15.79/5.02	WORST_CASE(Omega(n^1), O(n^1))
16.19/5.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.19/5.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.19/5.04	
16.19/5.04	
16.19/5.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.19/5.04	
16.19/5.04	(0) CpxTRS
16.19/5.04	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	(2) CpxTRS
16.19/5.04	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	    (4) CpxWeightedTrs
16.19/5.04	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	    (6) CpxTypedWeightedTrs
16.19/5.04	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
16.19/5.04	    (8) CpxTypedWeightedCompleteTrs
16.19/5.04	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	    (10) CpxTypedWeightedCompleteTrs
16.19/5.04	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
16.19/5.04	    (12) CpxRNTS
16.19/5.04	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	    (14) CpxRNTS
16.19/5.04	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
16.19/5.04	    (16) CpxRNTS
16.19/5.04	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
16.19/5.04	    (18) CpxRNTS
16.19/5.04	    (19) IntTrsBoundProof [UPPER BOUND(ID), 595 ms]
16.19/5.04	    (20) CpxRNTS
16.19/5.04	    (21) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
16.19/5.04	    (22) CpxRNTS
16.19/5.04	    (23) FinalProof [FINISHED, 0 ms]
16.19/5.04	    (24) BOUNDS(1, n^1)
16.19/5.04	(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
16.19/5.04	(26) TRS for Loop Detection
16.19/5.04	    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
16.19/5.04	    (28) BEST
16.19/5.04	        (29) proven lower bound
16.19/5.04	            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
16.19/5.04	            (31) BOUNDS(n^1, INF)
16.19/5.04	        (32) TRS for Loop Detection
16.19/5.04	
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(0)
16.19/5.04	Obligation:
16.19/5.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y))
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y))
16.19/5.04	   average(0, 0) -> 0
16.19/5.04	   average(0, s(0)) -> 0
16.19/5.04	   average(0, s(s(0))) -> s(0)
16.19/5.04	
16.19/5.04	S is empty.
16.19/5.04	Rewrite Strategy: FULL
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Converted rc-obligation to irc-obligation.
16.19/5.04	
16.19/5.04	As the TRS does not nest defined symbols, we have rc = irc.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(2)
16.19/5.04	Obligation:
16.19/5.04	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y))
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y))
16.19/5.04	   average(0, 0) -> 0
16.19/5.04	   average(0, s(0)) -> 0
16.19/5.04	   average(0, s(s(0))) -> s(0)
16.19/5.04	
16.19/5.04	S is empty.
16.19/5.04	Rewrite Strategy: INNERMOST
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Transformed relative TRS to weighted TRS
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(4)
16.19/5.04	Obligation:
16.19/5.04	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y)) [1]
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y)) [1]
16.19/5.04	   average(0, 0) -> 0 [1]
16.19/5.04	   average(0, s(0)) -> 0 [1]
16.19/5.04	   average(0, s(s(0))) -> s(0) [1]
16.19/5.04	
16.19/5.04	Rewrite Strategy: INNERMOST
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Infered types.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(6)
16.19/5.04	Obligation:
16.19/5.04	Runtime Complexity Weighted TRS with Types.
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y)) [1]
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y)) [1]
16.19/5.04	   average(0, 0) -> 0 [1]
16.19/5.04	   average(0, s(0)) -> 0 [1]
16.19/5.04	   average(0, s(s(0))) -> s(0) [1]
16.19/5.04	
16.19/5.04	The TRS has the following type information:
16.19/5.04	average :: s:0 -> s:0 -> s:0
16.19/5.04	s :: s:0 -> s:0
16.19/5.04	0 :: s:0
16.19/5.04	
16.19/5.04	Rewrite Strategy: INNERMOST
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(7) CompletionProof (UPPER BOUND(ID))
16.19/5.04	The transformation into a RNTS is sound, since:
16.19/5.04	
16.19/5.04	(a) The obligation is a constructor system where every type has a constant constructor,
16.19/5.04	
16.19/5.04	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
16.19/5.04	
16.19/5.04	   average_2
16.19/5.04	
16.19/5.04	(c) The following functions are completely defined:
16.19/5.04	none
16.19/5.04	
16.19/5.04	Due to the following rules being added:
16.19/5.04	none
16.19/5.04	
16.19/5.04	And the following fresh constants: none
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(8)
16.19/5.04	Obligation:
16.19/5.04	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
16.19/5.04	
16.19/5.04	Runtime Complexity Weighted TRS with Types.
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y)) [1]
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y)) [1]
16.19/5.04	   average(0, 0) -> 0 [1]
16.19/5.04	   average(0, s(0)) -> 0 [1]
16.19/5.04	   average(0, s(s(0))) -> s(0) [1]
16.19/5.04	
16.19/5.04	The TRS has the following type information:
16.19/5.04	average :: s:0 -> s:0 -> s:0
16.19/5.04	s :: s:0 -> s:0
16.19/5.04	0 :: s:0
16.19/5.04	
16.19/5.04	Rewrite Strategy: INNERMOST
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(10)
16.19/5.04	Obligation:
16.19/5.04	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
16.19/5.04	
16.19/5.04	Runtime Complexity Weighted TRS with Types.
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y)) [1]
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y)) [1]
16.19/5.04	   average(0, 0) -> 0 [1]
16.19/5.04	   average(0, s(0)) -> 0 [1]
16.19/5.04	   average(0, s(s(0))) -> s(0) [1]
16.19/5.04	
16.19/5.04	The TRS has the following type information:
16.19/5.04	average :: s:0 -> s:0 -> s:0
16.19/5.04	s :: s:0 -> s:0
16.19/5.04	0 :: s:0
16.19/5.04	
16.19/5.04	Rewrite Strategy: INNERMOST
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
16.19/5.04	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
16.19/5.04	The constant constructors are abstracted as follows:
16.19/5.04	
16.19/5.04	   0 => 0
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(12)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + x, y) :|: x >= 0, y >= 0, z' = 1 + (1 + (1 + y)), z = x
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(14)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
16.19/5.04	Found the following analysis order by SCC decomposition:
16.19/5.04	
16.19/5.04	   { average }
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(16)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	Function symbols to be analyzed: {average}
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(17) ResultPropagationProof (UPPER BOUND(ID))
16.19/5.04	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(18)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	Function symbols to be analyzed: {average}
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(19) IntTrsBoundProof (UPPER BOUND(ID))
16.19/5.04	
16.19/5.04	Computed SIZE bound using KoAT for: average
16.19/5.04	after applying outer abstraction to obtain an ITS,
16.19/5.04	resulting in: O(n^1) with polynomial bound: z + z'
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(20)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	Function symbols to be analyzed: {average}
16.19/5.04	Previous analysis results are:
16.19/5.04	  average: runtime: ?, size: O(n^1) [z + z']
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(21) IntTrsBoundProof (UPPER BOUND(ID))
16.19/5.04	
16.19/5.04	Computed RUNTIME bound using CoFloCo for: average
16.19/5.04	after applying outer abstraction to obtain an ITS,
16.19/5.04	resulting in: O(n^1) with polynomial bound: 3 + 2*z + z'
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(22)
16.19/5.04	Obligation:
16.19/5.04	Complexity RNTS consisting of the following rules:
16.19/5.04	
16.19/5.04	   average(z, z') -{ 1 }-> average(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
16.19/5.04	   average(z, z') -{ 1 }-> 0 :|: z' = 1 + 0, z = 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + average(1 + z, z' - 3) :|: z >= 0, z' - 3 >= 0
16.19/5.04	   average(z, z') -{ 1 }-> 1 + 0 :|: z' = 1 + (1 + 0), z = 0
16.19/5.04	
16.19/5.04	Function symbols to be analyzed: 
16.19/5.04	Previous analysis results are:
16.19/5.04	  average: runtime: O(n^1) [3 + 2*z + z'], size: O(n^1) [z + z']
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(23) FinalProof (FINISHED)
16.19/5.04	Computed overall runtime complexity
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(24)
16.19/5.04	BOUNDS(1, n^1)
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
16.19/5.04	Transformed a relative TRS into a decreasing-loop problem.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(26)
16.19/5.04	Obligation:
16.19/5.04	Analyzing the following TRS for decreasing loops:
16.19/5.04	
16.19/5.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y))
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y))
16.19/5.04	   average(0, 0) -> 0
16.19/5.04	   average(0, s(0)) -> 0
16.19/5.04	   average(0, s(s(0))) -> s(0)
16.19/5.04	
16.19/5.04	S is empty.
16.19/5.04	Rewrite Strategy: FULL
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(27) DecreasingLoopProof (LOWER BOUND(ID))
16.19/5.04	The following loop(s) give(s) rise to the lower bound Omega(n^1):
16.19/5.04	
16.19/5.04	The rewrite sequence
16.19/5.04	
16.19/5.04	average(s(x), y) ->^+ average(x, s(y))
16.19/5.04	
16.19/5.04	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
16.19/5.04	
16.19/5.04	The pumping substitution is [x / s(x)].
16.19/5.04	
16.19/5.04	The result substitution is [y / s(y)].
16.19/5.04	
16.19/5.04	
16.19/5.04	
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(28)
16.19/5.04	Complex Obligation (BEST)
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(29)
16.19/5.04	Obligation:
16.19/5.04	Proved the lower bound n^1 for the following obligation:
16.19/5.04	
16.19/5.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y))
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y))
16.19/5.04	   average(0, 0) -> 0
16.19/5.04	   average(0, s(0)) -> 0
16.19/5.04	   average(0, s(s(0))) -> s(0)
16.19/5.04	
16.19/5.04	S is empty.
16.19/5.04	Rewrite Strategy: FULL
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(30) LowerBoundPropagationProof (FINISHED)
16.19/5.04	Propagated lower bound.
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(31)
16.19/5.04	BOUNDS(n^1, INF)
16.19/5.04	
16.19/5.04	----------------------------------------
16.19/5.04	
16.19/5.04	(32)
16.19/5.04	Obligation:
16.19/5.04	Analyzing the following TRS for decreasing loops:
16.19/5.04	
16.19/5.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.19/5.04	
16.19/5.04	
16.19/5.04	The TRS R consists of the following rules:
16.19/5.04	
16.19/5.04	   average(s(x), y) -> average(x, s(y))
16.19/5.04	   average(x, s(s(s(y)))) -> s(average(s(x), y))
16.19/5.04	   average(0, 0) -> 0
16.19/5.04	   average(0, s(0)) -> 0
16.19/5.04	   average(0, s(s(0))) -> s(0)
16.19/5.04	
16.19/5.04	S is empty.
16.19/5.04	Rewrite Strategy: FULL
16.19/5.08	EOF