15.13/5.57	WORST_CASE(Omega(n^1), O(n^1))
15.22/5.58	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
15.22/5.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.22/5.58	
15.22/5.58	
15.22/5.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.22/5.58	
15.22/5.58	(0) CpxTRS
15.22/5.58	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	(2) CpxTRS
15.22/5.58	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	    (4) CpxWeightedTrs
15.22/5.58	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	    (6) CpxTypedWeightedTrs
15.22/5.58	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
15.22/5.58	    (8) CpxTypedWeightedCompleteTrs
15.22/5.58	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	    (10) CpxTypedWeightedCompleteTrs
15.22/5.58	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
15.22/5.58	    (12) CpxRNTS
15.22/5.58	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	    (14) CpxRNTS
15.22/5.58	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
15.22/5.58	    (16) CpxRNTS
15.22/5.58	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
15.22/5.58	    (18) CpxRNTS
15.22/5.58	    (19) IntTrsBoundProof [UPPER BOUND(ID), 260 ms]
15.22/5.58	    (20) CpxRNTS
15.22/5.58	    (21) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
15.22/5.58	    (22) CpxRNTS
15.22/5.58	    (23) FinalProof [FINISHED, 0 ms]
15.22/5.58	    (24) BOUNDS(1, n^1)
15.22/5.58	(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
15.22/5.58	(26) TRS for Loop Detection
15.22/5.58	    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
15.22/5.58	    (28) BEST
15.22/5.58	        (29) proven lower bound
15.22/5.58	            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
15.22/5.58	            (31) BOUNDS(n^1, INF)
15.22/5.58	        (32) TRS for Loop Detection
15.22/5.58	
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(0)
15.22/5.58	Obligation:
15.22/5.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a
15.22/5.58	   f(g(x), y) -> f(x, y)
15.22/5.58	
15.22/5.58	S is empty.
15.22/5.58	Rewrite Strategy: FULL
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Converted rc-obligation to irc-obligation.
15.22/5.58	
15.22/5.58	As the TRS does not nest defined symbols, we have rc = irc.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(2)
15.22/5.58	Obligation:
15.22/5.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a
15.22/5.58	   f(g(x), y) -> f(x, y)
15.22/5.58	
15.22/5.58	S is empty.
15.22/5.58	Rewrite Strategy: INNERMOST
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Transformed relative TRS to weighted TRS
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(4)
15.22/5.58	Obligation:
15.22/5.58	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a [1]
15.22/5.58	   f(g(x), y) -> f(x, y) [1]
15.22/5.58	
15.22/5.58	Rewrite Strategy: INNERMOST
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Infered types.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(6)
15.22/5.58	Obligation:
15.22/5.58	Runtime Complexity Weighted TRS with Types.
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a [1]
15.22/5.58	   f(g(x), y) -> f(x, y) [1]
15.22/5.58	
15.22/5.58	The TRS has the following type information:
15.22/5.58	f :: g -> g -> a
15.22/5.58	a :: a
15.22/5.58	g :: g -> g
15.22/5.58	
15.22/5.58	Rewrite Strategy: INNERMOST
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(7) CompletionProof (UPPER BOUND(ID))
15.22/5.58	The transformation into a RNTS is sound, since:
15.22/5.58	
15.22/5.58	(a) The obligation is a constructor system where every type has a constant constructor,
15.22/5.58	
15.22/5.58	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
15.22/5.58	
15.22/5.58	   f_2
15.22/5.58	
15.22/5.58	(c) The following functions are completely defined:
15.22/5.58	none
15.22/5.58	
15.22/5.58	Due to the following rules being added:
15.22/5.58	none
15.22/5.58	
15.22/5.58	And the following fresh constants:    const
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(8)
15.22/5.58	Obligation:
15.22/5.58	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
15.22/5.58	
15.22/5.58	Runtime Complexity Weighted TRS with Types.
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a [1]
15.22/5.58	   f(g(x), y) -> f(x, y) [1]
15.22/5.58	
15.22/5.58	The TRS has the following type information:
15.22/5.58	f :: g -> g -> a
15.22/5.58	a :: a
15.22/5.58	g :: g -> g
15.22/5.58	const :: g
15.22/5.58	
15.22/5.58	Rewrite Strategy: INNERMOST
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(10)
15.22/5.58	Obligation:
15.22/5.58	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
15.22/5.58	
15.22/5.58	Runtime Complexity Weighted TRS with Types.
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a [1]
15.22/5.58	   f(g(x), y) -> f(x, y) [1]
15.22/5.58	
15.22/5.58	The TRS has the following type information:
15.22/5.58	f :: g -> g -> a
15.22/5.58	a :: a
15.22/5.58	g :: g -> g
15.22/5.58	const :: g
15.22/5.58	
15.22/5.58	Rewrite Strategy: INNERMOST
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
15.22/5.58	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
15.22/5.58	The constant constructors are abstracted as follows:
15.22/5.58	
15.22/5.58	   a => 0
15.22/5.58	   const => 0
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(12)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = x
15.22/5.58	
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(14)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
15.22/5.58	
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
15.22/5.58	Found the following analysis order by SCC decomposition:
15.22/5.58	
15.22/5.58	   { f }
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(16)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
15.22/5.58	
15.22/5.58	Function symbols to be analyzed: {f}
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(17) ResultPropagationProof (UPPER BOUND(ID))
15.22/5.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(18)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
15.22/5.58	
15.22/5.58	Function symbols to be analyzed: {f}
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(19) IntTrsBoundProof (UPPER BOUND(ID))
15.22/5.58	
15.22/5.58	Computed SIZE bound using CoFloCo for: f
15.22/5.58	after applying outer abstraction to obtain an ITS,
15.22/5.58	resulting in: O(1) with polynomial bound: 0
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(20)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
15.22/5.58	
15.22/5.58	Function symbols to be analyzed: {f}
15.22/5.58	Previous analysis results are:
15.22/5.58	  f: runtime: ?, size: O(1) [0]
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(21) IntTrsBoundProof (UPPER BOUND(ID))
15.22/5.58	
15.22/5.58	Computed RUNTIME bound using KoAT for: f
15.22/5.58	after applying outer abstraction to obtain an ITS,
15.22/5.58	resulting in: O(n^1) with polynomial bound: 1 + z
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(22)
15.22/5.58	Obligation:
15.22/5.58	Complexity RNTS consisting of the following rules:
15.22/5.58	
15.22/5.58	   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
15.22/5.58	   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
15.22/5.58	
15.22/5.58	Function symbols to be analyzed: 
15.22/5.58	Previous analysis results are:
15.22/5.58	  f: runtime: O(n^1) [1 + z], size: O(1) [0]
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(23) FinalProof (FINISHED)
15.22/5.58	Computed overall runtime complexity
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(24)
15.22/5.58	BOUNDS(1, n^1)
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
15.22/5.58	Transformed a relative TRS into a decreasing-loop problem.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(26)
15.22/5.58	Obligation:
15.22/5.58	Analyzing the following TRS for decreasing loops:
15.22/5.58	
15.22/5.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a
15.22/5.58	   f(g(x), y) -> f(x, y)
15.22/5.58	
15.22/5.58	S is empty.
15.22/5.58	Rewrite Strategy: FULL
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(27) DecreasingLoopProof (LOWER BOUND(ID))
15.22/5.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
15.22/5.58	
15.22/5.58	The rewrite sequence
15.22/5.58	
15.22/5.58	f(g(x), y) ->^+ f(x, y)
15.22/5.58	
15.22/5.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
15.22/5.58	
15.22/5.58	The pumping substitution is [x / g(x)].
15.22/5.58	
15.22/5.58	The result substitution is [ ].
15.22/5.58	
15.22/5.58	
15.22/5.58	
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(28)
15.22/5.58	Complex Obligation (BEST)
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(29)
15.22/5.58	Obligation:
15.22/5.58	Proved the lower bound n^1 for the following obligation:
15.22/5.58	
15.22/5.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a
15.22/5.58	   f(g(x), y) -> f(x, y)
15.22/5.58	
15.22/5.58	S is empty.
15.22/5.58	Rewrite Strategy: FULL
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(30) LowerBoundPropagationProof (FINISHED)
15.22/5.58	Propagated lower bound.
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(31)
15.22/5.58	BOUNDS(n^1, INF)
15.22/5.58	
15.22/5.58	----------------------------------------
15.22/5.58	
15.22/5.58	(32)
15.22/5.58	Obligation:
15.22/5.58	Analyzing the following TRS for decreasing loops:
15.22/5.58	
15.22/5.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.22/5.58	
15.22/5.58	
15.22/5.58	The TRS R consists of the following rules:
15.22/5.58	
15.22/5.58	   f(x, x) -> a
15.22/5.58	   f(g(x), y) -> f(x, y)
15.22/5.58	
15.22/5.58	S is empty.
15.22/5.58	Rewrite Strategy: FULL
15.22/5.61	EOF