1147.81/291.55	WORST_CASE(Omega(n^1), ?)
1153.96/293.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1153.96/293.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.96/293.12	
1153.96/293.12	(0) CpxTRS
1153.96/293.12	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1153.96/293.12	(2) CpxTRS
1153.96/293.12	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1153.96/293.12	(4) typed CpxTrs
1153.96/293.12	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1153.96/293.12	(6) typed CpxTrs
1153.96/293.12	(7) RewriteLemmaProof [LOWER BOUND(ID), 288 ms]
1153.96/293.12	(8) BEST
1153.96/293.12	    (9) proven lower bound
1153.96/293.12	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1153.96/293.12	        (11) BOUNDS(n^1, INF)
1153.96/293.12	    (12) typed CpxTrs
1153.96/293.12	        (13) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
1153.96/293.12	        (14) typed CpxTrs
1153.96/293.12	        (15) RewriteLemmaProof [LOWER BOUND(ID), 79 ms]
1153.96/293.12	        (16) typed CpxTrs
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(0)
1153.96/293.12	Obligation:
1153.96/293.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The TRS R consists of the following rules:
1153.96/293.12	
1153.96/293.12	   -(x, 0) -> x
1153.96/293.12	   -(s(x), s(y)) -> -(x, y)
1153.96/293.12	   min(x, 0) -> 0
1153.96/293.12	   min(0, y) -> 0
1153.96/293.12	   min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	   twice(0) -> 0
1153.96/293.12	   twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	S is empty.
1153.96/293.12	Rewrite Strategy: INNERMOST
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1153.96/293.12	Renamed function symbols to avoid clashes with predefined symbol.
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(2)
1153.96/293.12	Obligation:
1153.96/293.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The TRS R consists of the following rules:
1153.96/293.12	
1153.96/293.12	   -(x, 0') -> x
1153.96/293.12	   -(s(x), s(y)) -> -(x, y)
1153.96/293.12	   min(x, 0') -> 0'
1153.96/293.12	   min(0', y) -> 0'
1153.96/293.12	   min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	   twice(0') -> 0'
1153.96/293.12	   twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	S is empty.
1153.96/293.12	Rewrite Strategy: INNERMOST
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1153.96/293.12	Infered types.
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(4)
1153.96/293.12	Obligation:
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(5) OrderProof (LOWER BOUND(ID))
1153.96/293.12	Heuristically decided to analyse the following defined symbols:
1153.96/293.12	-, min, twice, f
1153.96/293.12	
1153.96/293.12	They will be analysed ascendingly in the following order:
1153.96/293.12	- < f
1153.96/293.12	min < f
1153.96/293.12	twice < f
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(6)
1153.96/293.12	Obligation:
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Generator Equations:
1153.96/293.12	gen_0':s3_0(0) <=> 0'
1153.96/293.12	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The following defined symbols remain to be analysed:
1153.96/293.12	-, min, twice, f
1153.96/293.12	
1153.96/293.12	They will be analysed ascendingly in the following order:
1153.96/293.12	- < f
1153.96/293.12	min < f
1153.96/293.12	twice < f
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(7) RewriteLemmaProof (LOWER BOUND(ID))
1153.96/293.12	Proved the following rewrite lemma:
1153.96/293.12	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
1153.96/293.12	
1153.96/293.12	Induction Base:
1153.96/293.12	-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
1153.96/293.12	gen_0':s3_0(0)
1153.96/293.12	
1153.96/293.12	Induction Step:
1153.96/293.12	-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
1153.96/293.12	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
1153.96/293.12	gen_0':s3_0(0)
1153.96/293.12	
1153.96/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(8)
1153.96/293.12	Complex Obligation (BEST)
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(9)
1153.96/293.12	Obligation:
1153.96/293.12	Proved the lower bound n^1 for the following obligation:
1153.96/293.12	
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Generator Equations:
1153.96/293.12	gen_0':s3_0(0) <=> 0'
1153.96/293.12	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The following defined symbols remain to be analysed:
1153.96/293.12	-, min, twice, f
1153.96/293.12	
1153.96/293.12	They will be analysed ascendingly in the following order:
1153.96/293.12	- < f
1153.96/293.12	min < f
1153.96/293.12	twice < f
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(10) LowerBoundPropagationProof (FINISHED)
1153.96/293.12	Propagated lower bound.
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(11)
1153.96/293.12	BOUNDS(n^1, INF)
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(12)
1153.96/293.12	Obligation:
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Lemmas:
1153.96/293.12	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Generator Equations:
1153.96/293.12	gen_0':s3_0(0) <=> 0'
1153.96/293.12	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The following defined symbols remain to be analysed:
1153.96/293.12	min, twice, f
1153.96/293.12	
1153.96/293.12	They will be analysed ascendingly in the following order:
1153.96/293.12	min < f
1153.96/293.12	twice < f
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(13) RewriteLemmaProof (LOWER BOUND(ID))
1153.96/293.12	Proved the following rewrite lemma:
1153.96/293.12	min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0)
1153.96/293.12	
1153.96/293.12	Induction Base:
1153.96/293.12	min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
1153.96/293.12	0'
1153.96/293.12	
1153.96/293.12	Induction Step:
1153.96/293.12	min(gen_0':s3_0(+(n297_0, 1)), gen_0':s3_0(+(n297_0, 1))) ->_R^Omega(1)
1153.96/293.12	s(min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0))) ->_IH
1153.96/293.12	s(gen_0':s3_0(c298_0))
1153.96/293.12	
1153.96/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(14)
1153.96/293.12	Obligation:
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Lemmas:
1153.96/293.12	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
1153.96/293.12	min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0)
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Generator Equations:
1153.96/293.12	gen_0':s3_0(0) <=> 0'
1153.96/293.12	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The following defined symbols remain to be analysed:
1153.96/293.12	twice, f
1153.96/293.12	
1153.96/293.12	They will be analysed ascendingly in the following order:
1153.96/293.12	twice < f
1153.96/293.12	
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(15) RewriteLemmaProof (LOWER BOUND(ID))
1153.96/293.12	Proved the following rewrite lemma:
1153.96/293.12	twice(gen_0':s3_0(n635_0)) -> gen_0':s3_0(*(2, n635_0)), rt in Omega(1 + n635_0)
1153.96/293.12	
1153.96/293.12	Induction Base:
1153.96/293.12	twice(gen_0':s3_0(0)) ->_R^Omega(1)
1153.96/293.12	0'
1153.96/293.12	
1153.96/293.12	Induction Step:
1153.96/293.12	twice(gen_0':s3_0(+(n635_0, 1))) ->_R^Omega(1)
1153.96/293.12	s(s(twice(gen_0':s3_0(n635_0)))) ->_IH
1153.96/293.12	s(s(gen_0':s3_0(*(2, c636_0))))
1153.96/293.12	
1153.96/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1153.96/293.12	----------------------------------------
1153.96/293.12	
1153.96/293.12	(16)
1153.96/293.12	Obligation:
1153.96/293.12	Innermost TRS:
1153.96/293.12	Rules:
1153.96/293.12	-(x, 0') -> x
1153.96/293.12	-(s(x), s(y)) -> -(x, y)
1153.96/293.12	min(x, 0') -> 0'
1153.96/293.12	min(0', y) -> 0'
1153.96/293.12	min(s(x), s(y)) -> s(min(x, y))
1153.96/293.12	twice(0') -> 0'
1153.96/293.12	twice(s(x)) -> s(s(twice(x)))
1153.96/293.12	f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
1153.96/293.12	
1153.96/293.12	Types:
1153.96/293.12	- :: 0':s -> 0':s -> 0':s
1153.96/293.12	0' :: 0':s
1153.96/293.12	s :: 0':s -> 0':s
1153.96/293.12	min :: 0':s -> 0':s -> 0':s
1153.96/293.12	twice :: 0':s -> 0':s
1153.96/293.12	f :: 0':s -> 0':s -> f
1153.96/293.12	hole_0':s1_0 :: 0':s
1153.96/293.12	hole_f2_0 :: f
1153.96/293.12	gen_0':s3_0 :: Nat -> 0':s
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Lemmas:
1153.96/293.12	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
1153.96/293.12	min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0)
1153.96/293.12	twice(gen_0':s3_0(n635_0)) -> gen_0':s3_0(*(2, n635_0)), rt in Omega(1 + n635_0)
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	Generator Equations:
1153.96/293.12	gen_0':s3_0(0) <=> 0'
1153.96/293.12	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1153.96/293.12	
1153.96/293.12	
1153.96/293.12	The following defined symbols remain to be analysed:
1153.96/293.12	f
1154.23/293.19	EOF