1014.17/291.50	WORST_CASE(Omega(n^1), O(n^5))
1014.17/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1014.17/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).
1014.17/291.52	
1014.17/291.52	(0) CpxTRS
1014.17/291.52	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1014.17/291.52	(2) CpxWeightedTrs
1014.17/291.52	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1014.17/291.52	    (4) CpxTypedWeightedTrs
1014.17/291.52	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1014.17/291.52	    (6) CpxTypedWeightedCompleteTrs
1014.17/291.52	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
1014.17/291.52	    (8) CpxTypedWeightedCompleteTrs
1014.17/291.52	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1014.17/291.52	    (10) CpxRNTS
1014.17/291.52	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
1014.17/291.52	    (12) CpxRNTS
1014.17/291.52	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
1014.17/291.52	    (14) CpxRNTS
1014.17/291.52	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1014.17/291.52	    (16) CpxRNTS
1014.17/291.52	    (17) IntTrsBoundProof [UPPER BOUND(ID), 3592 ms]
1014.17/291.52	    (18) CpxRNTS
1014.17/291.52	    (19) IntTrsBoundProof [UPPER BOUND(ID), 3555 ms]
1014.17/291.52	    (20) CpxRNTS
1014.17/291.52	    (21) FinalProof [FINISHED, 0 ms]
1014.17/291.52	    (22) BOUNDS(1, n^5)
1014.17/291.52	(23) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1014.17/291.52	(24) TRS for Loop Detection
1014.17/291.52	    (25) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1014.17/291.52	    (26) BEST
1014.17/291.52	        (27) proven lower bound
1014.17/291.52	            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
1014.17/291.52	            (29) BOUNDS(n^1, INF)
1014.17/291.52	        (30) TRS for Loop Detection
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(0)
1014.17/291.52	Obligation:
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0
1014.17/291.52	
1014.17/291.52	S is empty.
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1014.17/291.52	Transformed relative TRS to weighted TRS
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(2)
1014.17/291.52	Obligation:
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^5).
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0 [1]
1014.17/291.52	
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1014.17/291.52	Infered types.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(4)
1014.17/291.52	Obligation:
1014.17/291.52	Runtime Complexity Weighted TRS with Types.
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0 [1]
1014.17/291.52	
1014.17/291.52	The TRS has the following type information:
1014.17/291.52	f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
1014.17/291.52	s :: s:0 -> s:0
1014.17/291.52	0 :: s:0
1014.17/291.52	
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(5) CompletionProof (UPPER BOUND(ID))
1014.17/291.52	The transformation into a RNTS is sound, since:
1014.17/291.52	
1014.17/291.52	(a) The obligation is a constructor system where every type has a constant constructor,
1014.17/291.52	
1014.17/291.52	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
1014.17/291.52	
1014.17/291.52	   f_5
1014.17/291.52	
1014.17/291.52	(c) The following functions are completely defined:
1014.17/291.52	none
1014.17/291.52	
1014.17/291.52	Due to the following rules being added:
1014.17/291.52	none
1014.17/291.52	
1014.17/291.52	And the following fresh constants: none
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(6)
1014.17/291.52	Obligation:
1014.17/291.52	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1014.17/291.52	
1014.17/291.52	Runtime Complexity Weighted TRS with Types.
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0 [1]
1014.17/291.52	
1014.17/291.52	The TRS has the following type information:
1014.17/291.52	f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
1014.17/291.52	s :: s:0 -> s:0
1014.17/291.52	0 :: s:0
1014.17/291.52	
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
1014.17/291.52	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(8)
1014.17/291.52	Obligation:
1014.17/291.52	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1014.17/291.52	
1014.17/291.52	Runtime Complexity Weighted TRS with Types.
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0 [1]
1014.17/291.52	
1014.17/291.52	The TRS has the following type information:
1014.17/291.52	f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
1014.17/291.52	s :: s:0 -> s:0
1014.17/291.52	0 :: s:0
1014.17/291.52	
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1014.17/291.52	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1014.17/291.52	The constant constructors are abstracted as follows:
1014.17/291.52	
1014.17/291.52	   0 => 0
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(10)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(x1, x2, x3, x4, x5) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = 1 + x1, z'' = x3, z2 = x5, x2 >= 0, x3 >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(x2, x2, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z' = 1 + x2, z'' = x3, z = 0, z2 = x5, x2 >= 0, x3 >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(x3, x3, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z'' = 1 + x3, z = 0, z2 = x5, x3 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(x4, x4, x4, x4, x5) :|: z'' = 0, x4 >= 0, x5 >= 0, z1 = 1 + x4, z = 0, z2 = x5, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(x5, x5, x5, x5, x5) :|: z'' = 0, z1 = 0, x5 >= 0, z = 0, z2 = 1 + x5, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
1014.17/291.52	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(12)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
1014.17/291.52	Found the following analysis order by SCC decomposition:
1014.17/291.52	
1014.17/291.52	   { f }
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(14)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	Function symbols to be analyzed: {f}
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(15) ResultPropagationProof (UPPER BOUND(ID))
1014.17/291.52	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(16)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	Function symbols to be analyzed: {f}
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(17) IntTrsBoundProof (UPPER BOUND(ID))
1014.17/291.52	
1014.17/291.52	Computed SIZE bound using CoFloCo for: f
1014.17/291.52	after applying outer abstraction to obtain an ITS,
1014.17/291.52	resulting in: O(1) with polynomial bound: 0
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(18)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	Function symbols to be analyzed: {f}
1014.17/291.52	Previous analysis results are:
1014.17/291.52	  f: runtime: ?, size: O(1) [0]
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(19) IntTrsBoundProof (UPPER BOUND(ID))
1014.17/291.52	
1014.17/291.52	Computed RUNTIME bound using KoAT for: f
1014.17/291.52	after applying outer abstraction to obtain an ITS,
1014.17/291.52	resulting in: O(n^5) with polynomial bound: 1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(20)
1014.17/291.52	Obligation:
1014.17/291.52	Complexity RNTS consisting of the following rules:
1014.17/291.52	
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
1014.17/291.52	   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0
1014.17/291.52	
1014.17/291.52	Function symbols to be analyzed: 
1014.17/291.52	Previous analysis results are:
1014.17/291.52	  f: runtime: O(n^5) [1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5], size: O(1) [0]
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(21) FinalProof (FINISHED)
1014.17/291.52	Computed overall runtime complexity
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(22)
1014.17/291.52	BOUNDS(1, n^5)
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(23) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1014.17/291.52	Transformed a relative TRS into a decreasing-loop problem.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(24)
1014.17/291.52	Obligation:
1014.17/291.52	Analyzing the following TRS for decreasing loops:
1014.17/291.52	
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0
1014.17/291.52	
1014.17/291.52	S is empty.
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(25) DecreasingLoopProof (LOWER BOUND(ID))
1014.17/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1014.17/291.52	
1014.17/291.52	The rewrite sequence
1014.17/291.52	
1014.17/291.52	f(s(x1), x2, x3, x4, x5) ->^+ f(x1, x2, x3, x4, x5)
1014.17/291.52	
1014.17/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1014.17/291.52	
1014.17/291.52	The pumping substitution is [x1 / s(x1)].
1014.17/291.52	
1014.17/291.52	The result substitution is [ ].
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(26)
1014.17/291.52	Complex Obligation (BEST)
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(27)
1014.17/291.52	Obligation:
1014.17/291.52	Proved the lower bound n^1 for the following obligation:
1014.17/291.52	
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0
1014.17/291.52	
1014.17/291.52	S is empty.
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(28) LowerBoundPropagationProof (FINISHED)
1014.17/291.52	Propagated lower bound.
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(29)
1014.17/291.52	BOUNDS(n^1, INF)
1014.17/291.52	
1014.17/291.52	----------------------------------------
1014.17/291.52	
1014.17/291.52	(30)
1014.17/291.52	Obligation:
1014.17/291.52	Analyzing the following TRS for decreasing loops:
1014.17/291.52	
1014.17/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).
1014.17/291.52	
1014.17/291.52	
1014.17/291.52	The TRS R consists of the following rules:
1014.17/291.52	
1014.17/291.52	   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
1014.17/291.52	   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
1014.17/291.52	   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
1014.17/291.52	   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
1014.17/291.52	   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
1014.17/291.52	   f(0, 0, 0, 0, 0) -> 0
1014.17/291.52	
1014.17/291.52	S is empty.
1014.17/291.52	Rewrite Strategy: INNERMOST
1014.44/291.57	EOF