311.30/291.50	WORST_CASE(Omega(n^1), ?)
311.37/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
311.37/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.37/291.51	
311.37/291.51	
311.37/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.37/291.51	
311.37/291.51	(0) CpxTRS
311.37/291.51	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
311.37/291.51	(2) CpxTRS
311.37/291.51	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
311.37/291.51	(4) typed CpxTrs
311.37/291.51	(5) OrderProof [LOWER BOUND(ID), 0 ms]
311.37/291.51	(6) typed CpxTrs
311.37/291.51	(7) RewriteLemmaProof [LOWER BOUND(ID), 346 ms]
311.37/291.51	(8) BEST
311.37/291.51	    (9) proven lower bound
311.37/291.51	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
311.37/291.51	        (11) BOUNDS(n^1, INF)
311.37/291.51	    (12) typed CpxTrs
311.37/291.51	        (13) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
311.37/291.51	        (14) typed CpxTrs
311.37/291.51	        (15) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
311.37/291.51	        (16) typed CpxTrs
311.37/291.51	
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(0)
311.37/291.51	Obligation:
311.37/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.37/291.51	
311.37/291.51	
311.37/291.51	The TRS R consists of the following rules:
311.37/291.51	
311.37/291.51	   min(x, 0) -> 0
311.37/291.51	   min(0, y) -> 0
311.37/291.51	   min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	   max(x, 0) -> x
311.37/291.51	   max(0, y) -> y
311.37/291.51	   max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	   -(x, 0) -> x
311.37/291.51	   -(s(x), s(y)) -> -(x, y)
311.37/291.51	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	   gcd(x, 0, 0) -> x
311.37/291.51	   gcd(0, y, 0) -> y
311.37/291.51	   gcd(0, 0, z) -> z
311.37/291.51	
311.37/291.51	S is empty.
311.37/291.51	Rewrite Strategy: FULL
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(1) RenamingProof (BOTH BOUNDS(ID, ID))
311.37/291.51	Renamed function symbols to avoid clashes with predefined symbol.
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(2)
311.37/291.51	Obligation:
311.37/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.37/291.51	
311.37/291.51	
311.37/291.51	The TRS R consists of the following rules:
311.37/291.51	
311.37/291.51	   min(x, 0') -> 0'
311.37/291.51	   min(0', y) -> 0'
311.37/291.51	   min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	   max(x, 0') -> x
311.37/291.51	   max(0', y) -> y
311.37/291.51	   max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	   -(x, 0') -> x
311.37/291.51	   -(s(x), s(y)) -> -(x, y)
311.37/291.51	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	   gcd(x, 0', 0') -> x
311.37/291.51	   gcd(0', y, 0') -> y
311.37/291.51	   gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	S is empty.
311.37/291.51	Rewrite Strategy: FULL
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
311.37/291.51	Infered types.
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(4)
311.37/291.51	Obligation:
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(5) OrderProof (LOWER BOUND(ID))
311.37/291.51	Heuristically decided to analyse the following defined symbols:
311.37/291.51	min, max, -, gcd
311.37/291.51	
311.37/291.51	They will be analysed ascendingly in the following order:
311.37/291.51	min < gcd
311.37/291.51	max < gcd
311.37/291.51	- < gcd
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(6)
311.37/291.51	Obligation:
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	
311.37/291.51	Generator Equations:
311.37/291.51	gen_0':s2_0(0) <=> 0'
311.37/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
311.37/291.51	
311.37/291.51	
311.37/291.51	The following defined symbols remain to be analysed:
311.37/291.51	min, max, -, gcd
311.37/291.51	
311.37/291.51	They will be analysed ascendingly in the following order:
311.37/291.51	min < gcd
311.37/291.51	max < gcd
311.37/291.51	- < gcd
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(7) RewriteLemmaProof (LOWER BOUND(ID))
311.37/291.51	Proved the following rewrite lemma:
311.37/291.51	min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
311.37/291.51	
311.37/291.51	Induction Base:
311.37/291.51	min(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
311.37/291.51	0'
311.37/291.51	
311.37/291.51	Induction Step:
311.37/291.51	min(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
311.37/291.51	s(min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0))) ->_IH
311.37/291.51	s(gen_0':s2_0(c5_0))
311.37/291.51	
311.37/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(8)
311.37/291.51	Complex Obligation (BEST)
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(9)
311.37/291.51	Obligation:
311.37/291.51	Proved the lower bound n^1 for the following obligation:
311.37/291.51	
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	
311.37/291.51	Generator Equations:
311.37/291.51	gen_0':s2_0(0) <=> 0'
311.37/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
311.37/291.51	
311.37/291.51	
311.37/291.51	The following defined symbols remain to be analysed:
311.37/291.51	min, max, -, gcd
311.37/291.51	
311.37/291.51	They will be analysed ascendingly in the following order:
311.37/291.51	min < gcd
311.37/291.51	max < gcd
311.37/291.51	- < gcd
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(10) LowerBoundPropagationProof (FINISHED)
311.37/291.51	Propagated lower bound.
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(11)
311.37/291.51	BOUNDS(n^1, INF)
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(12)
311.37/291.51	Obligation:
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	
311.37/291.51	Lemmas:
311.37/291.51	min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
311.37/291.51	
311.37/291.51	
311.37/291.51	Generator Equations:
311.37/291.51	gen_0':s2_0(0) <=> 0'
311.37/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
311.37/291.51	
311.37/291.51	
311.37/291.51	The following defined symbols remain to be analysed:
311.37/291.51	max, -, gcd
311.37/291.51	
311.37/291.51	They will be analysed ascendingly in the following order:
311.37/291.51	max < gcd
311.37/291.51	- < gcd
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(13) RewriteLemmaProof (LOWER BOUND(ID))
311.37/291.51	Proved the following rewrite lemma:
311.37/291.51	max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)
311.37/291.51	
311.37/291.51	Induction Base:
311.37/291.51	max(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
311.37/291.51	gen_0':s2_0(0)
311.37/291.51	
311.37/291.51	Induction Step:
311.37/291.51	max(gen_0':s2_0(+(n321_0, 1)), gen_0':s2_0(+(n321_0, 1))) ->_R^Omega(1)
311.37/291.51	s(max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0))) ->_IH
311.37/291.51	s(gen_0':s2_0(c322_0))
311.37/291.51	
311.37/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(14)
311.37/291.51	Obligation:
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	
311.37/291.51	Lemmas:
311.37/291.51	min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
311.37/291.51	max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)
311.37/291.51	
311.37/291.51	
311.37/291.51	Generator Equations:
311.37/291.51	gen_0':s2_0(0) <=> 0'
311.37/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
311.37/291.51	
311.37/291.51	
311.37/291.51	The following defined symbols remain to be analysed:
311.37/291.51	-, gcd
311.37/291.51	
311.37/291.51	They will be analysed ascendingly in the following order:
311.37/291.51	- < gcd
311.37/291.51	
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(15) RewriteLemmaProof (LOWER BOUND(ID))
311.37/291.51	Proved the following rewrite lemma:
311.37/291.51	-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) -> gen_0':s2_0(0), rt in Omega(1 + n724_0)
311.37/291.51	
311.37/291.51	Induction Base:
311.37/291.51	-(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
311.37/291.51	gen_0':s2_0(0)
311.37/291.51	
311.37/291.51	Induction Step:
311.37/291.51	-(gen_0':s2_0(+(n724_0, 1)), gen_0':s2_0(+(n724_0, 1))) ->_R^Omega(1)
311.37/291.51	-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) ->_IH
311.37/291.51	gen_0':s2_0(0)
311.37/291.51	
311.37/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.37/291.51	----------------------------------------
311.37/291.51	
311.37/291.51	(16)
311.37/291.51	Obligation:
311.37/291.51	TRS:
311.37/291.51	Rules:
311.37/291.51	min(x, 0') -> 0'
311.37/291.51	min(0', y) -> 0'
311.37/291.51	min(s(x), s(y)) -> s(min(x, y))
311.37/291.51	max(x, 0') -> x
311.37/291.51	max(0', y) -> y
311.37/291.51	max(s(x), s(y)) -> s(max(x, y))
311.37/291.51	-(x, 0') -> x
311.37/291.51	-(s(x), s(y)) -> -(x, y)
311.37/291.51	gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
311.37/291.51	gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
311.37/291.51	gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
311.37/291.51	gcd(x, 0', 0') -> x
311.37/291.51	gcd(0', y, 0') -> y
311.37/291.51	gcd(0', 0', z) -> z
311.37/291.51	
311.37/291.51	Types:
311.37/291.51	min :: 0':s -> 0':s -> 0':s
311.37/291.51	0' :: 0':s
311.37/291.51	s :: 0':s -> 0':s
311.37/291.51	max :: 0':s -> 0':s -> 0':s
311.37/291.51	- :: 0':s -> 0':s -> 0':s
311.37/291.51	gcd :: 0':s -> 0':s -> 0':s -> 0':s
311.37/291.51	hole_0':s1_0 :: 0':s
311.37/291.51	gen_0':s2_0 :: Nat -> 0':s
311.37/291.51	
311.37/291.51	
311.37/291.51	Lemmas:
311.37/291.51	min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
311.37/291.51	max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)
311.37/291.51	-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) -> gen_0':s2_0(0), rt in Omega(1 + n724_0)
311.37/291.51	
311.37/291.51	
311.37/291.51	Generator Equations:
311.37/291.51	gen_0':s2_0(0) <=> 0'
311.37/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
311.37/291.51	
311.37/291.51	
311.37/291.51	The following defined symbols remain to be analysed:
311.37/291.51	gcd
311.37/291.55	EOF