22.33/7.27	WORST_CASE(Omega(n^1), O(n^1))
22.45/7.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.45/7.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.45/7.88	
22.45/7.88	
22.45/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.45/7.88	
22.45/7.88	(0) CpxTRS
22.45/7.88	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	(2) CpxWeightedTrs
22.45/7.88	    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	    (4) CpxWeightedTrs
22.45/7.88	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	    (6) CpxTypedWeightedTrs
22.45/7.88	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
22.45/7.88	    (8) CpxTypedWeightedCompleteTrs
22.45/7.88	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	    (10) CpxTypedWeightedCompleteTrs
22.45/7.88	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
22.45/7.88	    (12) CpxRNTS
22.45/7.88	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	    (14) CpxRNTS
22.45/7.88	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
22.45/7.88	    (16) CpxRNTS
22.45/7.88	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.45/7.88	    (18) CpxRNTS
22.45/7.88	    (19) IntTrsBoundProof [UPPER BOUND(ID), 991 ms]
22.45/7.88	    (20) CpxRNTS
22.45/7.88	    (21) IntTrsBoundProof [UPPER BOUND(ID), 219 ms]
22.45/7.88	    (22) CpxRNTS
22.45/7.88	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.45/7.88	    (24) CpxRNTS
22.45/7.88	    (25) IntTrsBoundProof [UPPER BOUND(ID), 211 ms]
22.45/7.88	    (26) CpxRNTS
22.45/7.88	    (27) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
22.45/7.88	    (28) CpxRNTS
22.45/7.88	    (29) FinalProof [FINISHED, 0 ms]
22.45/7.88	    (30) BOUNDS(1, n^1)
22.45/7.88	(31) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
22.45/7.88	(32) TRS for Loop Detection
22.45/7.88	    (33) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
22.45/7.88	    (34) BEST
22.45/7.88	        (35) proven lower bound
22.45/7.88	            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
22.45/7.88	            (37) BOUNDS(n^1, INF)
22.45/7.88	        (38) TRS for Loop Detection
22.45/7.88	
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(0)
22.45/7.88	Obligation:
22.45/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   -(0, y) -> 0
22.45/7.88	   -(x, 0) -> x
22.45/7.88	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
22.45/7.88	   p(0) -> 0
22.45/7.88	   p(s(x)) -> x
22.45/7.88	
22.45/7.88	S is empty.
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Transformed relative TRS to weighted TRS
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(2)
22.45/7.88	Obligation:
22.45/7.88	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   -(0, y) -> 0 [1]
22.45/7.88	   -(x, 0) -> x [1]
22.45/7.88	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) [1]
22.45/7.88	   p(0) -> 0 [1]
22.45/7.88	   p(s(x)) -> x [1]
22.45/7.88	
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Renamed defined symbols to avoid conflicts with arithmetic symbols:
22.45/7.88	
22.45/7.88	- => minus
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(4)
22.45/7.88	Obligation:
22.45/7.88	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   minus(0, y) -> 0 [1]
22.45/7.88	   minus(x, 0) -> x [1]
22.45/7.88	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
22.45/7.88	   p(0) -> 0 [1]
22.45/7.88	   p(s(x)) -> x [1]
22.45/7.88	
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Infered types.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(6)
22.45/7.88	Obligation:
22.45/7.88	Runtime Complexity Weighted TRS with Types.
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   minus(0, y) -> 0 [1]
22.45/7.88	   minus(x, 0) -> x [1]
22.45/7.88	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
22.45/7.88	   p(0) -> 0 [1]
22.45/7.88	   p(s(x)) -> x [1]
22.45/7.88	
22.45/7.88	The TRS has the following type information:
22.45/7.88	minus :: 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	0 :: 0:s:if
22.45/7.88	s :: 0:s:if -> 0:s:if
22.45/7.88	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	greater :: 0:s:if -> 0:s:if -> greater
22.45/7.88	p :: 0:s:if -> 0:s:if
22.45/7.88	
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(7) CompletionProof (UPPER BOUND(ID))
22.45/7.88	The transformation into a RNTS is sound, since:
22.45/7.88	
22.45/7.88	(a) The obligation is a constructor system where every type has a constant constructor,
22.45/7.88	
22.45/7.88	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
22.45/7.88	
22.45/7.88	   minus_2
22.45/7.88	
22.45/7.88	(c) The following functions are completely defined:
22.45/7.88	
22.45/7.88	   p_1
22.45/7.88	
22.45/7.88	Due to the following rules being added:
22.45/7.88	
22.45/7.88	   p(v0) -> 0 [0]
22.45/7.88	
22.45/7.88	And the following fresh constants:    const
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(8)
22.45/7.88	Obligation:
22.45/7.88	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.45/7.88	
22.45/7.88	Runtime Complexity Weighted TRS with Types.
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   minus(0, y) -> 0 [1]
22.45/7.88	   minus(x, 0) -> x [1]
22.45/7.88	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
22.45/7.88	   p(0) -> 0 [1]
22.45/7.88	   p(s(x)) -> x [1]
22.45/7.88	   p(v0) -> 0 [0]
22.45/7.88	
22.45/7.88	The TRS has the following type information:
22.45/7.88	minus :: 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	0 :: 0:s:if
22.45/7.88	s :: 0:s:if -> 0:s:if
22.45/7.88	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	greater :: 0:s:if -> 0:s:if -> greater
22.45/7.88	p :: 0:s:if -> 0:s:if
22.45/7.88	const :: greater
22.45/7.88	
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(10)
22.45/7.88	Obligation:
22.45/7.88	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.45/7.88	
22.45/7.88	Runtime Complexity Weighted TRS with Types.
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   minus(0, y) -> 0 [1]
22.45/7.88	   minus(x, 0) -> x [1]
22.45/7.88	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, y)), 0) [2]
22.45/7.88	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, 0)), 0) [1]
22.45/7.88	   p(0) -> 0 [1]
22.45/7.88	   p(s(x)) -> x [1]
22.45/7.88	   p(v0) -> 0 [0]
22.45/7.88	
22.45/7.88	The TRS has the following type information:
22.45/7.88	minus :: 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	0 :: 0:s:if
22.45/7.88	s :: 0:s:if -> 0:s:if
22.45/7.88	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
22.45/7.88	greater :: 0:s:if -> 0:s:if -> greater
22.45/7.88	p :: 0:s:if -> 0:s:if
22.45/7.88	const :: greater
22.45/7.88	
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
22.45/7.88	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
22.45/7.88	The constant constructors are abstracted as follows:
22.45/7.88	
22.45/7.88	   0 => 0
22.45/7.88	   const => 0
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(12)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, y)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, 0)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
22.45/7.88	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
22.45/7.88	
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(14)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
22.45/7.88	Found the following analysis order by SCC decomposition:
22.45/7.88	
22.45/7.88	   { minus }
22.45/7.88	   { p }
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(16)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {minus}, {p}
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(17) ResultPropagationProof (UPPER BOUND(ID))
22.45/7.88	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(18)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {minus}, {p}
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(19) IntTrsBoundProof (UPPER BOUND(ID))
22.45/7.88	
22.45/7.88	Computed SIZE bound using CoFloCo for: minus
22.45/7.88	after applying outer abstraction to obtain an ITS,
22.45/7.88	resulting in: O(n^2) with polynomial bound: 3 + 2*z + z*z' + 3*z' + z'^2
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(20)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {minus}, {p}
22.45/7.88	Previous analysis results are:
22.45/7.88	  minus: runtime: ?, size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(21) IntTrsBoundProof (UPPER BOUND(ID))
22.45/7.88	
22.45/7.88	Computed RUNTIME bound using CoFloCo for: minus
22.45/7.88	after applying outer abstraction to obtain an ITS,
22.45/7.88	resulting in: O(n^1) with polynomial bound: 4 + 2*z'
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(22)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {p}
22.45/7.88	Previous analysis results are:
22.45/7.88	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(23) ResultPropagationProof (UPPER BOUND(ID))
22.45/7.88	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(24)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {p}
22.45/7.88	Previous analysis results are:
22.45/7.88	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(25) IntTrsBoundProof (UPPER BOUND(ID))
22.45/7.88	
22.45/7.88	Computed SIZE bound using KoAT for: p
22.45/7.88	after applying outer abstraction to obtain an ITS,
22.45/7.88	resulting in: O(n^1) with polynomial bound: z
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(26)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: {p}
22.45/7.88	Previous analysis results are:
22.45/7.88	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
22.45/7.88	  p: runtime: ?, size: O(n^1) [z]
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(27) IntTrsBoundProof (UPPER BOUND(ID))
22.45/7.88	
22.45/7.88	Computed RUNTIME bound using CoFloCo for: p
22.45/7.88	after applying outer abstraction to obtain an ITS,
22.45/7.88	resulting in: O(1) with polynomial bound: 1
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(28)
22.45/7.88	Obligation:
22.45/7.88	Complexity RNTS consisting of the following rules:
22.45/7.88	
22.45/7.88	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
22.45/7.88	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.45/7.88	   minus(z, z') -{ 4 + 2*z' }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s) + 0 :|: s >= 0, s <= (z' - 1) * (z' - 1) + 3 * (z' - 1) + 2 * z + 3 + (z' - 1) * z, z >= 0, z' - 1 >= 0
22.45/7.88	   minus(z, z') -{ 5 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + s') + 0 :|: s' >= 0, s' <= 0 * 0 + 3 * 0 + 2 * z + 3 + 0 * z, z >= 0, z' - 1 >= 0
22.45/7.88	   p(z) -{ 1 }-> 0 :|: z = 0
22.45/7.88	   p(z) -{ 0 }-> 0 :|: z >= 0
22.45/7.88	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
22.45/7.88	
22.45/7.88	Function symbols to be analyzed: 
22.45/7.88	Previous analysis results are:
22.45/7.88	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
22.45/7.88	  p: runtime: O(1) [1], size: O(n^1) [z]
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(29) FinalProof (FINISHED)
22.45/7.88	Computed overall runtime complexity
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(30)
22.45/7.88	BOUNDS(1, n^1)
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(31) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
22.45/7.88	Transformed a relative TRS into a decreasing-loop problem.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(32)
22.45/7.88	Obligation:
22.45/7.88	Analyzing the following TRS for decreasing loops:
22.45/7.88	
22.45/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   -(0, y) -> 0
22.45/7.88	   -(x, 0) -> x
22.45/7.88	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
22.45/7.88	   p(0) -> 0
22.45/7.88	   p(s(x)) -> x
22.45/7.88	
22.45/7.88	S is empty.
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(33) DecreasingLoopProof (LOWER BOUND(ID))
22.45/7.88	The following loop(s) give(s) rise to the lower bound Omega(n^1):
22.45/7.88	
22.45/7.88	The rewrite sequence
22.45/7.88	
22.45/7.88	-(x, s(y)) ->^+ if(greater(x, s(y)), s(-(x, y)), 0)
22.45/7.88	
22.45/7.88	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
22.45/7.88	
22.45/7.88	The pumping substitution is [y / s(y)].
22.45/7.88	
22.45/7.88	The result substitution is [ ].
22.45/7.88	
22.45/7.88	
22.45/7.88	
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(34)
22.45/7.88	Complex Obligation (BEST)
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(35)
22.45/7.88	Obligation:
22.45/7.88	Proved the lower bound n^1 for the following obligation:
22.45/7.88	
22.45/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   -(0, y) -> 0
22.45/7.88	   -(x, 0) -> x
22.45/7.88	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
22.45/7.88	   p(0) -> 0
22.45/7.88	   p(s(x)) -> x
22.45/7.88	
22.45/7.88	S is empty.
22.45/7.88	Rewrite Strategy: INNERMOST
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(36) LowerBoundPropagationProof (FINISHED)
22.45/7.88	Propagated lower bound.
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(37)
22.45/7.88	BOUNDS(n^1, INF)
22.45/7.88	
22.45/7.88	----------------------------------------
22.45/7.88	
22.45/7.88	(38)
22.45/7.88	Obligation:
22.45/7.88	Analyzing the following TRS for decreasing loops:
22.45/7.88	
22.45/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.45/7.88	
22.45/7.88	
22.45/7.88	The TRS R consists of the following rules:
22.45/7.88	
22.45/7.88	   -(0, y) -> 0
22.45/7.88	   -(x, 0) -> x
22.45/7.88	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
22.45/7.88	   p(0) -> 0
22.45/7.88	   p(s(x)) -> x
22.45/7.88	
22.45/7.88	S is empty.
22.45/7.88	Rewrite Strategy: INNERMOST
22.81/9.13	EOF