308.61/291.48	WORST_CASE(Omega(n^1), ?)
308.61/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
308.61/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.61/291.49	
308.61/291.49	
308.61/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.61/291.49	
308.61/291.49	(0) CpxTRS
308.61/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
308.61/291.49	(2) CpxTRS
308.61/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
308.61/291.49	(4) typed CpxTrs
308.61/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
308.61/291.49	(6) typed CpxTrs
308.61/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 211 ms]
308.61/291.49	(8) BEST
308.61/291.49	    (9) proven lower bound
308.61/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
308.61/291.49	        (11) BOUNDS(n^1, INF)
308.61/291.49	    (12) typed CpxTrs
308.61/291.49	
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(0)
308.61/291.49	Obligation:
308.61/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.61/291.49	
308.61/291.49	
308.61/291.49	The TRS R consists of the following rules:
308.61/291.49	
308.61/291.49	   f(0) -> true
308.61/291.49	   f(1) -> false
308.61/291.49	   f(s(x)) -> f(x)
308.61/291.49	   if(true, s(x), s(y)) -> s(x)
308.61/291.49	   if(false, s(x), s(y)) -> s(y)
308.61/291.49	   g(x, c(y)) -> c(g(x, y))
308.61/291.49	   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	S is empty.
308.61/291.49	Rewrite Strategy: FULL
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
308.61/291.49	Renamed function symbols to avoid clashes with predefined symbol.
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(2)
308.61/291.49	Obligation:
308.61/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.61/291.49	
308.61/291.49	
308.61/291.49	The TRS R consists of the following rules:
308.61/291.49	
308.61/291.49	   f(0') -> true
308.61/291.49	   f(1') -> false
308.61/291.49	   f(s(x)) -> f(x)
308.61/291.49	   if(true, s(x), s(y)) -> s(x)
308.61/291.49	   if(false, s(x), s(y)) -> s(y)
308.61/291.49	   g(x, c(y)) -> c(g(x, y))
308.61/291.49	   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	S is empty.
308.61/291.49	Rewrite Strategy: FULL
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
308.61/291.49	Infered types.
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(4)
308.61/291.49	Obligation:
308.61/291.49	TRS:
308.61/291.49	Rules:
308.61/291.49	f(0') -> true
308.61/291.49	f(1') -> false
308.61/291.49	f(s(x)) -> f(x)
308.61/291.49	if(true, s(x), s(y)) -> s(x)
308.61/291.49	if(false, s(x), s(y)) -> s(y)
308.61/291.49	g(x, c(y)) -> c(g(x, y))
308.61/291.49	g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	Types:
308.61/291.49	f :: 0':1':s:c -> true:false
308.61/291.49	0' :: 0':1':s:c
308.61/291.49	true :: true:false
308.61/291.49	1' :: 0':1':s:c
308.61/291.49	false :: true:false
308.61/291.49	s :: 0':1':s:c -> 0':1':s:c
308.61/291.49	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	c :: 0':1':s:c -> 0':1':s:c
308.61/291.49	hole_true:false1_0 :: true:false
308.61/291.49	hole_0':1':s:c2_0 :: 0':1':s:c
308.61/291.49	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(5) OrderProof (LOWER BOUND(ID))
308.61/291.49	Heuristically decided to analyse the following defined symbols:
308.61/291.49	f, g
308.61/291.49	
308.61/291.49	They will be analysed ascendingly in the following order:
308.61/291.49	f < g
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(6)
308.61/291.49	Obligation:
308.61/291.49	TRS:
308.61/291.49	Rules:
308.61/291.49	f(0') -> true
308.61/291.49	f(1') -> false
308.61/291.49	f(s(x)) -> f(x)
308.61/291.49	if(true, s(x), s(y)) -> s(x)
308.61/291.49	if(false, s(x), s(y)) -> s(y)
308.61/291.49	g(x, c(y)) -> c(g(x, y))
308.61/291.49	g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	Types:
308.61/291.49	f :: 0':1':s:c -> true:false
308.61/291.49	0' :: 0':1':s:c
308.61/291.49	true :: true:false
308.61/291.49	1' :: 0':1':s:c
308.61/291.49	false :: true:false
308.61/291.49	s :: 0':1':s:c -> 0':1':s:c
308.61/291.49	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	c :: 0':1':s:c -> 0':1':s:c
308.61/291.49	hole_true:false1_0 :: true:false
308.61/291.49	hole_0':1':s:c2_0 :: 0':1':s:c
308.61/291.49	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
308.61/291.49	
308.61/291.49	
308.61/291.49	Generator Equations:
308.61/291.49	gen_0':1':s:c3_0(0) <=> 0'
308.61/291.49	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
308.61/291.49	
308.61/291.49	
308.61/291.49	The following defined symbols remain to be analysed:
308.61/291.49	f, g
308.61/291.49	
308.61/291.49	They will be analysed ascendingly in the following order:
308.61/291.49	f < g
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
308.61/291.49	Proved the following rewrite lemma:
308.61/291.49	f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
308.61/291.49	
308.61/291.49	Induction Base:
308.61/291.49	f(gen_0':1':s:c3_0(0)) ->_R^Omega(1)
308.61/291.49	true
308.61/291.49	
308.61/291.49	Induction Step:
308.61/291.49	f(gen_0':1':s:c3_0(+(n5_0, 1))) ->_R^Omega(1)
308.61/291.49	f(gen_0':1':s:c3_0(n5_0)) ->_IH
308.61/291.49	true
308.61/291.49	
308.61/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(8)
308.61/291.49	Complex Obligation (BEST)
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(9)
308.61/291.49	Obligation:
308.61/291.49	Proved the lower bound n^1 for the following obligation:
308.61/291.49	
308.61/291.49	TRS:
308.61/291.49	Rules:
308.61/291.49	f(0') -> true
308.61/291.49	f(1') -> false
308.61/291.49	f(s(x)) -> f(x)
308.61/291.49	if(true, s(x), s(y)) -> s(x)
308.61/291.49	if(false, s(x), s(y)) -> s(y)
308.61/291.49	g(x, c(y)) -> c(g(x, y))
308.61/291.49	g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	Types:
308.61/291.49	f :: 0':1':s:c -> true:false
308.61/291.49	0' :: 0':1':s:c
308.61/291.49	true :: true:false
308.61/291.49	1' :: 0':1':s:c
308.61/291.49	false :: true:false
308.61/291.49	s :: 0':1':s:c -> 0':1':s:c
308.61/291.49	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	c :: 0':1':s:c -> 0':1':s:c
308.61/291.49	hole_true:false1_0 :: true:false
308.61/291.49	hole_0':1':s:c2_0 :: 0':1':s:c
308.61/291.49	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
308.61/291.49	
308.61/291.49	
308.61/291.49	Generator Equations:
308.61/291.49	gen_0':1':s:c3_0(0) <=> 0'
308.61/291.49	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
308.61/291.49	
308.61/291.49	
308.61/291.49	The following defined symbols remain to be analysed:
308.61/291.49	f, g
308.61/291.49	
308.61/291.49	They will be analysed ascendingly in the following order:
308.61/291.49	f < g
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(10) LowerBoundPropagationProof (FINISHED)
308.61/291.49	Propagated lower bound.
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(11)
308.61/291.49	BOUNDS(n^1, INF)
308.61/291.49	
308.61/291.49	----------------------------------------
308.61/291.49	
308.61/291.49	(12)
308.61/291.49	Obligation:
308.61/291.49	TRS:
308.61/291.49	Rules:
308.61/291.49	f(0') -> true
308.61/291.49	f(1') -> false
308.61/291.49	f(s(x)) -> f(x)
308.61/291.49	if(true, s(x), s(y)) -> s(x)
308.61/291.49	if(false, s(x), s(y)) -> s(y)
308.61/291.49	g(x, c(y)) -> c(g(x, y))
308.61/291.49	g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
308.61/291.49	
308.61/291.49	Types:
308.61/291.49	f :: 0':1':s:c -> true:false
308.61/291.49	0' :: 0':1':s:c
308.61/291.49	true :: true:false
308.61/291.49	1' :: 0':1':s:c
308.61/291.49	false :: true:false
308.61/291.49	s :: 0':1':s:c -> 0':1':s:c
308.61/291.49	if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
308.61/291.49	c :: 0':1':s:c -> 0':1':s:c
308.61/291.49	hole_true:false1_0 :: true:false
308.61/291.49	hole_0':1':s:c2_0 :: 0':1':s:c
308.61/291.49	gen_0':1':s:c3_0 :: Nat -> 0':1':s:c
308.61/291.49	
308.61/291.49	
308.61/291.49	Lemmas:
308.61/291.49	f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
308.61/291.49	
308.61/291.49	
308.61/291.49	Generator Equations:
308.61/291.49	gen_0':1':s:c3_0(0) <=> 0'
308.61/291.49	gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))
308.61/291.49	
308.61/291.49	
308.61/291.49	The following defined symbols remain to be analysed:
308.61/291.49	g
308.61/291.53	EOF