300.85/291.62	WORST_CASE(Omega(n^1), ?)
300.85/291.63	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
300.85/291.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
300.85/291.63	
300.85/291.63	
300.85/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
300.85/291.63	
300.85/291.63	(0) CpxTRS
300.85/291.63	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
300.85/291.63	(2) CpxTRS
300.85/291.63	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
300.85/291.63	(4) typed CpxTrs
300.85/291.63	(5) OrderProof [LOWER BOUND(ID), 0 ms]
300.85/291.63	(6) typed CpxTrs
300.85/291.63	(7) RewriteLemmaProof [LOWER BOUND(ID), 162 ms]
300.85/291.63	(8) BEST
300.85/291.63	    (9) proven lower bound
300.85/291.63	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
300.85/291.63	        (11) BOUNDS(n^1, INF)
300.85/291.63	    (12) typed CpxTrs
300.85/291.63	
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(0)
300.85/291.63	Obligation:
300.85/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
300.85/291.63	
300.85/291.63	
300.85/291.63	The TRS R consists of the following rules:
300.85/291.63	
300.85/291.63	   f(0) -> true
300.85/291.63	   f(1) -> false
300.85/291.63	   f(s(x)) -> f(x)
300.85/291.63	   if(true, x, y) -> x
300.85/291.63	   if(false, x, y) -> y
300.85/291.63	   g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	   g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	S is empty.
300.85/291.63	Rewrite Strategy: FULL
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(1) RenamingProof (BOTH BOUNDS(ID, ID))
300.85/291.63	Renamed function symbols to avoid clashes with predefined symbol.
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(2)
300.85/291.63	Obligation:
300.85/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
300.85/291.63	
300.85/291.63	
300.85/291.63	The TRS R consists of the following rules:
300.85/291.63	
300.85/291.63	   f(0') -> true
300.85/291.63	   f(1') -> false
300.85/291.63	   f(s(x)) -> f(x)
300.85/291.63	   if(true, x, y) -> x
300.85/291.63	   if(false, x, y) -> y
300.85/291.63	   g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	   g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	S is empty.
300.85/291.63	Rewrite Strategy: FULL
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
300.85/291.63	Infered types.
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(4)
300.85/291.63	Obligation:
300.85/291.63	TRS:
300.85/291.63	Rules:
300.85/291.63	f(0') -> true
300.85/291.63	f(1') -> false
300.85/291.63	f(s(x)) -> f(x)
300.85/291.63	if(true, x, y) -> x
300.85/291.63	if(false, x, y) -> y
300.85/291.63	g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	Types:
300.85/291.63	f :: 0':1':s:c -> true:false
300.85/291.63	0' :: 0':1':s:c
300.85/291.63	true :: true:false
300.85/291.63	1' :: 0':1':s:c
300.85/291.63	false :: true:false
300.85/291.63	s :: 0':1':s:c -> 0':1':s:c
300.85/291.63	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	c :: 0':1':s:c -> 0':1':s:c
300.85/291.63	hole_true:false1_0 :: true:false
300.85/291.63	hole_0':1':s:c2_0 :: 0':1':s:c
300.85/291.63	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(5) OrderProof (LOWER BOUND(ID))
300.85/291.63	Heuristically decided to analyse the following defined symbols:
300.85/291.63	f, g
300.85/291.63	
300.85/291.63	They will be analysed ascendingly in the following order:
300.85/291.63	f < g
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(6)
300.85/291.63	Obligation:
300.85/291.63	TRS:
300.85/291.63	Rules:
300.85/291.63	f(0') -> true
300.85/291.63	f(1') -> false
300.85/291.63	f(s(x)) -> f(x)
300.85/291.63	if(true, x, y) -> x
300.85/291.63	if(false, x, y) -> y
300.85/291.63	g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	Types:
300.85/291.63	f :: 0':1':s:c -> true:false
300.85/291.63	0' :: 0':1':s:c
300.85/291.63	true :: true:false
300.85/291.63	1' :: 0':1':s:c
300.85/291.63	false :: true:false
300.85/291.63	s :: 0':1':s:c -> 0':1':s:c
300.85/291.63	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	c :: 0':1':s:c -> 0':1':s:c
300.85/291.63	hole_true:false1_0 :: true:false
300.85/291.63	hole_0':1':s:c2_0 :: 0':1':s:c
300.85/291.63	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
300.85/291.63	
300.85/291.63	
300.85/291.63	Generator Equations:
300.85/291.63	gen_0':1':s:c3_0(0) <=> 0'
300.85/291.63	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
300.85/291.63	
300.85/291.63	
300.85/291.63	The following defined symbols remain to be analysed:
300.85/291.63	f, g
300.85/291.63	
300.85/291.63	They will be analysed ascendingly in the following order:
300.85/291.63	f < g
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(7) RewriteLemmaProof (LOWER BOUND(ID))
300.85/291.63	Proved the following rewrite lemma:
300.85/291.63	f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
300.85/291.63	
300.85/291.63	Induction Base:
300.85/291.63	f(gen_0':1':s:c3_0(0)) ->_R^Omega(1)
300.85/291.63	true
300.85/291.63	
300.85/291.63	Induction Step:
300.85/291.63	f(gen_0':1':s:c3_0(+(n5_0, 1))) ->_R^Omega(1)
300.85/291.63	f(gen_0':1':s:c3_0(n5_0)) ->_IH
300.85/291.63	true
300.85/291.63	
300.85/291.63	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(8)
300.85/291.63	Complex Obligation (BEST)
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(9)
300.85/291.63	Obligation:
300.85/291.63	Proved the lower bound n^1 for the following obligation:
300.85/291.63	
300.85/291.63	TRS:
300.85/291.63	Rules:
300.85/291.63	f(0') -> true
300.85/291.63	f(1') -> false
300.85/291.63	f(s(x)) -> f(x)
300.85/291.63	if(true, x, y) -> x
300.85/291.63	if(false, x, y) -> y
300.85/291.63	g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	Types:
300.85/291.63	f :: 0':1':s:c -> true:false
300.85/291.63	0' :: 0':1':s:c
300.85/291.63	true :: true:false
300.85/291.63	1' :: 0':1':s:c
300.85/291.63	false :: true:false
300.85/291.63	s :: 0':1':s:c -> 0':1':s:c
300.85/291.63	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	c :: 0':1':s:c -> 0':1':s:c
300.85/291.63	hole_true:false1_0 :: true:false
300.85/291.63	hole_0':1':s:c2_0 :: 0':1':s:c
300.85/291.63	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
300.85/291.63	
300.85/291.63	
300.85/291.63	Generator Equations:
300.85/291.63	gen_0':1':s:c3_0(0) <=> 0'
300.85/291.63	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
300.85/291.63	
300.85/291.63	
300.85/291.63	The following defined symbols remain to be analysed:
300.85/291.63	f, g
300.85/291.63	
300.85/291.63	They will be analysed ascendingly in the following order:
300.85/291.63	f < g
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(10) LowerBoundPropagationProof (FINISHED)
300.85/291.63	Propagated lower bound.
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(11)
300.85/291.63	BOUNDS(n^1, INF)
300.85/291.63	
300.85/291.63	----------------------------------------
300.85/291.63	
300.85/291.63	(12)
300.85/291.63	Obligation:
300.85/291.63	TRS:
300.85/291.63	Rules:
300.85/291.63	f(0') -> true
300.85/291.63	f(1') -> false
300.85/291.63	f(s(x)) -> f(x)
300.85/291.63	if(true, x, y) -> x
300.85/291.63	if(false, x, y) -> y
300.85/291.63	g(s(x), s(y)) -> if(f(x), s(x), s(y))
300.85/291.63	g(x, c(y)) -> g(x, g(s(c(y)), y))
300.85/291.63	
300.85/291.63	Types:
300.85/291.63	f :: 0':1':s:c -> true:false
300.85/291.63	0' :: 0':1':s:c
300.85/291.63	true :: true:false
300.85/291.63	1' :: 0':1':s:c
300.85/291.63	false :: true:false
300.85/291.63	s :: 0':1':s:c -> 0':1':s:c
300.85/291.63	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
300.85/291.63	c :: 0':1':s:c -> 0':1':s:c
300.85/291.63	hole_true:false1_0 :: true:false
300.85/291.63	hole_0':1':s:c2_0 :: 0':1':s:c
300.85/291.63	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
300.85/291.63	
300.85/291.63	
300.85/291.63	Lemmas:
300.85/291.63	f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
300.85/291.63	
300.85/291.63	
300.85/291.63	Generator Equations:
300.85/291.63	gen_0':1':s:c3_0(0) <=> 0'
300.85/291.63	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
300.85/291.63	
300.85/291.63	
300.85/291.63	The following defined symbols remain to be analysed:
300.85/291.63	g
300.90/291.67	EOF