314.84/291.47	WORST_CASE(Omega(n^2), ?)
314.84/291.48	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
314.84/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.84/291.48	
314.84/291.48	
314.84/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
314.84/291.48	
314.84/291.48	(0) CpxTRS
314.84/291.48	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
314.84/291.48	(2) CpxTRS
314.84/291.48	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
314.84/291.48	(4) typed CpxTrs
314.84/291.48	(5) OrderProof [LOWER BOUND(ID), 0 ms]
314.84/291.48	(6) typed CpxTrs
314.84/291.48	(7) RewriteLemmaProof [LOWER BOUND(ID), 278 ms]
314.84/291.48	(8) BEST
314.84/291.48	    (9) proven lower bound
314.84/291.48	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
314.84/291.48	        (11) BOUNDS(n^1, INF)
314.84/291.48	    (12) typed CpxTrs
314.84/291.48	        (13) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
314.84/291.48	        (14) typed CpxTrs
314.84/291.48	        (15) RewriteLemmaProof [LOWER BOUND(ID), 87 ms]
314.84/291.48	        (16) BEST
314.84/291.48	            (17) proven lower bound
314.84/291.48	                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
314.84/291.48	                (19) BOUNDS(n^2, INF)
314.84/291.48	            (20) typed CpxTrs
314.84/291.48	
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(0)
314.84/291.48	Obligation:
314.84/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
314.84/291.48	
314.84/291.48	
314.84/291.48	The TRS R consists of the following rules:
314.84/291.48	
314.84/291.48	   -(x, 0) -> x
314.84/291.48	   -(s(x), s(y)) -> -(x, y)
314.84/291.48	   +(0, y) -> y
314.84/291.48	   +(s(x), y) -> s(+(x, y))
314.84/291.48	   *(x, 0) -> 0
314.84/291.48	   *(x, s(y)) -> +(x, *(x, y))
314.84/291.48	   f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	S is empty.
314.84/291.48	Rewrite Strategy: FULL
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(1) RenamingProof (BOTH BOUNDS(ID, ID))
314.84/291.48	Renamed function symbols to avoid clashes with predefined symbol.
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(2)
314.84/291.48	Obligation:
314.84/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
314.84/291.48	
314.84/291.48	
314.84/291.48	The TRS R consists of the following rules:
314.84/291.48	
314.84/291.48	   -(x, 0') -> x
314.84/291.48	   -(s(x), s(y)) -> -(x, y)
314.84/291.48	   +'(0', y) -> y
314.84/291.48	   +'(s(x), y) -> s(+'(x, y))
314.84/291.48	   *'(x, 0') -> 0'
314.84/291.48	   *'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	   f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	S is empty.
314.84/291.48	Rewrite Strategy: FULL
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
314.84/291.48	Infered types.
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(4)
314.84/291.48	Obligation:
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(5) OrderProof (LOWER BOUND(ID))
314.84/291.48	Heuristically decided to analyse the following defined symbols:
314.84/291.48	-, +', *', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	- < f
314.84/291.48	+' < *'
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(6)
314.84/291.48	Obligation:
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	-, +', *', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	- < f
314.84/291.48	+' < *'
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(7) RewriteLemmaProof (LOWER BOUND(ID))
314.84/291.48	Proved the following rewrite lemma:
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
314.84/291.48	
314.84/291.48	Induction Base:
314.84/291.48	-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
314.84/291.48	gen_0':s3_0(0)
314.84/291.48	
314.84/291.48	Induction Step:
314.84/291.48	-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
314.84/291.48	gen_0':s3_0(0)
314.84/291.48	
314.84/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(8)
314.84/291.48	Complex Obligation (BEST)
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(9)
314.84/291.48	Obligation:
314.84/291.48	Proved the lower bound n^1 for the following obligation:
314.84/291.48	
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	-, +', *', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	- < f
314.84/291.48	+' < *'
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(10) LowerBoundPropagationProof (FINISHED)
314.84/291.48	Propagated lower bound.
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(11)
314.84/291.48	BOUNDS(n^1, INF)
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(12)
314.84/291.48	Obligation:
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Lemmas:
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	+', *', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	+' < *'
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(13) RewriteLemmaProof (LOWER BOUND(ID))
314.84/291.48	Proved the following rewrite lemma:
314.84/291.48	+'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0)
314.84/291.48	
314.84/291.48	Induction Base:
314.84/291.48	+'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
314.84/291.48	gen_0':s3_0(b)
314.84/291.48	
314.84/291.48	Induction Step:
314.84/291.48	+'(gen_0':s3_0(+(n225_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
314.84/291.48	s(+'(gen_0':s3_0(n225_0), gen_0':s3_0(b))) ->_IH
314.84/291.48	s(gen_0':s3_0(+(b, c226_0)))
314.84/291.48	
314.84/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(14)
314.84/291.48	Obligation:
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Lemmas:
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
314.84/291.48	+'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0)
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	*', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(15) RewriteLemmaProof (LOWER BOUND(ID))
314.84/291.48	Proved the following rewrite lemma:
314.84/291.48	*'(gen_0':s3_0(a), gen_0':s3_0(n696_0)) -> gen_0':s3_0(*(n696_0, a)), rt in Omega(1 + a*n696_0 + n696_0)
314.84/291.48	
314.84/291.48	Induction Base:
314.84/291.48	*'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
314.84/291.48	0'
314.84/291.48	
314.84/291.48	Induction Step:
314.84/291.48	*'(gen_0':s3_0(a), gen_0':s3_0(+(n696_0, 1))) ->_R^Omega(1)
314.84/291.48	+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n696_0))) ->_IH
314.84/291.48	+'(gen_0':s3_0(a), gen_0':s3_0(*(c697_0, a))) ->_L^Omega(1 + a)
314.84/291.48	gen_0':s3_0(+(a, *(n696_0, a)))
314.84/291.48	
314.84/291.48	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(16)
314.84/291.48	Complex Obligation (BEST)
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(17)
314.84/291.48	Obligation:
314.84/291.48	Proved the lower bound n^2 for the following obligation:
314.84/291.48	
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Lemmas:
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
314.84/291.48	+'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0)
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	*', f
314.84/291.48	
314.84/291.48	They will be analysed ascendingly in the following order:
314.84/291.48	*' < f
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(18) LowerBoundPropagationProof (FINISHED)
314.84/291.48	Propagated lower bound.
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(19)
314.84/291.48	BOUNDS(n^2, INF)
314.84/291.48	
314.84/291.48	----------------------------------------
314.84/291.48	
314.84/291.48	(20)
314.84/291.48	Obligation:
314.84/291.48	TRS:
314.84/291.48	Rules:
314.84/291.48	-(x, 0') -> x
314.84/291.48	-(s(x), s(y)) -> -(x, y)
314.84/291.48	+'(0', y) -> y
314.84/291.48	+'(s(x), y) -> s(+'(x, y))
314.84/291.48	*'(x, 0') -> 0'
314.84/291.48	*'(x, s(y)) -> +'(x, *'(x, y))
314.84/291.48	f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(s(x))))
314.84/291.48	
314.84/291.48	Types:
314.84/291.48	- :: 0':s -> 0':s -> 0':s
314.84/291.48	0' :: 0':s
314.84/291.48	s :: 0':s -> 0':s
314.84/291.48	+' :: 0':s -> 0':s -> 0':s
314.84/291.48	*' :: 0':s -> 0':s -> 0':s
314.84/291.48	f :: 0':s -> f
314.84/291.48	hole_0':s1_0 :: 0':s
314.84/291.48	hole_f2_0 :: f
314.84/291.48	gen_0':s3_0 :: Nat -> 0':s
314.84/291.48	
314.84/291.48	
314.84/291.48	Lemmas:
314.84/291.48	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
314.84/291.48	+'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0)
314.84/291.48	*'(gen_0':s3_0(a), gen_0':s3_0(n696_0)) -> gen_0':s3_0(*(n696_0, a)), rt in Omega(1 + a*n696_0 + n696_0)
314.84/291.48	
314.84/291.48	
314.84/291.48	Generator Equations:
314.84/291.48	gen_0':s3_0(0) <=> 0'
314.84/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.84/291.48	
314.84/291.48	
314.84/291.48	The following defined symbols remain to be analysed:
314.84/291.48	f
315.14/291.62	EOF