628.79/291.51	WORST_CASE(Omega(n^1), O(n^2))
628.79/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
628.79/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
628.79/291.54	
628.79/291.54	
628.79/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
628.79/291.54	
628.79/291.54	(0) CpxTRS
628.79/291.54	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
628.79/291.54	(2) CpxWeightedTrs
628.79/291.54	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
628.79/291.54	    (4) CpxTypedWeightedTrs
628.79/291.54	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
628.79/291.54	    (6) CpxTypedWeightedCompleteTrs
628.79/291.54	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
628.79/291.54	    (8) CpxTypedWeightedCompleteTrs
628.79/291.54	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
628.79/291.54	    (10) CpxRNTS
628.79/291.54	    (11) InliningProof [UPPER BOUND(ID), 243 ms]
628.79/291.54	    (12) CpxRNTS
628.79/291.54	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
628.79/291.54	    (14) CpxRNTS
628.79/291.54	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
628.79/291.54	    (16) CpxRNTS
628.79/291.54	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
628.79/291.54	    (18) CpxRNTS
628.79/291.54	    (19) IntTrsBoundProof [UPPER BOUND(ID), 280 ms]
628.79/291.54	    (20) CpxRNTS
628.79/291.54	    (21) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
628.79/291.54	    (22) CpxRNTS
628.79/291.54	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
628.79/291.54	    (24) CpxRNTS
628.79/291.54	    (25) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
628.79/291.54	    (26) CpxRNTS
628.79/291.54	    (27) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
628.79/291.54	    (28) CpxRNTS
628.79/291.54	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
628.79/291.54	    (30) CpxRNTS
628.79/291.54	    (31) IntTrsBoundProof [UPPER BOUND(ID), 528 ms]
628.79/291.54	    (32) CpxRNTS
628.79/291.54	    (33) IntTrsBoundProof [UPPER BOUND(ID), 289 ms]
628.79/291.54	    (34) CpxRNTS
628.79/291.54	    (35) FinalProof [FINISHED, 0 ms]
628.79/291.54	    (36) BOUNDS(1, n^2)
628.79/291.54	(37) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
628.79/291.54	(38) TRS for Loop Detection
628.79/291.54	    (39) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
628.79/291.54	    (40) BEST
628.79/291.54	        (41) proven lower bound
628.79/291.54	            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
628.79/291.54	            (43) BOUNDS(n^1, INF)
628.79/291.54	        (44) TRS for Loop Detection
628.79/291.54	
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(0)
628.79/291.54	Obligation:
628.79/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
628.79/291.54	
628.79/291.54	
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x))))
628.79/291.54	   odd(0) -> false
628.79/291.54	   odd(s(0)) -> true
628.79/291.54	   odd(s(s(x))) -> odd(x)
628.79/291.54	   p(0) -> 0
628.79/291.54	   p(s(x)) -> x
628.79/291.54	
628.79/291.54	S is empty.
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
628.79/291.54	Transformed relative TRS to weighted TRS
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(2)
628.79/291.54	Obligation:
628.79/291.54	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
628.79/291.54	
628.79/291.54	
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
628.79/291.54	   odd(0) -> false [1]
628.79/291.54	   odd(s(0)) -> true [1]
628.79/291.54	   odd(s(s(x))) -> odd(x) [1]
628.79/291.54	   p(0) -> 0 [1]
628.79/291.54	   p(s(x)) -> x [1]
628.79/291.54	
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
628.79/291.54	Infered types.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(4)
628.79/291.54	Obligation:
628.79/291.54	Runtime Complexity Weighted TRS with Types.
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
628.79/291.54	   odd(0) -> false [1]
628.79/291.54	   odd(s(0)) -> true [1]
628.79/291.54	   odd(s(s(x))) -> odd(x) [1]
628.79/291.54	   p(0) -> 0 [1]
628.79/291.54	   p(s(x)) -> x [1]
628.79/291.54	
628.79/291.54	The TRS has the following type information:
628.79/291.54	cond :: true:false -> 0:s -> cond
628.79/291.54	true :: true:false
628.79/291.54	odd :: 0:s -> true:false
628.79/291.54	p :: 0:s -> 0:s
628.79/291.54	0 :: 0:s
628.79/291.54	false :: true:false
628.79/291.54	s :: 0:s -> 0:s
628.79/291.54	
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(5) CompletionProof (UPPER BOUND(ID))
628.79/291.54	The transformation into a RNTS is sound, since:
628.79/291.54	
628.79/291.54	(a) The obligation is a constructor system where every type has a constant constructor,
628.79/291.54	
628.79/291.54	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
628.79/291.54	
628.79/291.54	   cond_2
628.79/291.54	
628.79/291.54	(c) The following functions are completely defined:
628.79/291.54	
628.79/291.54	   odd_1
628.79/291.54	   p_1
628.79/291.54	
628.79/291.54	Due to the following rules being added:
628.79/291.54	none
628.79/291.54	
628.79/291.54	And the following fresh constants:    const
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(6)
628.79/291.54	Obligation:
628.79/291.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
628.79/291.54	
628.79/291.54	Runtime Complexity Weighted TRS with Types.
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
628.79/291.54	   odd(0) -> false [1]
628.79/291.54	   odd(s(0)) -> true [1]
628.79/291.54	   odd(s(s(x))) -> odd(x) [1]
628.79/291.54	   p(0) -> 0 [1]
628.79/291.54	   p(s(x)) -> x [1]
628.79/291.54	
628.79/291.54	The TRS has the following type information:
628.79/291.54	cond :: true:false -> 0:s -> cond
628.79/291.54	true :: true:false
628.79/291.54	odd :: 0:s -> true:false
628.79/291.54	p :: 0:s -> 0:s
628.79/291.54	0 :: 0:s
628.79/291.54	false :: true:false
628.79/291.54	s :: 0:s -> 0:s
628.79/291.54	const :: cond
628.79/291.54	
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
628.79/291.54	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(8)
628.79/291.54	Obligation:
628.79/291.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
628.79/291.54	
628.79/291.54	Runtime Complexity Weighted TRS with Types.
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, 0) -> cond(false, p(p(0))) [3]
628.79/291.54	   cond(true, s(0)) -> cond(true, p(p(0))) [3]
628.79/291.54	   cond(true, s(s(x'))) -> cond(odd(x'), p(p(s(x')))) [3]
628.79/291.54	   odd(0) -> false [1]
628.79/291.54	   odd(s(0)) -> true [1]
628.79/291.54	   odd(s(s(x))) -> odd(x) [1]
628.79/291.54	   p(0) -> 0 [1]
628.79/291.54	   p(s(x)) -> x [1]
628.79/291.54	
628.79/291.54	The TRS has the following type information:
628.79/291.54	cond :: true:false -> 0:s -> cond
628.79/291.54	true :: true:false
628.79/291.54	odd :: 0:s -> true:false
628.79/291.54	p :: 0:s -> 0:s
628.79/291.54	0 :: 0:s
628.79/291.54	false :: true:false
628.79/291.54	s :: 0:s -> 0:s
628.79/291.54	const :: cond
628.79/291.54	
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
628.79/291.54	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
628.79/291.54	The constant constructors are abstracted as follows:
628.79/291.54	
628.79/291.54	   true => 1
628.79/291.54	   0 => 0
628.79/291.54	   false => 0
628.79/291.54	   const => 0
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(10)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 3 }-> cond(odd(x'), p(p(1 + x'))) :|: z' = 1 + (1 + x'), z = 1, x' >= 0
628.79/291.54	   cond(z, z') -{ 3 }-> cond(1, p(p(0))) :|: z = 1, z' = 1 + 0
628.79/291.54	   cond(z, z') -{ 3 }-> cond(0, p(p(0))) :|: z = 1, z' = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x)
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(11) InliningProof (UPPER BOUND(ID))
628.79/291.54	Inlined the following terminating rules on right-hand sides where appropriate:
628.79/291.54	
628.79/291.54	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(12)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(x'), x'') :|: z' = 1 + (1 + x'), z = 1, x' >= 0, x >= 0, 1 + x' = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(x'), 0) :|: z' = 1 + (1 + x'), z = 1, x' >= 0, x >= 0, 1 + x' = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x)
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
628.79/291.54	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(14)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
628.79/291.54	Found the following analysis order by SCC decomposition:
628.79/291.54	
628.79/291.54	   { odd }
628.79/291.54	   { p }
628.79/291.54	   { cond }
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(16)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {odd}, {p}, {cond}
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(17) ResultPropagationProof (UPPER BOUND(ID))
628.79/291.54	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(18)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {odd}, {p}, {cond}
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(19) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed SIZE bound using CoFloCo for: odd
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(1) with polynomial bound: 1
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(20)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {odd}, {p}, {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: ?, size: O(1) [1]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(21) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed RUNTIME bound using CoFloCo for: odd
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(n^1) with polynomial bound: 2 + z
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(22)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {p}, {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(23) ResultPropagationProof (UPPER BOUND(ID))
628.79/291.54	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(24)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {p}, {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(25) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed SIZE bound using KoAT for: p
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(n^1) with polynomial bound: z
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(26)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {p}, {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	  p: runtime: ?, size: O(n^1) [z]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(27) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed RUNTIME bound using CoFloCo for: p
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(1) with polynomial bound: 1
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(28)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	  p: runtime: O(1) [1], size: O(n^1) [z]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(29) ResultPropagationProof (UPPER BOUND(ID))
628.79/291.54	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(30)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	  p: runtime: O(1) [1], size: O(n^1) [z]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(31) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed SIZE bound using CoFloCo for: cond
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(1) with polynomial bound: 0
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(32)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: {cond}
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	  p: runtime: O(1) [1], size: O(n^1) [z]
628.79/291.54	  cond: runtime: ?, size: O(1) [0]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(33) IntTrsBoundProof (UPPER BOUND(ID))
628.79/291.54	
628.79/291.54	Computed RUNTIME bound using CoFloCo for: cond
628.79/291.54	after applying outer abstraction to obtain an ITS,
628.79/291.54	resulting in: O(n^2) with polynomial bound: 22 + 7*z' + z'^2
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(34)
628.79/291.54	Obligation:
628.79/291.54	Complexity RNTS consisting of the following rules:
628.79/291.54	
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x''
628.79/291.54	   cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0
628.79/291.54	   cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0
628.79/291.54	   odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0
628.79/291.54	   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
628.79/291.54	   odd(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> 0 :|: z = 0
628.79/291.54	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
628.79/291.54	
628.79/291.54	Function symbols to be analyzed: 
628.79/291.54	Previous analysis results are:
628.79/291.54	  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
628.79/291.54	  p: runtime: O(1) [1], size: O(n^1) [z]
628.79/291.54	  cond: runtime: O(n^2) [22 + 7*z' + z'^2], size: O(1) [0]
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(35) FinalProof (FINISHED)
628.79/291.54	Computed overall runtime complexity
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(36)
628.79/291.54	BOUNDS(1, n^2)
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(37) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
628.79/291.54	Transformed a relative TRS into a decreasing-loop problem.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(38)
628.79/291.54	Obligation:
628.79/291.54	Analyzing the following TRS for decreasing loops:
628.79/291.54	
628.79/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
628.79/291.54	
628.79/291.54	
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x))))
628.79/291.54	   odd(0) -> false
628.79/291.54	   odd(s(0)) -> true
628.79/291.54	   odd(s(s(x))) -> odd(x)
628.79/291.54	   p(0) -> 0
628.79/291.54	   p(s(x)) -> x
628.79/291.54	
628.79/291.54	S is empty.
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(39) DecreasingLoopProof (LOWER BOUND(ID))
628.79/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
628.79/291.54	
628.79/291.54	The rewrite sequence
628.79/291.54	
628.79/291.54	odd(s(s(x))) ->^+ odd(x)
628.79/291.54	
628.79/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
628.79/291.54	
628.79/291.54	The pumping substitution is [x / s(s(x))].
628.79/291.54	
628.79/291.54	The result substitution is [ ].
628.79/291.54	
628.79/291.54	
628.79/291.54	
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(40)
628.79/291.54	Complex Obligation (BEST)
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(41)
628.79/291.54	Obligation:
628.79/291.54	Proved the lower bound n^1 for the following obligation:
628.79/291.54	
628.79/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
628.79/291.54	
628.79/291.54	
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x))))
628.79/291.54	   odd(0) -> false
628.79/291.54	   odd(s(0)) -> true
628.79/291.54	   odd(s(s(x))) -> odd(x)
628.79/291.54	   p(0) -> 0
628.79/291.54	   p(s(x)) -> x
628.79/291.54	
628.79/291.54	S is empty.
628.79/291.54	Rewrite Strategy: INNERMOST
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(42) LowerBoundPropagationProof (FINISHED)
628.79/291.54	Propagated lower bound.
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(43)
628.79/291.54	BOUNDS(n^1, INF)
628.79/291.54	
628.79/291.54	----------------------------------------
628.79/291.54	
628.79/291.54	(44)
628.79/291.54	Obligation:
628.79/291.54	Analyzing the following TRS for decreasing loops:
628.79/291.54	
628.79/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
628.79/291.54	
628.79/291.54	
628.79/291.54	The TRS R consists of the following rules:
628.79/291.54	
628.79/291.54	   cond(true, x) -> cond(odd(x), p(p(p(x))))
628.79/291.54	   odd(0) -> false
628.79/291.54	   odd(s(0)) -> true
628.79/291.54	   odd(s(s(x))) -> odd(x)
628.79/291.54	   p(0) -> 0
628.79/291.54	   p(s(x)) -> x
628.79/291.54	
628.79/291.54	S is empty.
628.79/291.54	Rewrite Strategy: INNERMOST
628.90/291.59	EOF