305.36/291.51	WORST_CASE(Omega(n^1), ?)
305.36/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
305.36/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.36/291.52	
305.36/291.52	
305.36/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.36/291.52	
305.36/291.52	(0) CpxTRS
305.36/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
305.36/291.52	(2) CpxTRS
305.36/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
305.36/291.52	(4) typed CpxTrs
305.36/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
305.36/291.52	(6) typed CpxTrs
305.36/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 292 ms]
305.36/291.52	(8) BEST
305.36/291.52	    (9) proven lower bound
305.36/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
305.36/291.52	        (11) BOUNDS(n^1, INF)
305.36/291.52	    (12) typed CpxTrs
305.36/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 150 ms]
305.36/291.52	        (14) typed CpxTrs
305.36/291.52	
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(0)
305.36/291.52	Obligation:
305.36/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.36/291.52	
305.36/291.52	
305.36/291.52	The TRS R consists of the following rules:
305.36/291.52	
305.36/291.52	   f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	   trunc(0) -> 0
305.36/291.52	   trunc(s(0)) -> 0
305.36/291.52	   trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	   gt(0, v) -> false
305.36/291.52	   gt(s(u), 0) -> true
305.36/291.52	   gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	S is empty.
305.36/291.52	Rewrite Strategy: FULL
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
305.36/291.52	Renamed function symbols to avoid clashes with predefined symbol.
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(2)
305.36/291.52	Obligation:
305.36/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.36/291.52	
305.36/291.52	
305.36/291.52	The TRS R consists of the following rules:
305.36/291.52	
305.36/291.52	   f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	   trunc(0') -> 0'
305.36/291.52	   trunc(s(0')) -> 0'
305.36/291.52	   trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	   gt(0', v) -> false
305.36/291.52	   gt(s(u), 0') -> true
305.36/291.52	   gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	S is empty.
305.36/291.52	Rewrite Strategy: FULL
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
305.36/291.52	Infered types.
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(4)
305.36/291.52	Obligation:
305.36/291.52	TRS:
305.36/291.52	Rules:
305.36/291.52	f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	trunc(0') -> 0'
305.36/291.52	trunc(s(0')) -> 0'
305.36/291.52	trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	gt(0', v) -> false
305.36/291.52	gt(s(u), 0') -> true
305.36/291.52	gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	Types:
305.36/291.52	f :: true:false -> s:0' -> s:0' -> f
305.36/291.52	true :: true:false
305.36/291.52	gt :: s:0' -> s:0' -> true:false
305.36/291.52	trunc :: s:0' -> s:0'
305.36/291.52	s :: s:0' -> s:0'
305.36/291.52	0' :: s:0'
305.36/291.52	false :: true:false
305.36/291.52	hole_f1_0 :: f
305.36/291.52	hole_true:false2_0 :: true:false
305.36/291.52	hole_s:0'3_0 :: s:0'
305.36/291.52	gen_s:0'4_0 :: Nat -> s:0'
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(5) OrderProof (LOWER BOUND(ID))
305.36/291.52	Heuristically decided to analyse the following defined symbols:
305.36/291.52	f, gt, trunc
305.36/291.52	
305.36/291.52	They will be analysed ascendingly in the following order:
305.36/291.52	gt < f
305.36/291.52	trunc < f
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(6)
305.36/291.52	Obligation:
305.36/291.52	TRS:
305.36/291.52	Rules:
305.36/291.52	f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	trunc(0') -> 0'
305.36/291.52	trunc(s(0')) -> 0'
305.36/291.52	trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	gt(0', v) -> false
305.36/291.52	gt(s(u), 0') -> true
305.36/291.52	gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	Types:
305.36/291.52	f :: true:false -> s:0' -> s:0' -> f
305.36/291.52	true :: true:false
305.36/291.52	gt :: s:0' -> s:0' -> true:false
305.36/291.52	trunc :: s:0' -> s:0'
305.36/291.52	s :: s:0' -> s:0'
305.36/291.52	0' :: s:0'
305.36/291.52	false :: true:false
305.36/291.52	hole_f1_0 :: f
305.36/291.52	hole_true:false2_0 :: true:false
305.36/291.52	hole_s:0'3_0 :: s:0'
305.36/291.52	gen_s:0'4_0 :: Nat -> s:0'
305.36/291.52	
305.36/291.52	
305.36/291.52	Generator Equations:
305.36/291.52	gen_s:0'4_0(0) <=> 0'
305.36/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
305.36/291.52	
305.36/291.52	
305.36/291.52	The following defined symbols remain to be analysed:
305.36/291.52	gt, f, trunc
305.36/291.52	
305.36/291.52	They will be analysed ascendingly in the following order:
305.36/291.52	gt < f
305.36/291.52	trunc < f
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
305.36/291.52	Proved the following rewrite lemma:
305.36/291.52	gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
305.36/291.52	
305.36/291.52	Induction Base:
305.36/291.52	gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
305.36/291.52	false
305.36/291.52	
305.36/291.52	Induction Step:
305.36/291.52	gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
305.36/291.52	gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
305.36/291.52	false
305.36/291.52	
305.36/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(8)
305.36/291.52	Complex Obligation (BEST)
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(9)
305.36/291.52	Obligation:
305.36/291.52	Proved the lower bound n^1 for the following obligation:
305.36/291.52	
305.36/291.52	TRS:
305.36/291.52	Rules:
305.36/291.52	f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	trunc(0') -> 0'
305.36/291.52	trunc(s(0')) -> 0'
305.36/291.52	trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	gt(0', v) -> false
305.36/291.52	gt(s(u), 0') -> true
305.36/291.52	gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	Types:
305.36/291.52	f :: true:false -> s:0' -> s:0' -> f
305.36/291.52	true :: true:false
305.36/291.52	gt :: s:0' -> s:0' -> true:false
305.36/291.52	trunc :: s:0' -> s:0'
305.36/291.52	s :: s:0' -> s:0'
305.36/291.52	0' :: s:0'
305.36/291.52	false :: true:false
305.36/291.52	hole_f1_0 :: f
305.36/291.52	hole_true:false2_0 :: true:false
305.36/291.52	hole_s:0'3_0 :: s:0'
305.36/291.52	gen_s:0'4_0 :: Nat -> s:0'
305.36/291.52	
305.36/291.52	
305.36/291.52	Generator Equations:
305.36/291.52	gen_s:0'4_0(0) <=> 0'
305.36/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
305.36/291.52	
305.36/291.52	
305.36/291.52	The following defined symbols remain to be analysed:
305.36/291.52	gt, f, trunc
305.36/291.52	
305.36/291.52	They will be analysed ascendingly in the following order:
305.36/291.52	gt < f
305.36/291.52	trunc < f
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(10) LowerBoundPropagationProof (FINISHED)
305.36/291.52	Propagated lower bound.
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(11)
305.36/291.52	BOUNDS(n^1, INF)
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(12)
305.36/291.52	Obligation:
305.36/291.52	TRS:
305.36/291.52	Rules:
305.36/291.52	f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	trunc(0') -> 0'
305.36/291.52	trunc(s(0')) -> 0'
305.36/291.52	trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	gt(0', v) -> false
305.36/291.52	gt(s(u), 0') -> true
305.36/291.52	gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	Types:
305.36/291.52	f :: true:false -> s:0' -> s:0' -> f
305.36/291.52	true :: true:false
305.36/291.52	gt :: s:0' -> s:0' -> true:false
305.36/291.52	trunc :: s:0' -> s:0'
305.36/291.52	s :: s:0' -> s:0'
305.36/291.52	0' :: s:0'
305.36/291.52	false :: true:false
305.36/291.52	hole_f1_0 :: f
305.36/291.52	hole_true:false2_0 :: true:false
305.36/291.52	hole_s:0'3_0 :: s:0'
305.36/291.52	gen_s:0'4_0 :: Nat -> s:0'
305.36/291.52	
305.36/291.52	
305.36/291.52	Lemmas:
305.36/291.52	gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
305.36/291.52	
305.36/291.52	
305.36/291.52	Generator Equations:
305.36/291.52	gen_s:0'4_0(0) <=> 0'
305.36/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
305.36/291.52	
305.36/291.52	
305.36/291.52	The following defined symbols remain to be analysed:
305.36/291.52	trunc, f
305.36/291.52	
305.36/291.52	They will be analysed ascendingly in the following order:
305.36/291.52	trunc < f
305.36/291.52	
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
305.36/291.52	Proved the following rewrite lemma:
305.36/291.52	trunc(gen_s:0'4_0(*(2, n235_0))) -> gen_s:0'4_0(*(2, n235_0)), rt in Omega(1 + n235_0)
305.36/291.52	
305.36/291.52	Induction Base:
305.36/291.52	trunc(gen_s:0'4_0(*(2, 0))) ->_R^Omega(1)
305.36/291.52	0'
305.36/291.52	
305.36/291.52	Induction Step:
305.36/291.52	trunc(gen_s:0'4_0(*(2, +(n235_0, 1)))) ->_R^Omega(1)
305.36/291.52	s(s(trunc(gen_s:0'4_0(*(2, n235_0))))) ->_IH
305.36/291.52	s(s(gen_s:0'4_0(*(2, c236_0))))
305.36/291.52	
305.36/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
305.36/291.52	----------------------------------------
305.36/291.52	
305.36/291.52	(14)
305.36/291.52	Obligation:
305.36/291.52	TRS:
305.36/291.52	Rules:
305.36/291.52	f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
305.36/291.52	trunc(0') -> 0'
305.36/291.52	trunc(s(0')) -> 0'
305.36/291.52	trunc(s(s(x))) -> s(s(trunc(x)))
305.36/291.52	gt(0', v) -> false
305.36/291.52	gt(s(u), 0') -> true
305.36/291.52	gt(s(u), s(v)) -> gt(u, v)
305.36/291.52	
305.36/291.52	Types:
305.36/291.52	f :: true:false -> s:0' -> s:0' -> f
305.36/291.52	true :: true:false
305.36/291.52	gt :: s:0' -> s:0' -> true:false
305.36/291.52	trunc :: s:0' -> s:0'
305.36/291.52	s :: s:0' -> s:0'
305.36/291.52	0' :: s:0'
305.36/291.52	false :: true:false
305.36/291.52	hole_f1_0 :: f
305.36/291.52	hole_true:false2_0 :: true:false
305.36/291.52	hole_s:0'3_0 :: s:0'
305.36/291.52	gen_s:0'4_0 :: Nat -> s:0'
305.36/291.52	
305.36/291.52	
305.36/291.52	Lemmas:
305.36/291.52	gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
305.36/291.52	trunc(gen_s:0'4_0(*(2, n235_0))) -> gen_s:0'4_0(*(2, n235_0)), rt in Omega(1 + n235_0)
305.36/291.52	
305.36/291.52	
305.36/291.52	Generator Equations:
305.36/291.52	gen_s:0'4_0(0) <=> 0'
305.36/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
305.36/291.52	
305.36/291.52	
305.36/291.52	The following defined symbols remain to be analysed:
305.36/291.52	f
305.51/291.55	EOF