325.79/291.48	WORST_CASE(Omega(n^1), ?)
325.87/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
325.87/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
325.87/291.49	
325.87/291.49	
325.87/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
325.87/291.49	
325.87/291.49	(0) CpxTRS
325.87/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
325.87/291.49	(2) CpxTRS
325.87/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
325.87/291.49	(4) typed CpxTrs
325.87/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
325.87/291.49	(6) typed CpxTrs
325.87/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
325.87/291.49	(8) BEST
325.87/291.49	    (9) proven lower bound
325.87/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
325.87/291.49	        (11) BOUNDS(n^1, INF)
325.87/291.49	    (12) typed CpxTrs
325.87/291.49	        (13) RewriteLemmaProof [LOWER BOUND(ID), 95 ms]
325.87/291.49	        (14) typed CpxTrs
325.87/291.49	        (15) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
325.87/291.49	        (16) BOUNDS(1, INF)
325.87/291.49	
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(0)
325.87/291.49	Obligation:
325.87/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
325.87/291.49	
325.87/291.49	
325.87/291.49	The TRS R consists of the following rules:
325.87/291.49	
325.87/291.49	   minus(x, 0) -> x
325.87/291.49	   minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	   double(0) -> 0
325.87/291.49	   double(s(x)) -> s(s(double(x)))
325.87/291.49	   plus(0, y) -> y
325.87/291.49	   plus(s(x), y) -> s(plus(x, y))
325.87/291.49	   plus(s(x), y) -> plus(x, s(y))
325.87/291.49	   plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	   plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	S is empty.
325.87/291.49	Rewrite Strategy: FULL
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
325.87/291.49	Renamed function symbols to avoid clashes with predefined symbol.
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(2)
325.87/291.49	Obligation:
325.87/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
325.87/291.49	
325.87/291.49	
325.87/291.49	The TRS R consists of the following rules:
325.87/291.49	
325.87/291.49	   minus(x, 0') -> x
325.87/291.49	   minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	   double(0') -> 0'
325.87/291.49	   double(s(x)) -> s(s(double(x)))
325.87/291.49	   plus(0', y) -> y
325.87/291.49	   plus(s(x), y) -> s(plus(x, y))
325.87/291.49	   plus(s(x), y) -> plus(x, s(y))
325.87/291.49	   plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	   plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	S is empty.
325.87/291.49	Rewrite Strategy: FULL
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
325.87/291.49	Infered types.
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(4)
325.87/291.49	Obligation:
325.87/291.49	TRS:
325.87/291.49	Rules:
325.87/291.49	minus(x, 0') -> x
325.87/291.49	minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	double(0') -> 0'
325.87/291.49	double(s(x)) -> s(s(double(x)))
325.87/291.49	plus(0', y) -> y
325.87/291.49	plus(s(x), y) -> s(plus(x, y))
325.87/291.49	plus(s(x), y) -> plus(x, s(y))
325.87/291.49	plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	Types:
325.87/291.49	minus :: 0':s -> 0':s -> 0':s
325.87/291.49	0' :: 0':s
325.87/291.49	s :: 0':s -> 0':s
325.87/291.49	double :: 0':s -> 0':s
325.87/291.49	plus :: 0':s -> 0':s -> 0':s
325.87/291.49	hole_0':s1_0 :: 0':s
325.87/291.49	gen_0':s2_0 :: Nat -> 0':s
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(5) OrderProof (LOWER BOUND(ID))
325.87/291.49	Heuristically decided to analyse the following defined symbols:
325.87/291.49	minus, double, plus
325.87/291.49	
325.87/291.49	They will be analysed ascendingly in the following order:
325.87/291.49	minus < plus
325.87/291.49	double < plus
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(6)
325.87/291.49	Obligation:
325.87/291.49	TRS:
325.87/291.49	Rules:
325.87/291.49	minus(x, 0') -> x
325.87/291.49	minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	double(0') -> 0'
325.87/291.49	double(s(x)) -> s(s(double(x)))
325.87/291.49	plus(0', y) -> y
325.87/291.49	plus(s(x), y) -> s(plus(x, y))
325.87/291.49	plus(s(x), y) -> plus(x, s(y))
325.87/291.49	plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	Types:
325.87/291.49	minus :: 0':s -> 0':s -> 0':s
325.87/291.49	0' :: 0':s
325.87/291.49	s :: 0':s -> 0':s
325.87/291.49	double :: 0':s -> 0':s
325.87/291.49	plus :: 0':s -> 0':s -> 0':s
325.87/291.49	hole_0':s1_0 :: 0':s
325.87/291.49	gen_0':s2_0 :: Nat -> 0':s
325.87/291.49	
325.87/291.49	
325.87/291.49	Generator Equations:
325.87/291.49	gen_0':s2_0(0) <=> 0'
325.87/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
325.87/291.49	
325.87/291.49	
325.87/291.49	The following defined symbols remain to be analysed:
325.87/291.49	minus, double, plus
325.87/291.49	
325.87/291.49	They will be analysed ascendingly in the following order:
325.87/291.49	minus < plus
325.87/291.49	double < plus
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
325.87/291.49	Proved the following rewrite lemma:
325.87/291.49	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
325.87/291.49	
325.87/291.49	Induction Base:
325.87/291.49	minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
325.87/291.49	gen_0':s2_0(0)
325.87/291.49	
325.87/291.49	Induction Step:
325.87/291.49	minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
325.87/291.49	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
325.87/291.49	gen_0':s2_0(0)
325.87/291.49	
325.87/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(8)
325.87/291.49	Complex Obligation (BEST)
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(9)
325.87/291.49	Obligation:
325.87/291.49	Proved the lower bound n^1 for the following obligation:
325.87/291.49	
325.87/291.49	TRS:
325.87/291.49	Rules:
325.87/291.49	minus(x, 0') -> x
325.87/291.49	minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	double(0') -> 0'
325.87/291.49	double(s(x)) -> s(s(double(x)))
325.87/291.49	plus(0', y) -> y
325.87/291.49	plus(s(x), y) -> s(plus(x, y))
325.87/291.49	plus(s(x), y) -> plus(x, s(y))
325.87/291.49	plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	Types:
325.87/291.49	minus :: 0':s -> 0':s -> 0':s
325.87/291.49	0' :: 0':s
325.87/291.49	s :: 0':s -> 0':s
325.87/291.49	double :: 0':s -> 0':s
325.87/291.49	plus :: 0':s -> 0':s -> 0':s
325.87/291.49	hole_0':s1_0 :: 0':s
325.87/291.49	gen_0':s2_0 :: Nat -> 0':s
325.87/291.49	
325.87/291.49	
325.87/291.49	Generator Equations:
325.87/291.49	gen_0':s2_0(0) <=> 0'
325.87/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
325.87/291.49	
325.87/291.49	
325.87/291.49	The following defined symbols remain to be analysed:
325.87/291.49	minus, double, plus
325.87/291.49	
325.87/291.49	They will be analysed ascendingly in the following order:
325.87/291.49	minus < plus
325.87/291.49	double < plus
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(10) LowerBoundPropagationProof (FINISHED)
325.87/291.49	Propagated lower bound.
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(11)
325.87/291.49	BOUNDS(n^1, INF)
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(12)
325.87/291.49	Obligation:
325.87/291.49	TRS:
325.87/291.49	Rules:
325.87/291.49	minus(x, 0') -> x
325.87/291.49	minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	double(0') -> 0'
325.87/291.49	double(s(x)) -> s(s(double(x)))
325.87/291.49	plus(0', y) -> y
325.87/291.49	plus(s(x), y) -> s(plus(x, y))
325.87/291.49	plus(s(x), y) -> plus(x, s(y))
325.87/291.49	plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	Types:
325.87/291.49	minus :: 0':s -> 0':s -> 0':s
325.87/291.49	0' :: 0':s
325.87/291.49	s :: 0':s -> 0':s
325.87/291.49	double :: 0':s -> 0':s
325.87/291.49	plus :: 0':s -> 0':s -> 0':s
325.87/291.49	hole_0':s1_0 :: 0':s
325.87/291.49	gen_0':s2_0 :: Nat -> 0':s
325.87/291.49	
325.87/291.49	
325.87/291.49	Lemmas:
325.87/291.49	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
325.87/291.49	
325.87/291.49	
325.87/291.49	Generator Equations:
325.87/291.49	gen_0':s2_0(0) <=> 0'
325.87/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
325.87/291.49	
325.87/291.49	
325.87/291.49	The following defined symbols remain to be analysed:
325.87/291.49	double, plus
325.87/291.49	
325.87/291.49	They will be analysed ascendingly in the following order:
325.87/291.49	double < plus
325.87/291.49	
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(13) RewriteLemmaProof (LOWER BOUND(ID))
325.87/291.49	Proved the following rewrite lemma:
325.87/291.49	double(gen_0':s2_0(n240_0)) -> gen_0':s2_0(*(2, n240_0)), rt in Omega(1 + n240_0)
325.87/291.49	
325.87/291.49	Induction Base:
325.87/291.49	double(gen_0':s2_0(0)) ->_R^Omega(1)
325.87/291.49	0'
325.87/291.49	
325.87/291.49	Induction Step:
325.87/291.49	double(gen_0':s2_0(+(n240_0, 1))) ->_R^Omega(1)
325.87/291.49	s(s(double(gen_0':s2_0(n240_0)))) ->_IH
325.87/291.49	s(s(gen_0':s2_0(*(2, c241_0))))
325.87/291.49	
325.87/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
325.87/291.49	----------------------------------------
325.87/291.49	
325.87/291.49	(14)
325.87/291.49	Obligation:
325.87/291.49	TRS:
325.87/291.49	Rules:
325.87/291.49	minus(x, 0') -> x
325.87/291.49	minus(s(x), s(y)) -> minus(x, y)
325.87/291.49	double(0') -> 0'
325.87/291.49	double(s(x)) -> s(s(double(x)))
325.87/291.49	plus(0', y) -> y
325.87/291.49	plus(s(x), y) -> s(plus(x, y))
325.87/291.49	plus(s(x), y) -> plus(x, s(y))
325.87/291.49	plus(s(x), y) -> s(plus(minus(x, y), double(y)))
325.87/291.49	plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))
325.87/291.49	
325.87/291.49	Types:
325.87/291.49	minus :: 0':s -> 0':s -> 0':s
325.87/291.49	0' :: 0':s
325.87/291.49	s :: 0':s -> 0':s
325.87/291.49	double :: 0':s -> 0':s
325.87/291.49	plus :: 0':s -> 0':s -> 0':s
325.87/291.49	hole_0':s1_0 :: 0':s
325.87/291.49	gen_0':s2_0 :: Nat -> 0':s
325.87/291.49	
325.87/291.49	
325.87/291.49	Lemmas:
325.87/291.49	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
325.87/291.49	double(gen_0':s2_0(n240_0)) -> gen_0':s2_0(*(2, n240_0)), rt in Omega(1 + n240_0)
325.87/291.50	
325.87/291.50	
325.87/291.50	Generator Equations:
325.87/291.50	gen_0':s2_0(0) <=> 0'
325.87/291.50	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
325.87/291.50	
325.87/291.50	
325.87/291.50	The following defined symbols remain to be analysed:
325.87/291.50	plus
325.87/291.50	----------------------------------------
325.87/291.50	
325.87/291.50	(15) RewriteLemmaProof (LOWER BOUND(ID))
325.87/291.50	Proved the following rewrite lemma:
325.87/291.50	plus(gen_0':s2_0(n484_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n484_0, b)), rt in Omega(1 + n484_0)
325.87/291.50	
325.87/291.50	Induction Base:
325.87/291.50	plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
325.87/291.50	gen_0':s2_0(b)
325.87/291.50	
325.87/291.50	Induction Step:
325.87/291.50	plus(gen_0':s2_0(+(n484_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
325.87/291.50	s(plus(gen_0':s2_0(n484_0), gen_0':s2_0(b))) ->_IH
325.87/291.50	s(gen_0':s2_0(+(b, c485_0)))
325.87/291.50	
325.87/291.50	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
325.87/291.50	----------------------------------------
325.87/291.50	
325.87/291.50	(16)
325.87/291.50	BOUNDS(1, INF)
325.87/291.53	EOF