17.32/5.55	WORST_CASE(Omega(n^1), O(n^1))
17.64/5.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.64/5.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.64/5.56	
17.64/5.56	
17.64/5.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.64/5.56	
17.64/5.56	(0) CpxTRS
17.64/5.56	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	(2) CpxTRS
17.64/5.56	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (4) CpxWeightedTrs
17.64/5.56	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (6) CpxTypedWeightedTrs
17.64/5.56	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
17.64/5.56	    (8) CpxTypedWeightedCompleteTrs
17.64/5.56	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (10) CpxTypedWeightedCompleteTrs
17.64/5.56	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
17.64/5.56	    (12) CpxRNTS
17.64/5.56	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (14) CpxRNTS
17.64/5.56	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (16) CpxRNTS
17.64/5.56	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
17.64/5.56	    (18) CpxRNTS
17.64/5.56	    (19) IntTrsBoundProof [UPPER BOUND(ID), 514 ms]
17.64/5.56	    (20) CpxRNTS
17.64/5.56	    (21) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
17.64/5.56	    (22) CpxRNTS
17.64/5.56	    (23) FinalProof [FINISHED, 0 ms]
17.64/5.56	    (24) BOUNDS(1, n^1)
17.64/5.56	(25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	(26) CpxTRS
17.64/5.56	    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
17.64/5.56	    (28) typed CpxTrs
17.64/5.56	    (29) OrderProof [LOWER BOUND(ID), 0 ms]
17.64/5.56	    (30) typed CpxTrs
17.64/5.56	    (31) RewriteLemmaProof [LOWER BOUND(ID), 420 ms]
17.64/5.56	    (32) proven lower bound
17.64/5.56	    (33) LowerBoundPropagationProof [FINISHED, 0 ms]
17.64/5.56	    (34) BOUNDS(n^1, INF)
17.64/5.56	
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(0)
17.64/5.56	Obligation:
17.64/5.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.64/5.56	
17.64/5.56	
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	S is empty.
17.64/5.56	Rewrite Strategy: FULL
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Converted rc-obligation to irc-obligation.
17.64/5.56	
17.64/5.56	As the TRS does not nest defined symbols, we have rc = irc.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(2)
17.64/5.56	Obligation:
17.64/5.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
17.64/5.56	
17.64/5.56	
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	S is empty.
17.64/5.56	Rewrite Strategy: INNERMOST
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Transformed relative TRS to weighted TRS
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(4)
17.64/5.56	Obligation:
17.64/5.56	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
17.64/5.56	
17.64/5.56	
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0 [1]
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
17.64/5.56	
17.64/5.56	Rewrite Strategy: INNERMOST
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Infered types.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(6)
17.64/5.56	Obligation:
17.64/5.56	Runtime Complexity Weighted TRS with Types.
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0 [1]
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
17.64/5.56	
17.64/5.56	The TRS has the following type information:
17.64/5.56	quot :: 0:s -> 0:s -> 0:s -> 0:s
17.64/5.56	0 :: 0:s
17.64/5.56	s :: 0:s -> 0:s
17.64/5.56	
17.64/5.56	Rewrite Strategy: INNERMOST
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(7) CompletionProof (UPPER BOUND(ID))
17.64/5.56	The transformation into a RNTS is sound, since:
17.64/5.56	
17.64/5.56	(a) The obligation is a constructor system where every type has a constant constructor,
17.64/5.56	
17.64/5.56	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
17.64/5.56	
17.64/5.56	   quot_3
17.64/5.56	
17.64/5.56	(c) The following functions are completely defined:
17.64/5.56	none
17.64/5.56	
17.64/5.56	Due to the following rules being added:
17.64/5.56	none
17.64/5.56	
17.64/5.56	And the following fresh constants: none
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(8)
17.64/5.56	Obligation:
17.64/5.56	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
17.64/5.56	
17.64/5.56	Runtime Complexity Weighted TRS with Types.
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0 [1]
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
17.64/5.56	
17.64/5.56	The TRS has the following type information:
17.64/5.56	quot :: 0:s -> 0:s -> 0:s -> 0:s
17.64/5.56	0 :: 0:s
17.64/5.56	s :: 0:s -> 0:s
17.64/5.56	
17.64/5.56	Rewrite Strategy: INNERMOST
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(10)
17.64/5.56	Obligation:
17.64/5.56	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
17.64/5.56	
17.64/5.56	Runtime Complexity Weighted TRS with Types.
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0, s(y), s(z)) -> 0 [1]
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
17.64/5.56	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z))) [1]
17.64/5.56	
17.64/5.56	The TRS has the following type information:
17.64/5.56	quot :: 0:s -> 0:s -> 0:s -> 0:s
17.64/5.56	0 :: 0:s
17.64/5.56	s :: 0:s -> 0:s
17.64/5.56	
17.64/5.56	Rewrite Strategy: INNERMOST
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
17.64/5.56	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
17.64/5.56	The constant constructors are abstracted as follows:
17.64/5.56	
17.64/5.56	   0 => 0
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(12)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(x, 1 + z, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z
17.64/5.56	
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(14)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
17.64/5.56	
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Found the following analysis order by SCC decomposition:
17.64/5.56	
17.64/5.56	   { quot }
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(16)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
17.64/5.56	
17.64/5.56	Function symbols to be analyzed: {quot}
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(17) ResultPropagationProof (UPPER BOUND(ID))
17.64/5.56	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(18)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
17.64/5.56	
17.64/5.56	Function symbols to be analyzed: {quot}
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(19) IntTrsBoundProof (UPPER BOUND(ID))
17.64/5.56	
17.64/5.56	Computed SIZE bound using KoAT for: quot
17.64/5.56	after applying outer abstraction to obtain an ITS,
17.64/5.56	resulting in: O(n^1) with polynomial bound: 1 + z'
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(20)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
17.64/5.56	
17.64/5.56	Function symbols to be analyzed: {quot}
17.64/5.56	Previous analysis results are:
17.64/5.56	  quot: runtime: ?, size: O(n^1) [1 + z']
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(21) IntTrsBoundProof (UPPER BOUND(ID))
17.64/5.56	
17.64/5.56	Computed RUNTIME bound using KoAT for: quot
17.64/5.56	after applying outer abstraction to obtain an ITS,
17.64/5.56	resulting in: O(n^1) with polynomial bound: 2 + 2*z'
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(22)
17.64/5.56	Obligation:
17.64/5.56	Complexity RNTS consisting of the following rules:
17.64/5.56	
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
17.64/5.56	   quot(z', z'', z1) -{ 1 }-> 1 + quot(z', 1 + (z1 - 1), 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
17.64/5.56	
17.64/5.56	Function symbols to be analyzed: 
17.64/5.56	Previous analysis results are:
17.64/5.56	  quot: runtime: O(n^1) [2 + 2*z'], size: O(n^1) [1 + z']
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(23) FinalProof (FINISHED)
17.64/5.56	Computed overall runtime complexity
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(24)
17.64/5.56	BOUNDS(1, n^1)
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(25) RenamingProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Renamed function symbols to avoid clashes with predefined symbol.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(26)
17.64/5.56	Obligation:
17.64/5.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
17.64/5.56	
17.64/5.56	
17.64/5.56	The TRS R consists of the following rules:
17.64/5.56	
17.64/5.56	   quot(0', s(y), s(z)) -> 0'
17.64/5.56	   quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	   quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	S is empty.
17.64/5.56	Rewrite Strategy: FULL
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(27) TypeInferenceProof (BOTH BOUNDS(ID, ID))
17.64/5.56	Infered types.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(28)
17.64/5.56	Obligation:
17.64/5.56	TRS:
17.64/5.56	Rules:
17.64/5.56	quot(0', s(y), s(z)) -> 0'
17.64/5.56	quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	Types:
17.64/5.56	quot :: 0':s -> 0':s -> 0':s -> 0':s
17.64/5.56	0' :: 0':s
17.64/5.56	s :: 0':s -> 0':s
17.64/5.56	hole_0':s1_0 :: 0':s
17.64/5.56	gen_0':s2_0 :: Nat -> 0':s
17.64/5.56	
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(29) OrderProof (LOWER BOUND(ID))
17.64/5.56	Heuristically decided to analyse the following defined symbols:
17.64/5.56	quot
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(30)
17.64/5.56	Obligation:
17.64/5.56	TRS:
17.64/5.56	Rules:
17.64/5.56	quot(0', s(y), s(z)) -> 0'
17.64/5.56	quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	Types:
17.64/5.56	quot :: 0':s -> 0':s -> 0':s -> 0':s
17.64/5.56	0' :: 0':s
17.64/5.56	s :: 0':s -> 0':s
17.64/5.56	hole_0':s1_0 :: 0':s
17.64/5.56	gen_0':s2_0 :: Nat -> 0':s
17.64/5.56	
17.64/5.56	
17.64/5.56	Generator Equations:
17.64/5.56	gen_0':s2_0(0) <=> 0'
17.64/5.56	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
17.64/5.56	
17.64/5.56	
17.64/5.56	The following defined symbols remain to be analysed:
17.64/5.56	quot
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(31) RewriteLemmaProof (LOWER BOUND(ID))
17.64/5.56	Proved the following rewrite lemma:
17.64/5.56	quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(1)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
17.64/5.56	
17.64/5.56	Induction Base:
17.64/5.56	quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(1)) ->_R^Omega(1)
17.64/5.56	0'
17.64/5.56	
17.64/5.56	Induction Step:
17.64/5.56	quot(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(1)) ->_R^Omega(1)
17.64/5.56	quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(1)) ->_IH
17.64/5.56	gen_0':s2_0(0)
17.64/5.56	
17.64/5.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(32)
17.64/5.56	Obligation:
17.64/5.56	Proved the lower bound n^1 for the following obligation:
17.64/5.56	
17.64/5.56	TRS:
17.64/5.56	Rules:
17.64/5.56	quot(0', s(y), s(z)) -> 0'
17.64/5.56	quot(s(x), s(y), z) -> quot(x, y, z)
17.64/5.56	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
17.64/5.56	
17.64/5.56	Types:
17.64/5.56	quot :: 0':s -> 0':s -> 0':s -> 0':s
17.64/5.56	0' :: 0':s
17.64/5.56	s :: 0':s -> 0':s
17.64/5.56	hole_0':s1_0 :: 0':s
17.64/5.56	gen_0':s2_0 :: Nat -> 0':s
17.64/5.56	
17.64/5.56	
17.64/5.56	Generator Equations:
17.64/5.56	gen_0':s2_0(0) <=> 0'
17.64/5.56	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
17.64/5.56	
17.64/5.56	
17.64/5.56	The following defined symbols remain to be analysed:
17.64/5.56	quot
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(33) LowerBoundPropagationProof (FINISHED)
17.64/5.56	Propagated lower bound.
17.64/5.56	----------------------------------------
17.64/5.56	
17.64/5.56	(34)
17.64/5.56	BOUNDS(n^1, INF)
18.01/5.93	EOF