302.40/291.49	WORST_CASE(Omega(n^1), ?)
302.52/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
302.52/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
302.52/291.53	
302.52/291.53	
302.52/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.52/291.53	
302.52/291.53	(0) CpxTRS
302.52/291.53	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
302.52/291.53	(2) CpxTRS
302.52/291.53	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
302.52/291.53	(4) typed CpxTrs
302.52/291.53	(5) OrderProof [LOWER BOUND(ID), 0 ms]
302.52/291.53	(6) typed CpxTrs
302.52/291.53	(7) RewriteLemmaProof [LOWER BOUND(ID), 294 ms]
302.52/291.53	(8) BEST
302.52/291.53	    (9) proven lower bound
302.52/291.53	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
302.52/291.53	        (11) BOUNDS(n^1, INF)
302.52/291.53	    (12) typed CpxTrs
302.52/291.53	        (13) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
302.52/291.53	        (14) typed CpxTrs
302.52/291.53	
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(0)
302.52/291.53	Obligation:
302.52/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.52/291.53	
302.52/291.53	
302.52/291.53	The TRS R consists of the following rules:
302.52/291.53	
302.52/291.53	   minus(0, x) -> 0
302.52/291.53	   minus(s(x), 0) -> s(x)
302.52/291.53	   minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	   mod(x, 0) -> 0
302.52/291.53	   mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	   if(true, x, y) -> x
302.52/291.53	   if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	   gcd(x, 0) -> x
302.52/291.53	   gcd(0, s(y)) -> s(y)
302.52/291.53	   gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	   lt(x, 0) -> false
302.52/291.53	   lt(0, s(x)) -> true
302.52/291.53	   lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	S is empty.
302.52/291.53	Rewrite Strategy: FULL
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(1) RenamingProof (BOTH BOUNDS(ID, ID))
302.52/291.53	Renamed function symbols to avoid clashes with predefined symbol.
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(2)
302.52/291.53	Obligation:
302.52/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.52/291.53	
302.52/291.53	
302.52/291.53	The TRS R consists of the following rules:
302.52/291.53	
302.52/291.53	   minus(0', x) -> 0'
302.52/291.53	   minus(s(x), 0') -> s(x)
302.52/291.53	   minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	   mod(x, 0') -> 0'
302.52/291.53	   mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	   if(true, x, y) -> x
302.52/291.53	   if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	   gcd(x, 0') -> x
302.52/291.53	   gcd(0', s(y)) -> s(y)
302.52/291.53	   gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	   lt(x, 0') -> false
302.52/291.53	   lt(0', s(x)) -> true
302.52/291.53	   lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	S is empty.
302.52/291.53	Rewrite Strategy: FULL
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
302.52/291.53	Infered types.
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(4)
302.52/291.53	Obligation:
302.52/291.53	TRS:
302.52/291.53	Rules:
302.52/291.53	minus(0', x) -> 0'
302.52/291.53	minus(s(x), 0') -> s(x)
302.52/291.53	minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	mod(x, 0') -> 0'
302.52/291.53	mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	if(true, x, y) -> x
302.52/291.53	if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	gcd(x, 0') -> x
302.52/291.53	gcd(0', s(y)) -> s(y)
302.52/291.53	gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	lt(x, 0') -> false
302.52/291.53	lt(0', s(x)) -> true
302.52/291.53	lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	Types:
302.52/291.53	minus :: 0':s -> 0':s -> 0':s
302.52/291.53	0' :: 0':s
302.52/291.53	s :: 0':s -> 0':s
302.52/291.53	mod :: 0':s -> 0':s -> 0':s
302.52/291.53	if :: true:false -> 0':s -> 0':s -> 0':s
302.52/291.53	lt :: 0':s -> 0':s -> true:false
302.52/291.53	true :: true:false
302.52/291.53	false :: true:false
302.52/291.53	gcd :: 0':s -> 0':s -> 0':s
302.52/291.53	hole_0':s1_0 :: 0':s
302.52/291.53	hole_true:false2_0 :: true:false
302.52/291.53	gen_0':s3_0 :: Nat -> 0':s
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(5) OrderProof (LOWER BOUND(ID))
302.52/291.53	Heuristically decided to analyse the following defined symbols:
302.52/291.53	minus, mod, lt, gcd
302.52/291.53	
302.52/291.53	They will be analysed ascendingly in the following order:
302.52/291.53	minus < mod
302.52/291.53	lt < mod
302.52/291.53	mod < gcd
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(6)
302.52/291.53	Obligation:
302.52/291.53	TRS:
302.52/291.53	Rules:
302.52/291.53	minus(0', x) -> 0'
302.52/291.53	minus(s(x), 0') -> s(x)
302.52/291.53	minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	mod(x, 0') -> 0'
302.52/291.53	mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	if(true, x, y) -> x
302.52/291.53	if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	gcd(x, 0') -> x
302.52/291.53	gcd(0', s(y)) -> s(y)
302.52/291.53	gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	lt(x, 0') -> false
302.52/291.53	lt(0', s(x)) -> true
302.52/291.53	lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	Types:
302.52/291.53	minus :: 0':s -> 0':s -> 0':s
302.52/291.53	0' :: 0':s
302.52/291.53	s :: 0':s -> 0':s
302.52/291.53	mod :: 0':s -> 0':s -> 0':s
302.52/291.53	if :: true:false -> 0':s -> 0':s -> 0':s
302.52/291.53	lt :: 0':s -> 0':s -> true:false
302.52/291.53	true :: true:false
302.52/291.53	false :: true:false
302.52/291.53	gcd :: 0':s -> 0':s -> 0':s
302.52/291.53	hole_0':s1_0 :: 0':s
302.52/291.53	hole_true:false2_0 :: true:false
302.52/291.53	gen_0':s3_0 :: Nat -> 0':s
302.52/291.53	
302.52/291.53	
302.52/291.53	Generator Equations:
302.52/291.53	gen_0':s3_0(0) <=> 0'
302.52/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.52/291.53	
302.52/291.53	
302.52/291.53	The following defined symbols remain to be analysed:
302.52/291.53	minus, mod, lt, gcd
302.52/291.53	
302.52/291.53	They will be analysed ascendingly in the following order:
302.52/291.53	minus < mod
302.52/291.53	lt < mod
302.52/291.53	mod < gcd
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(7) RewriteLemmaProof (LOWER BOUND(ID))
302.52/291.53	Proved the following rewrite lemma:
302.52/291.53	minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
302.52/291.53	
302.52/291.53	Induction Base:
302.52/291.53	minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
302.52/291.53	0'
302.52/291.53	
302.52/291.53	Induction Step:
302.52/291.53	minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
302.52/291.53	minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
302.52/291.53	gen_0':s3_0(0)
302.52/291.53	
302.52/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(8)
302.52/291.53	Complex Obligation (BEST)
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(9)
302.52/291.53	Obligation:
302.52/291.53	Proved the lower bound n^1 for the following obligation:
302.52/291.53	
302.52/291.53	TRS:
302.52/291.53	Rules:
302.52/291.53	minus(0', x) -> 0'
302.52/291.53	minus(s(x), 0') -> s(x)
302.52/291.53	minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	mod(x, 0') -> 0'
302.52/291.53	mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	if(true, x, y) -> x
302.52/291.53	if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	gcd(x, 0') -> x
302.52/291.53	gcd(0', s(y)) -> s(y)
302.52/291.53	gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	lt(x, 0') -> false
302.52/291.53	lt(0', s(x)) -> true
302.52/291.53	lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	Types:
302.52/291.53	minus :: 0':s -> 0':s -> 0':s
302.52/291.53	0' :: 0':s
302.52/291.53	s :: 0':s -> 0':s
302.52/291.53	mod :: 0':s -> 0':s -> 0':s
302.52/291.53	if :: true:false -> 0':s -> 0':s -> 0':s
302.52/291.53	lt :: 0':s -> 0':s -> true:false
302.52/291.53	true :: true:false
302.52/291.53	false :: true:false
302.52/291.53	gcd :: 0':s -> 0':s -> 0':s
302.52/291.53	hole_0':s1_0 :: 0':s
302.52/291.53	hole_true:false2_0 :: true:false
302.52/291.53	gen_0':s3_0 :: Nat -> 0':s
302.52/291.53	
302.52/291.53	
302.52/291.53	Generator Equations:
302.52/291.53	gen_0':s3_0(0) <=> 0'
302.52/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.52/291.53	
302.52/291.53	
302.52/291.53	The following defined symbols remain to be analysed:
302.52/291.53	minus, mod, lt, gcd
302.52/291.53	
302.52/291.53	They will be analysed ascendingly in the following order:
302.52/291.53	minus < mod
302.52/291.53	lt < mod
302.52/291.53	mod < gcd
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(10) LowerBoundPropagationProof (FINISHED)
302.52/291.53	Propagated lower bound.
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(11)
302.52/291.53	BOUNDS(n^1, INF)
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(12)
302.52/291.53	Obligation:
302.52/291.53	TRS:
302.52/291.53	Rules:
302.52/291.53	minus(0', x) -> 0'
302.52/291.53	minus(s(x), 0') -> s(x)
302.52/291.53	minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	mod(x, 0') -> 0'
302.52/291.53	mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	if(true, x, y) -> x
302.52/291.53	if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	gcd(x, 0') -> x
302.52/291.53	gcd(0', s(y)) -> s(y)
302.52/291.53	gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	lt(x, 0') -> false
302.52/291.53	lt(0', s(x)) -> true
302.52/291.53	lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	Types:
302.52/291.53	minus :: 0':s -> 0':s -> 0':s
302.52/291.53	0' :: 0':s
302.52/291.53	s :: 0':s -> 0':s
302.52/291.53	mod :: 0':s -> 0':s -> 0':s
302.52/291.53	if :: true:false -> 0':s -> 0':s -> 0':s
302.52/291.53	lt :: 0':s -> 0':s -> true:false
302.52/291.53	true :: true:false
302.52/291.53	false :: true:false
302.52/291.53	gcd :: 0':s -> 0':s -> 0':s
302.52/291.53	hole_0':s1_0 :: 0':s
302.52/291.53	hole_true:false2_0 :: true:false
302.52/291.53	gen_0':s3_0 :: Nat -> 0':s
302.52/291.53	
302.52/291.53	
302.52/291.53	Lemmas:
302.52/291.53	minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
302.52/291.53	
302.52/291.53	
302.52/291.53	Generator Equations:
302.52/291.53	gen_0':s3_0(0) <=> 0'
302.52/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.52/291.53	
302.52/291.53	
302.52/291.53	The following defined symbols remain to be analysed:
302.52/291.53	lt, mod, gcd
302.52/291.53	
302.52/291.53	They will be analysed ascendingly in the following order:
302.52/291.53	lt < mod
302.52/291.53	mod < gcd
302.52/291.53	
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(13) RewriteLemmaProof (LOWER BOUND(ID))
302.52/291.53	Proved the following rewrite lemma:
302.52/291.53	lt(gen_0':s3_0(n404_0), gen_0':s3_0(n404_0)) -> false, rt in Omega(1 + n404_0)
302.52/291.53	
302.52/291.53	Induction Base:
302.52/291.53	lt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
302.52/291.53	false
302.52/291.53	
302.52/291.53	Induction Step:
302.52/291.53	lt(gen_0':s3_0(+(n404_0, 1)), gen_0':s3_0(+(n404_0, 1))) ->_R^Omega(1)
302.52/291.53	lt(gen_0':s3_0(n404_0), gen_0':s3_0(n404_0)) ->_IH
302.52/291.53	false
302.52/291.53	
302.52/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
302.52/291.53	----------------------------------------
302.52/291.53	
302.52/291.53	(14)
302.52/291.53	Obligation:
302.52/291.53	TRS:
302.52/291.53	Rules:
302.52/291.53	minus(0', x) -> 0'
302.52/291.53	minus(s(x), 0') -> s(x)
302.52/291.53	minus(s(x), s(y)) -> minus(x, y)
302.52/291.53	mod(x, 0') -> 0'
302.52/291.53	mod(x, s(y)) -> if(lt(x, s(y)), x, s(y))
302.52/291.53	if(true, x, y) -> x
302.52/291.53	if(false, x, y) -> mod(minus(x, y), y)
302.52/291.53	gcd(x, 0') -> x
302.52/291.53	gcd(0', s(y)) -> s(y)
302.52/291.53	gcd(s(x), s(y)) -> gcd(mod(s(x), s(y)), mod(s(y), s(x)))
302.52/291.53	lt(x, 0') -> false
302.52/291.53	lt(0', s(x)) -> true
302.52/291.53	lt(s(x), s(y)) -> lt(x, y)
302.52/291.53	
302.52/291.53	Types:
302.52/291.53	minus :: 0':s -> 0':s -> 0':s
302.52/291.53	0' :: 0':s
302.52/291.53	s :: 0':s -> 0':s
302.52/291.53	mod :: 0':s -> 0':s -> 0':s
302.52/291.53	if :: true:false -> 0':s -> 0':s -> 0':s
302.52/291.53	lt :: 0':s -> 0':s -> true:false
302.52/291.53	true :: true:false
302.52/291.53	false :: true:false
302.52/291.53	gcd :: 0':s -> 0':s -> 0':s
302.52/291.53	hole_0':s1_0 :: 0':s
302.52/291.53	hole_true:false2_0 :: true:false
302.52/291.53	gen_0':s3_0 :: Nat -> 0':s
302.52/291.53	
302.52/291.53	
302.52/291.53	Lemmas:
302.52/291.53	minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
302.52/291.53	lt(gen_0':s3_0(n404_0), gen_0':s3_0(n404_0)) -> false, rt in Omega(1 + n404_0)
302.52/291.53	
302.52/291.53	
302.52/291.53	Generator Equations:
302.52/291.53	gen_0':s3_0(0) <=> 0'
302.52/291.53	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.52/291.53	
302.52/291.53	
302.52/291.53	The following defined symbols remain to be analysed:
302.52/291.53	mod, gcd
302.52/291.53	
302.52/291.53	They will be analysed ascendingly in the following order:
302.52/291.53	mod < gcd
302.52/291.57	EOF