815.54/291.48	WORST_CASE(Omega(n^1), ?)
815.54/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
815.54/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
815.54/291.49	
815.54/291.49	
815.54/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
815.54/291.49	
815.54/291.49	(0) CpxTRS
815.54/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
815.54/291.49	(2) CpxTRS
815.54/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
815.54/291.49	(4) typed CpxTrs
815.54/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
815.54/291.49	(6) typed CpxTrs
815.54/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
815.54/291.49	(8) BEST
815.54/291.49	    (9) proven lower bound
815.54/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
815.54/291.49	        (11) BOUNDS(n^1, INF)
815.54/291.49	    (12) typed CpxTrs
815.54/291.49	        (13) RewriteLemmaProof [LOWER BOUND(ID), 334 ms]
815.54/291.49	        (14) typed CpxTrs
815.54/291.49	
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(0)
815.54/291.49	Obligation:
815.54/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
815.54/291.49	
815.54/291.49	
815.54/291.49	The TRS R consists of the following rules:
815.54/291.49	
815.54/291.49	   minus(x, 0) -> x
815.54/291.49	   minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	   quot(0, s(y)) -> 0
815.54/291.49	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	   plus(0, y) -> y
815.54/291.49	   plus(s(x), y) -> s(plus(x, y))
815.54/291.49	   minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	S is empty.
815.54/291.49	Rewrite Strategy: FULL
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
815.54/291.49	Renamed function symbols to avoid clashes with predefined symbol.
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(2)
815.54/291.49	Obligation:
815.54/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
815.54/291.49	
815.54/291.49	
815.54/291.49	The TRS R consists of the following rules:
815.54/291.49	
815.54/291.49	   minus(x, 0') -> x
815.54/291.49	   minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	   quot(0', s(y)) -> 0'
815.54/291.49	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	   plus(0', y) -> y
815.54/291.49	   plus(s(x), y) -> s(plus(x, y))
815.54/291.49	   minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	S is empty.
815.54/291.49	Rewrite Strategy: FULL
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
815.54/291.49	Infered types.
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(4)
815.54/291.49	Obligation:
815.54/291.49	TRS:
815.54/291.49	Rules:
815.54/291.49	minus(x, 0') -> x
815.54/291.49	minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	quot(0', s(y)) -> 0'
815.54/291.49	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	plus(0', y) -> y
815.54/291.49	plus(s(x), y) -> s(plus(x, y))
815.54/291.49	minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	Types:
815.54/291.49	minus :: 0':s -> 0':s -> 0':s
815.54/291.49	0' :: 0':s
815.54/291.49	s :: 0':s -> 0':s
815.54/291.49	quot :: 0':s -> 0':s -> 0':s
815.54/291.49	plus :: 0':s -> 0':s -> 0':s
815.54/291.49	hole_0':s1_0 :: 0':s
815.54/291.49	gen_0':s2_0 :: Nat -> 0':s
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(5) OrderProof (LOWER BOUND(ID))
815.54/291.49	Heuristically decided to analyse the following defined symbols:
815.54/291.49	minus, quot, plus
815.54/291.49	
815.54/291.49	They will be analysed ascendingly in the following order:
815.54/291.49	minus < quot
815.54/291.49	plus < minus
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(6)
815.54/291.49	Obligation:
815.54/291.49	TRS:
815.54/291.49	Rules:
815.54/291.49	minus(x, 0') -> x
815.54/291.49	minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	quot(0', s(y)) -> 0'
815.54/291.49	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	plus(0', y) -> y
815.54/291.49	plus(s(x), y) -> s(plus(x, y))
815.54/291.49	minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	Types:
815.54/291.49	minus :: 0':s -> 0':s -> 0':s
815.54/291.49	0' :: 0':s
815.54/291.49	s :: 0':s -> 0':s
815.54/291.49	quot :: 0':s -> 0':s -> 0':s
815.54/291.49	plus :: 0':s -> 0':s -> 0':s
815.54/291.49	hole_0':s1_0 :: 0':s
815.54/291.49	gen_0':s2_0 :: Nat -> 0':s
815.54/291.49	
815.54/291.49	
815.54/291.49	Generator Equations:
815.54/291.49	gen_0':s2_0(0) <=> 0'
815.54/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
815.54/291.49	
815.54/291.49	
815.54/291.49	The following defined symbols remain to be analysed:
815.54/291.49	plus, minus, quot
815.54/291.49	
815.54/291.49	They will be analysed ascendingly in the following order:
815.54/291.49	minus < quot
815.54/291.49	plus < minus
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
815.54/291.49	Proved the following rewrite lemma:
815.54/291.49	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
815.54/291.49	
815.54/291.49	Induction Base:
815.54/291.49	plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
815.54/291.49	gen_0':s2_0(b)
815.54/291.49	
815.54/291.49	Induction Step:
815.54/291.49	plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
815.54/291.49	s(plus(gen_0':s2_0(n4_0), gen_0':s2_0(b))) ->_IH
815.54/291.49	s(gen_0':s2_0(+(b, c5_0)))
815.54/291.49	
815.54/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(8)
815.54/291.49	Complex Obligation (BEST)
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(9)
815.54/291.49	Obligation:
815.54/291.49	Proved the lower bound n^1 for the following obligation:
815.54/291.49	
815.54/291.49	TRS:
815.54/291.49	Rules:
815.54/291.49	minus(x, 0') -> x
815.54/291.49	minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	quot(0', s(y)) -> 0'
815.54/291.49	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	plus(0', y) -> y
815.54/291.49	plus(s(x), y) -> s(plus(x, y))
815.54/291.49	minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	Types:
815.54/291.49	minus :: 0':s -> 0':s -> 0':s
815.54/291.49	0' :: 0':s
815.54/291.49	s :: 0':s -> 0':s
815.54/291.49	quot :: 0':s -> 0':s -> 0':s
815.54/291.49	plus :: 0':s -> 0':s -> 0':s
815.54/291.49	hole_0':s1_0 :: 0':s
815.54/291.49	gen_0':s2_0 :: Nat -> 0':s
815.54/291.49	
815.54/291.49	
815.54/291.49	Generator Equations:
815.54/291.49	gen_0':s2_0(0) <=> 0'
815.54/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
815.54/291.49	
815.54/291.49	
815.54/291.49	The following defined symbols remain to be analysed:
815.54/291.49	plus, minus, quot
815.54/291.49	
815.54/291.49	They will be analysed ascendingly in the following order:
815.54/291.49	minus < quot
815.54/291.49	plus < minus
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(10) LowerBoundPropagationProof (FINISHED)
815.54/291.49	Propagated lower bound.
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(11)
815.54/291.49	BOUNDS(n^1, INF)
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(12)
815.54/291.49	Obligation:
815.54/291.49	TRS:
815.54/291.49	Rules:
815.54/291.49	minus(x, 0') -> x
815.54/291.49	minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	quot(0', s(y)) -> 0'
815.54/291.49	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	plus(0', y) -> y
815.54/291.49	plus(s(x), y) -> s(plus(x, y))
815.54/291.49	minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	Types:
815.54/291.49	minus :: 0':s -> 0':s -> 0':s
815.54/291.49	0' :: 0':s
815.54/291.49	s :: 0':s -> 0':s
815.54/291.49	quot :: 0':s -> 0':s -> 0':s
815.54/291.49	plus :: 0':s -> 0':s -> 0':s
815.54/291.49	hole_0':s1_0 :: 0':s
815.54/291.49	gen_0':s2_0 :: Nat -> 0':s
815.54/291.49	
815.54/291.49	
815.54/291.49	Lemmas:
815.54/291.49	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
815.54/291.49	
815.54/291.49	
815.54/291.49	Generator Equations:
815.54/291.49	gen_0':s2_0(0) <=> 0'
815.54/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
815.54/291.49	
815.54/291.49	
815.54/291.49	The following defined symbols remain to be analysed:
815.54/291.49	minus, quot
815.54/291.49	
815.54/291.49	They will be analysed ascendingly in the following order:
815.54/291.49	minus < quot
815.54/291.49	
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(13) RewriteLemmaProof (LOWER BOUND(ID))
815.54/291.49	Proved the following rewrite lemma:
815.54/291.49	minus(gen_0':s2_0(+(1, n457_0)), gen_0':s2_0(+(1, n457_0))) -> *3_0, rt in Omega(n457_0)
815.54/291.49	
815.54/291.49	Induction Base:
815.54/291.49	minus(gen_0':s2_0(+(1, 0)), gen_0':s2_0(+(1, 0)))
815.54/291.49	
815.54/291.49	Induction Step:
815.54/291.49	minus(gen_0':s2_0(+(1, +(n457_0, 1))), gen_0':s2_0(+(1, +(n457_0, 1)))) ->_R^Omega(1)
815.54/291.49	minus(gen_0':s2_0(+(1, n457_0)), gen_0':s2_0(+(1, n457_0))) ->_IH
815.54/291.49	*3_0
815.54/291.49	
815.54/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
815.54/291.49	----------------------------------------
815.54/291.49	
815.54/291.49	(14)
815.54/291.49	Obligation:
815.54/291.49	TRS:
815.54/291.49	Rules:
815.54/291.49	minus(x, 0') -> x
815.54/291.49	minus(s(x), s(y)) -> minus(x, y)
815.54/291.49	quot(0', s(y)) -> 0'
815.54/291.49	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
815.54/291.49	plus(0', y) -> y
815.54/291.49	plus(s(x), y) -> s(plus(x, y))
815.54/291.49	minus(minus(x, y), z) -> minus(x, plus(y, z))
815.54/291.49	
815.54/291.49	Types:
815.54/291.49	minus :: 0':s -> 0':s -> 0':s
815.54/291.49	0' :: 0':s
815.54/291.49	s :: 0':s -> 0':s
815.54/291.49	quot :: 0':s -> 0':s -> 0':s
815.54/291.49	plus :: 0':s -> 0':s -> 0':s
815.54/291.49	hole_0':s1_0 :: 0':s
815.54/291.49	gen_0':s2_0 :: Nat -> 0':s
815.54/291.49	
815.54/291.49	
815.54/291.49	Lemmas:
815.54/291.49	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
815.54/291.49	minus(gen_0':s2_0(+(1, n457_0)), gen_0':s2_0(+(1, n457_0))) -> *3_0, rt in Omega(n457_0)
815.54/291.49	
815.54/291.49	
815.54/291.49	Generator Equations:
815.54/291.49	gen_0':s2_0(0) <=> 0'
815.54/291.49	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
815.54/291.49	
815.54/291.49	
815.54/291.49	The following defined symbols remain to be analysed:
815.54/291.49	quot
815.71/291.57	EOF