16.36/5.40	WORST_CASE(Omega(n^1), O(n^1))
16.36/5.41	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.36/5.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.36/5.41	
16.36/5.41	
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.36/5.41	
16.36/5.41	(0) CpxTRS
16.36/5.41	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
16.36/5.41	(2) CpxTRS
16.36/5.41	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (4) CpxTRS
16.36/5.41	    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (6) CpxWeightedTrs
16.36/5.41	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (8) CpxTypedWeightedTrs
16.36/5.41	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
16.36/5.41	    (10) CpxTypedWeightedCompleteTrs
16.36/5.41	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (12) CpxTypedWeightedCompleteTrs
16.36/5.41	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
16.36/5.41	    (14) CpxRNTS
16.36/5.41	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (16) CpxRNTS
16.36/5.41	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
16.36/5.41	    (18) CpxRNTS
16.36/5.41	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
16.36/5.41	    (20) CpxRNTS
16.36/5.41	    (21) IntTrsBoundProof [UPPER BOUND(ID), 209 ms]
16.36/5.41	    (22) CpxRNTS
16.36/5.41	    (23) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
16.36/5.41	    (24) CpxRNTS
16.36/5.41	    (25) FinalProof [FINISHED, 0 ms]
16.36/5.41	    (26) BOUNDS(1, n^1)
16.36/5.41	(27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
16.36/5.41	(28) TRS for Loop Detection
16.36/5.41	    (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
16.36/5.41	    (30) BEST
16.36/5.41	        (31) proven lower bound
16.36/5.41	            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
16.36/5.41	            (33) BOUNDS(n^1, INF)
16.36/5.41	        (34) TRS for Loop Detection
16.36/5.41	
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(0)
16.36/5.41	Obligation:
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: FULL
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
16.36/5.41	The following defined symbols can occur below the 1th argument of plus: times
16.36/5.41	
16.36/5.41	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
16.36/5.41	plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(2)
16.36/5.41	Obligation:
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: FULL
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Converted rc-obligation to irc-obligation.
16.36/5.41	
16.36/5.41	As the TRS does not nest defined symbols, we have rc = irc.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(4)
16.36/5.41	Obligation:
16.36/5.41	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: INNERMOST
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Transformed relative TRS to weighted TRS
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(6)
16.36/5.41	Obligation:
16.36/5.41	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X)) [1]
16.36/5.41	
16.36/5.41	Rewrite Strategy: INNERMOST
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Infered types.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(8)
16.36/5.41	Obligation:
16.36/5.41	Runtime Complexity Weighted TRS with Types.
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X)) [1]
16.36/5.41	
16.36/5.41	The TRS has the following type information:
16.36/5.41	times :: s -> s -> plus
16.36/5.41	s :: s -> s
16.36/5.41	plus :: s -> plus -> plus
16.36/5.41	
16.36/5.41	Rewrite Strategy: INNERMOST
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(9) CompletionProof (UPPER BOUND(ID))
16.36/5.41	The transformation into a RNTS is sound, since:
16.36/5.41	
16.36/5.41	(a) The obligation is a constructor system where every type has a constant constructor,
16.36/5.41	
16.36/5.41	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
16.36/5.41	
16.36/5.41	   times_2
16.36/5.41	
16.36/5.41	(c) The following functions are completely defined:
16.36/5.41	none
16.36/5.41	
16.36/5.41	Due to the following rules being added:
16.36/5.41	none
16.36/5.41	
16.36/5.41	And the following fresh constants:    const, const1
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(10)
16.36/5.41	Obligation:
16.36/5.41	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
16.36/5.41	
16.36/5.41	Runtime Complexity Weighted TRS with Types.
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X)) [1]
16.36/5.41	
16.36/5.41	The TRS has the following type information:
16.36/5.41	times :: s -> s -> plus
16.36/5.41	s :: s -> s
16.36/5.41	plus :: s -> plus -> plus
16.36/5.41	const :: plus
16.36/5.41	const1 :: s
16.36/5.41	
16.36/5.41	Rewrite Strategy: INNERMOST
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(12)
16.36/5.41	Obligation:
16.36/5.41	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
16.36/5.41	
16.36/5.41	Runtime Complexity Weighted TRS with Types.
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X)) [1]
16.36/5.41	
16.36/5.41	The TRS has the following type information:
16.36/5.41	times :: s -> s -> plus
16.36/5.41	s :: s -> s
16.36/5.41	plus :: s -> plus -> plus
16.36/5.41	const :: plus
16.36/5.41	const1 :: s
16.36/5.41	
16.36/5.41	Rewrite Strategy: INNERMOST
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
16.36/5.41	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
16.36/5.41	The constant constructors are abstracted as follows:
16.36/5.41	
16.36/5.41	   const => 0
16.36/5.41	   const1 => 0
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(14)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + X + times(Y, X) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
16.36/5.41	
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(16)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0
16.36/5.41	
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
16.36/5.41	Found the following analysis order by SCC decomposition:
16.36/5.41	
16.36/5.41	   { times }
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(18)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0
16.36/5.41	
16.36/5.41	Function symbols to be analyzed: {times}
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(19) ResultPropagationProof (UPPER BOUND(ID))
16.36/5.41	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(20)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0
16.36/5.41	
16.36/5.41	Function symbols to be analyzed: {times}
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(21) IntTrsBoundProof (UPPER BOUND(ID))
16.36/5.41	
16.36/5.41	Computed SIZE bound using CoFloCo for: times
16.36/5.41	after applying outer abstraction to obtain an ITS,
16.36/5.41	resulting in: O(1) with polynomial bound: 0
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(22)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0
16.36/5.41	
16.36/5.41	Function symbols to be analyzed: {times}
16.36/5.41	Previous analysis results are:
16.36/5.41	  times: runtime: ?, size: O(1) [0]
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(23) IntTrsBoundProof (UPPER BOUND(ID))
16.36/5.41	
16.36/5.41	Computed RUNTIME bound using CoFloCo for: times
16.36/5.41	after applying outer abstraction to obtain an ITS,
16.36/5.41	resulting in: O(n^1) with polynomial bound: z + z'
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(24)
16.36/5.41	Obligation:
16.36/5.41	Complexity RNTS consisting of the following rules:
16.36/5.41	
16.36/5.41	   times(z, z') -{ 1 }-> 1 + z + times(z' - 1, z) :|: z' - 1 >= 0, z >= 0
16.36/5.41	
16.36/5.41	Function symbols to be analyzed: 
16.36/5.41	Previous analysis results are:
16.36/5.41	  times: runtime: O(n^1) [z + z'], size: O(1) [0]
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(25) FinalProof (FINISHED)
16.36/5.41	Computed overall runtime complexity
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(26)
16.36/5.41	BOUNDS(1, n^1)
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
16.36/5.41	Transformed a relative TRS into a decreasing-loop problem.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(28)
16.36/5.41	Obligation:
16.36/5.41	Analyzing the following TRS for decreasing loops:
16.36/5.41	
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: FULL
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(29) DecreasingLoopProof (LOWER BOUND(ID))
16.36/5.41	The following loop(s) give(s) rise to the lower bound Omega(n^1):
16.36/5.41	
16.36/5.41	The rewrite sequence
16.36/5.41	
16.36/5.41	times(s(Y2_0), s(Y)) ->^+ plus(s(Y2_0), plus(Y, times(Y2_0, Y)))
16.36/5.41	
16.36/5.41	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
16.36/5.41	
16.36/5.41	The pumping substitution is [Y2_0 / s(Y2_0), Y / s(Y)].
16.36/5.41	
16.36/5.41	The result substitution is [ ].
16.36/5.41	
16.36/5.41	
16.36/5.41	
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(30)
16.36/5.41	Complex Obligation (BEST)
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(31)
16.36/5.41	Obligation:
16.36/5.41	Proved the lower bound n^1 for the following obligation:
16.36/5.41	
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: FULL
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(32) LowerBoundPropagationProof (FINISHED)
16.36/5.41	Propagated lower bound.
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(33)
16.36/5.41	BOUNDS(n^1, INF)
16.36/5.41	
16.36/5.41	----------------------------------------
16.36/5.41	
16.36/5.41	(34)
16.36/5.41	Obligation:
16.36/5.41	Analyzing the following TRS for decreasing loops:
16.36/5.41	
16.36/5.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.36/5.41	
16.36/5.41	
16.36/5.41	The TRS R consists of the following rules:
16.36/5.41	
16.36/5.41	   plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
16.36/5.41	   times(X, s(Y)) -> plus(X, times(Y, X))
16.36/5.41	
16.36/5.41	S is empty.
16.36/5.41	Rewrite Strategy: FULL
16.36/5.44	EOF