52	g(f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))) ->_IH
644.63/291.52	g(*3_0)
644.63/291.52	
644.63/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(8)
644.63/291.52	Obligation:
644.63/291.52	Proved the lower bound n^1 for the following obligation:
644.63/291.52	
644.63/291.52	Innermost TRS:
644.63/291.52	Rules:
644.63/291.52	f(0', 1', x) -> f(g(x), g(x), x)
644.63/291.52	f(g(x), y, z) -> g(f(x, y, z))
644.63/291.52	f(x, g(y), z) -> g(f(x, y, z))
644.63/291.52	f(x, y, g(z)) -> g(f(x, y, z))
644.63/291.52	
644.63/291.52	Types:
644.63/291.52	f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
644.63/291.52	0' :: 0':1':g
644.63/291.52	1' :: 0':1':g
644.63/291.52	g :: 0':1':g -> 0':1':g
644.63/291.52	hole_0':1':g1_0 :: 0':1':g
644.63/291.52	gen_0':1':g2_0 :: Nat -> 0':1':g
644.63/291.52	
644.63/291.52	
644.63/291.52	Generator Equations:
644.63/291.52	gen_0':1':g2_0(0) <=> 1'
644.63/291.52	gen_0':1':g2_0(+(x, 1)) <=> g(gen_0':1':g2_0(x))
644.63/291.52	
644.63/291.52	
644.63/291.52	The following defined symbols remain to be analysed:
644.63/291.52	f
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(9) LowerBoundPropagationProof (FINISHED)
644.63/291.52	Propagated lower bound.
644.63/291.52	----------------------------------------
644.63/291.52	
644.63/291.52	(10)
644.63/291.52	BOUNDS(n^1, INF)
644.80/291.57	EOF