21.48/6.95	WORST_CASE(Omega(n^1), O(n^1))
21.48/6.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.48/6.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.48/6.96	
21.48/6.96	
21.48/6.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.48/6.96	
21.48/6.96	(0) CpxTRS
21.48/6.96	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	(2) CpxTRS
21.48/6.96	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	    (4) CpxWeightedTrs
21.48/6.96	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	    (6) CpxTypedWeightedTrs
21.48/6.96	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
21.48/6.96	    (8) CpxTypedWeightedCompleteTrs
21.48/6.96	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	    (10) CpxTypedWeightedCompleteTrs
21.48/6.96	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
21.48/6.96	    (12) CpxRNTS
21.48/6.96	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	    (14) CpxRNTS
21.48/6.96	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
21.48/6.96	    (16) CpxRNTS
21.48/6.96	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
21.48/6.96	    (18) CpxRNTS
21.48/6.96	    (19) IntTrsBoundProof [UPPER BOUND(ID), 1198 ms]
21.48/6.96	    (20) CpxRNTS
21.48/6.96	    (21) IntTrsBoundProof [UPPER BOUND(ID), 481 ms]
21.48/6.96	    (22) CpxRNTS
21.48/6.96	    (23) FinalProof [FINISHED, 0 ms]
21.48/6.96	    (24) BOUNDS(1, n^1)
21.48/6.96	(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
21.48/6.96	(26) TRS for Loop Detection
21.48/6.96	    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
21.48/6.96	    (28) BEST
21.48/6.96	        (29) proven lower bound
21.48/6.96	            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
21.48/6.96	            (31) BOUNDS(n^1, INF)
21.48/6.96	        (32) TRS for Loop Detection
21.48/6.96	
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(0)
21.48/6.96	Obligation:
21.48/6.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.48/6.96	
21.48/6.96	
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0
21.48/6.96	   div(x, y) -> quot(x, y, y)
21.48/6.96	   quot(0, s(y), z) -> 0
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z)
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z)))
21.48/6.96	
21.48/6.96	S is empty.
21.48/6.96	Rewrite Strategy: FULL
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Converted rc-obligation to irc-obligation.
21.48/6.96	
21.48/6.96	As the TRS does not nest defined symbols, we have rc = irc.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(2)
21.48/6.96	Obligation:
21.48/6.96	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
21.48/6.96	
21.48/6.96	
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0
21.48/6.96	   div(x, y) -> quot(x, y, y)
21.48/6.96	   quot(0, s(y), z) -> 0
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z)
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z)))
21.48/6.96	
21.48/6.96	S is empty.
21.48/6.96	Rewrite Strategy: INNERMOST
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Transformed relative TRS to weighted TRS
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(4)
21.48/6.96	Obligation:
21.48/6.96	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
21.48/6.96	
21.48/6.96	
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0 [1]
21.48/6.96	   div(x, y) -> quot(x, y, y) [1]
21.48/6.96	   quot(0, s(y), z) -> 0 [1]
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
21.48/6.96	
21.48/6.96	Rewrite Strategy: INNERMOST
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Infered types.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(6)
21.48/6.96	Obligation:
21.48/6.96	Runtime Complexity Weighted TRS with Types.
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0 [1]
21.48/6.96	   div(x, y) -> quot(x, y, y) [1]
21.48/6.96	   quot(0, s(y), z) -> 0 [1]
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
21.48/6.96	
21.48/6.96	The TRS has the following type information:
21.48/6.96	div :: 0:s -> 0:s -> 0:s
21.48/6.96	0 :: 0:s
21.48/6.96	quot :: 0:s -> 0:s -> 0:s -> 0:s
21.48/6.96	s :: 0:s -> 0:s
21.48/6.96	
21.48/6.96	Rewrite Strategy: INNERMOST
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(7) CompletionProof (UPPER BOUND(ID))
21.48/6.96	The transformation into a RNTS is sound, since:
21.48/6.96	
21.48/6.96	(a) The obligation is a constructor system where every type has a constant constructor,
21.48/6.96	
21.48/6.96	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
21.48/6.96	
21.48/6.96	   div_2
21.48/6.96	   quot_3
21.48/6.96	
21.48/6.96	(c) The following functions are completely defined:
21.48/6.96	none
21.48/6.96	
21.48/6.96	Due to the following rules being added:
21.48/6.96	none
21.48/6.96	
21.48/6.96	And the following fresh constants: none
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(8)
21.48/6.96	Obligation:
21.48/6.96	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
21.48/6.96	
21.48/6.96	Runtime Complexity Weighted TRS with Types.
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0 [1]
21.48/6.96	   div(x, y) -> quot(x, y, y) [1]
21.48/6.96	   quot(0, s(y), z) -> 0 [1]
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
21.48/6.96	
21.48/6.96	The TRS has the following type information:
21.48/6.96	div :: 0:s -> 0:s -> 0:s
21.48/6.96	0 :: 0:s
21.48/6.96	quot :: 0:s -> 0:s -> 0:s -> 0:s
21.48/6.96	s :: 0:s -> 0:s
21.48/6.96	
21.48/6.96	Rewrite Strategy: INNERMOST
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(10)
21.48/6.96	Obligation:
21.48/6.96	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
21.48/6.96	
21.48/6.96	Runtime Complexity Weighted TRS with Types.
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0 [1]
21.48/6.96	   div(x, y) -> quot(x, y, y) [1]
21.48/6.96	   quot(0, s(y), z) -> 0 [1]
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z) [1]
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
21.48/6.96	
21.48/6.96	The TRS has the following type information:
21.48/6.96	div :: 0:s -> 0:s -> 0:s
21.48/6.96	0 :: 0:s
21.48/6.96	quot :: 0:s -> 0:s -> 0:s -> 0:s
21.48/6.96	s :: 0:s -> 0:s
21.48/6.96	
21.48/6.96	Rewrite Strategy: INNERMOST
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
21.48/6.96	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
21.48/6.96	The constant constructors are abstracted as follows:
21.48/6.96	
21.48/6.96	   0 => 0
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(12)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z
21.48/6.96	
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(14)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
21.48/6.96	
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
21.48/6.96	Found the following analysis order by SCC decomposition:
21.48/6.96	
21.48/6.96	   { div, quot }
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(16)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
21.48/6.96	
21.48/6.96	Function symbols to be analyzed: {div,quot}
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(17) ResultPropagationProof (UPPER BOUND(ID))
21.48/6.96	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(18)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
21.48/6.96	
21.48/6.96	Function symbols to be analyzed: {div,quot}
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(19) IntTrsBoundProof (UPPER BOUND(ID))
21.48/6.96	
21.48/6.96	Computed SIZE bound using KoAT for: div
21.48/6.96	after applying outer abstraction to obtain an ITS,
21.48/6.96	resulting in: O(n^1) with polynomial bound: z'
21.48/6.96	
21.48/6.96	Computed SIZE bound using KoAT for: quot
21.48/6.96	after applying outer abstraction to obtain an ITS,
21.48/6.96	resulting in: O(n^1) with polynomial bound: 1 + z'
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(20)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
21.48/6.96	
21.48/6.96	Function symbols to be analyzed: {div,quot}
21.48/6.96	Previous analysis results are:
21.48/6.96	  div: runtime: ?, size: O(n^1) [z']
21.48/6.96	  quot: runtime: ?, size: O(n^1) [1 + z']
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(21) IntTrsBoundProof (UPPER BOUND(ID))
21.48/6.96	
21.48/6.96	Computed RUNTIME bound using KoAT for: div
21.48/6.96	after applying outer abstraction to obtain an ITS,
21.48/6.96	resulting in: O(n^1) with polynomial bound: 3 + 3*z'
21.48/6.96	
21.48/6.96	Computed RUNTIME bound using KoAT for: quot
21.48/6.96	after applying outer abstraction to obtain an ITS,
21.48/6.96	resulting in: O(n^1) with polynomial bound: 5 + 3*z' + z''
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(22)
21.48/6.96	Obligation:
21.48/6.96	Complexity RNTS consisting of the following rules:
21.48/6.96	
21.48/6.96	   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
21.48/6.96	   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
21.48/6.96	   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
21.48/6.96	
21.48/6.96	Function symbols to be analyzed: 
21.48/6.96	Previous analysis results are:
21.48/6.96	  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
21.48/6.96	  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(23) FinalProof (FINISHED)
21.48/6.96	Computed overall runtime complexity
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(24)
21.48/6.96	BOUNDS(1, n^1)
21.48/6.96	
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
21.48/6.96	Transformed a relative TRS into a decreasing-loop problem.
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(26)
21.48/6.96	Obligation:
21.48/6.96	Analyzing the following TRS for decreasing loops:
21.48/6.96	
21.48/6.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.48/6.96	
21.48/6.96	
21.48/6.96	The TRS R consists of the following rules:
21.48/6.96	
21.48/6.96	   div(0, y) -> 0
21.48/6.96	   div(x, y) -> quot(x, y, y)
21.48/6.96	   quot(0, s(y), z) -> 0
21.48/6.96	   quot(s(x), s(y), z) -> quot(x, y, z)
21.48/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z)))
21.48/6.96	
21.48/6.96	S is empty.
21.48/6.96	Rewrite Strategy: FULL
21.48/6.96	----------------------------------------
21.48/6.96	
21.48/6.96	(27) DecreasingLoopProof (LOWER BOUND(ID))
21.48/6.96	The following loop(s) give(s) rise to the lower bound Omega(n^1):
21.48/6.96	
21.48/6.96	The rewrite sequence
21.48/6.96	
21.48/6.96	quot(s(x), s(y), z) ->^+ quot(x, y, z)
21.48/6.96	
21.48/6.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
21.68/6.96	
21.68/6.96	The pumping substitution is [x / s(x), y / s(y)].
21.68/6.96	
21.68/6.96	The result substitution is [ ].
21.68/6.96	
21.68/6.96	
21.68/6.96	
21.68/6.96	
21.68/6.96	----------------------------------------
21.68/6.96	
21.68/6.96	(28)
21.68/6.96	Complex Obligation (BEST)
21.68/6.96	
21.68/6.96	----------------------------------------
21.68/6.96	
21.68/6.96	(29)
21.68/6.96	Obligation:
21.68/6.96	Proved the lower bound n^1 for the following obligation:
21.68/6.96	
21.68/6.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.68/6.96	
21.68/6.96	
21.68/6.96	The TRS R consists of the following rules:
21.68/6.96	
21.68/6.96	   div(0, y) -> 0
21.68/6.96	   div(x, y) -> quot(x, y, y)
21.68/6.96	   quot(0, s(y), z) -> 0
21.68/6.96	   quot(s(x), s(y), z) -> quot(x, y, z)
21.68/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z)))
21.68/6.96	
21.68/6.96	S is empty.
21.68/6.96	Rewrite Strategy: FULL
21.68/6.96	----------------------------------------
21.68/6.96	
21.68/6.96	(30) LowerBoundPropagationProof (FINISHED)
21.68/6.96	Propagated lower bound.
21.68/6.96	----------------------------------------
21.68/6.96	
21.68/6.96	(31)
21.68/6.96	BOUNDS(n^1, INF)
21.68/6.96	
21.68/6.96	----------------------------------------
21.68/6.96	
21.68/6.96	(32)
21.68/6.96	Obligation:
21.68/6.96	Analyzing the following TRS for decreasing loops:
21.68/6.96	
21.68/6.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.68/6.96	
21.68/6.96	
21.68/6.96	The TRS R consists of the following rules:
21.68/6.96	
21.68/6.96	   div(0, y) -> 0
21.68/6.96	   div(x, y) -> quot(x, y, y)
21.68/6.96	   quot(0, s(y), z) -> 0
21.68/6.96	   quot(s(x), s(y), z) -> quot(x, y, z)
21.68/6.96	   quot(x, 0, s(z)) -> s(div(x, s(z)))
21.68/6.96	
21.68/6.96	S is empty.
21.68/6.96	Rewrite Strategy: FULL
21.72/7.24	EOF