303.11/291.48	WORST_CASE(Omega(n^1), ?)
303.11/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
303.11/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.11/291.49	
303.11/291.49	
303.11/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.11/291.49	
303.11/291.49	(0) CpxTRS
303.11/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
303.11/291.49	(2) CpxTRS
303.11/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
303.11/291.49	(4) typed CpxTrs
303.11/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
303.11/291.49	(6) typed CpxTrs
303.11/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 255 ms]
303.11/291.49	(8) BEST
303.11/291.49	    (9) proven lower bound
303.11/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
303.11/291.49	        (11) BOUNDS(n^1, INF)
303.11/291.49	    (12) typed CpxTrs
303.11/291.49	
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(0)
303.11/291.49	Obligation:
303.11/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.11/291.49	
303.11/291.49	
303.11/291.49	The TRS R consists of the following rules:
303.11/291.49	
303.11/291.49	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	   gr(0, x) -> false
303.11/291.49	   gr(s(x), 0) -> true
303.11/291.49	   gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	   p(0) -> 0
303.11/291.49	   p(s(x)) -> x
303.11/291.49	
303.11/291.49	S is empty.
303.11/291.49	Rewrite Strategy: FULL
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
303.11/291.49	Renamed function symbols to avoid clashes with predefined symbol.
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(2)
303.11/291.49	Obligation:
303.11/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.11/291.49	
303.11/291.49	
303.11/291.49	The TRS R consists of the following rules:
303.11/291.49	
303.11/291.49	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	   gr(0', x) -> false
303.11/291.49	   gr(s(x), 0') -> true
303.11/291.49	   gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	   p(0') -> 0'
303.11/291.49	   p(s(x)) -> x
303.11/291.49	
303.11/291.49	S is empty.
303.11/291.49	Rewrite Strategy: FULL
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
303.11/291.49	Infered types.
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(4)
303.11/291.49	Obligation:
303.11/291.49	TRS:
303.11/291.49	Rules:
303.11/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	gr(0', x) -> false
303.11/291.49	gr(s(x), 0') -> true
303.11/291.49	gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	p(0') -> 0'
303.11/291.49	p(s(x)) -> x
303.11/291.49	
303.11/291.49	Types:
303.11/291.49	cond :: true:false -> s:0' -> s:0' -> cond
303.11/291.49	true :: true:false
303.11/291.49	gr :: s:0' -> s:0' -> true:false
303.11/291.49	p :: s:0' -> s:0'
303.11/291.49	s :: s:0' -> s:0'
303.11/291.49	0' :: s:0'
303.11/291.49	false :: true:false
303.11/291.49	hole_cond1_0 :: cond
303.11/291.49	hole_true:false2_0 :: true:false
303.11/291.49	hole_s:0'3_0 :: s:0'
303.11/291.49	gen_s:0'4_0 :: Nat -> s:0'
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(5) OrderProof (LOWER BOUND(ID))
303.11/291.49	Heuristically decided to analyse the following defined symbols:
303.11/291.49	cond, gr
303.11/291.49	
303.11/291.49	They will be analysed ascendingly in the following order:
303.11/291.49	gr < cond
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(6)
303.11/291.49	Obligation:
303.11/291.49	TRS:
303.11/291.49	Rules:
303.11/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	gr(0', x) -> false
303.11/291.49	gr(s(x), 0') -> true
303.11/291.49	gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	p(0') -> 0'
303.11/291.49	p(s(x)) -> x
303.11/291.49	
303.11/291.49	Types:
303.11/291.49	cond :: true:false -> s:0' -> s:0' -> cond
303.11/291.49	true :: true:false
303.11/291.49	gr :: s:0' -> s:0' -> true:false
303.11/291.49	p :: s:0' -> s:0'
303.11/291.49	s :: s:0' -> s:0'
303.11/291.49	0' :: s:0'
303.11/291.49	false :: true:false
303.11/291.49	hole_cond1_0 :: cond
303.11/291.49	hole_true:false2_0 :: true:false
303.11/291.49	hole_s:0'3_0 :: s:0'
303.11/291.49	gen_s:0'4_0 :: Nat -> s:0'
303.11/291.49	
303.11/291.49	
303.11/291.49	Generator Equations:
303.11/291.49	gen_s:0'4_0(0) <=> 0'
303.11/291.49	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
303.11/291.49	
303.11/291.49	
303.11/291.49	The following defined symbols remain to be analysed:
303.11/291.49	gr, cond
303.11/291.49	
303.11/291.49	They will be analysed ascendingly in the following order:
303.11/291.49	gr < cond
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
303.11/291.49	Proved the following rewrite lemma:
303.11/291.49	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
303.11/291.49	
303.11/291.49	Induction Base:
303.11/291.49	gr(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
303.11/291.49	false
303.11/291.49	
303.11/291.49	Induction Step:
303.11/291.49	gr(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
303.11/291.49	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
303.11/291.49	false
303.11/291.49	
303.11/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(8)
303.11/291.49	Complex Obligation (BEST)
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(9)
303.11/291.49	Obligation:
303.11/291.49	Proved the lower bound n^1 for the following obligation:
303.11/291.49	
303.11/291.49	TRS:
303.11/291.49	Rules:
303.11/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	gr(0', x) -> false
303.11/291.49	gr(s(x), 0') -> true
303.11/291.49	gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	p(0') -> 0'
303.11/291.49	p(s(x)) -> x
303.11/291.49	
303.11/291.49	Types:
303.11/291.49	cond :: true:false -> s:0' -> s:0' -> cond
303.11/291.49	true :: true:false
303.11/291.49	gr :: s:0' -> s:0' -> true:false
303.11/291.49	p :: s:0' -> s:0'
303.11/291.49	s :: s:0' -> s:0'
303.11/291.49	0' :: s:0'
303.11/291.49	false :: true:false
303.11/291.49	hole_cond1_0 :: cond
303.11/291.49	hole_true:false2_0 :: true:false
303.11/291.49	hole_s:0'3_0 :: s:0'
303.11/291.49	gen_s:0'4_0 :: Nat -> s:0'
303.11/291.49	
303.11/291.49	
303.11/291.49	Generator Equations:
303.11/291.49	gen_s:0'4_0(0) <=> 0'
303.11/291.49	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
303.11/291.49	
303.11/291.49	
303.11/291.49	The following defined symbols remain to be analysed:
303.11/291.49	gr, cond
303.11/291.49	
303.11/291.49	They will be analysed ascendingly in the following order:
303.11/291.49	gr < cond
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(10) LowerBoundPropagationProof (FINISHED)
303.11/291.49	Propagated lower bound.
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(11)
303.11/291.49	BOUNDS(n^1, INF)
303.11/291.49	
303.11/291.49	----------------------------------------
303.11/291.49	
303.11/291.49	(12)
303.11/291.49	Obligation:
303.11/291.49	TRS:
303.11/291.49	Rules:
303.11/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
303.11/291.49	gr(0', x) -> false
303.11/291.49	gr(s(x), 0') -> true
303.11/291.49	gr(s(x), s(y)) -> gr(x, y)
303.11/291.49	p(0') -> 0'
303.11/291.49	p(s(x)) -> x
303.11/291.49	
303.11/291.49	Types:
303.11/291.49	cond :: true:false -> s:0' -> s:0' -> cond
303.11/291.49	true :: true:false
303.11/291.49	gr :: s:0' -> s:0' -> true:false
303.11/291.49	p :: s:0' -> s:0'
303.11/291.49	s :: s:0' -> s:0'
303.11/291.49	0' :: s:0'
303.11/291.49	false :: true:false
303.11/291.49	hole_cond1_0 :: cond
303.11/291.49	hole_true:false2_0 :: true:false
303.11/291.49	hole_s:0'3_0 :: s:0'
303.11/291.49	gen_s:0'4_0 :: Nat -> s:0'
303.11/291.49	
303.11/291.49	
303.11/291.49	Lemmas:
303.11/291.49	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
303.11/291.49	
303.11/291.49	
303.11/291.49	Generator Equations:
303.11/291.49	gen_s:0'4_0(0) <=> 0'
303.11/291.49	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
303.11/291.49	
303.11/291.49	
303.11/291.49	The following defined symbols remain to be analysed:
303.11/291.49	cond
303.11/291.53	EOF