20.52/7.00	WORST_CASE(Omega(n^1), O(n^1))
20.62/7.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.62/7.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.62/7.02	
20.62/7.02	
20.62/7.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.62/7.02	
20.62/7.02	(0) CpxTRS
20.62/7.02	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 4 ms]
20.62/7.02	(2) CpxTRS
20.62/7.02	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (4) CpxWeightedTrs
20.62/7.02	    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (6) CpxWeightedTrs
20.62/7.02	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (8) CpxTypedWeightedTrs
20.62/7.02	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
20.62/7.02	    (10) CpxTypedWeightedCompleteTrs
20.62/7.02	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (12) CpxTypedWeightedCompleteTrs
20.62/7.02	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.62/7.02	    (14) CpxRNTS
20.62/7.02	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (16) CpxRNTS
20.62/7.02	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (18) CpxRNTS
20.62/7.02	    (19) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
20.62/7.02	    (20) CpxRNTS
20.62/7.02	    (21) IntTrsBoundProof [UPPER BOUND(ID), 1020 ms]
20.62/7.02	    (22) CpxRNTS
20.62/7.02	    (23) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
20.62/7.02	    (24) CpxRNTS
20.62/7.02	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.62/7.02	    (26) CpxRNTS
20.62/7.02	    (27) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
20.62/7.02	    (28) CpxRNTS
20.62/7.02	    (29) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
20.62/7.02	    (30) CpxRNTS
20.62/7.02	    (31) FinalProof [FINISHED, 0 ms]
20.62/7.02	    (32) BOUNDS(1, n^1)
20.62/7.02	(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	(34) CpxTRS
20.62/7.02	    (35) SlicingProof [LOWER BOUND(ID), 0 ms]
20.62/7.02	    (36) CpxTRS
20.62/7.02	    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.62/7.02	    (38) typed CpxTrs
20.62/7.02	    (39) OrderProof [LOWER BOUND(ID), 0 ms]
20.62/7.02	    (40) typed CpxTrs
20.62/7.02	    (41) RewriteLemmaProof [LOWER BOUND(ID), 1073 ms]
20.62/7.02	    (42) proven lower bound
20.62/7.02	    (43) LowerBoundPropagationProof [FINISHED, 0 ms]
20.62/7.02	    (44) BOUNDS(n^1, INF)
20.62/7.02	
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(0)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   -(0, y) -> 0
20.62/7.02	   -(x, 0) -> x
20.62/7.02	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
20.62/7.02	   p(0) -> 0
20.62/7.02	   p(s(x)) -> x
20.62/7.02	
20.62/7.02	S is empty.
20.62/7.02	Rewrite Strategy: FULL
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Converted rc-obligation to irc-obligation.
20.62/7.02	
20.62/7.02	The duplicating contexts are:
20.62/7.02	-([], s(y))
20.62/7.02	-(x, s([]))
20.62/7.02	
20.62/7.02	
20.62/7.02	The defined contexts are:
20.62/7.02	-(x0, [])
20.62/7.02	p(s([]))
20.62/7.02	
20.62/7.02	
20.62/7.02	[] just represents basic- or constructor-terms in the following defined contexts:
20.62/7.02	-(x0, [])
20.62/7.02	
20.62/7.02	
20.62/7.02	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(2)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   -(0, y) -> 0
20.62/7.02	   -(x, 0) -> x
20.62/7.02	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
20.62/7.02	   p(0) -> 0
20.62/7.02	   p(s(x)) -> x
20.62/7.02	
20.62/7.02	S is empty.
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Transformed relative TRS to weighted TRS
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(4)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   -(0, y) -> 0 [1]
20.62/7.02	   -(x, 0) -> x [1]
20.62/7.02	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0) [1]
20.62/7.02	   p(0) -> 0 [1]
20.62/7.02	   p(s(x)) -> x [1]
20.62/7.02	
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Renamed defined symbols to avoid conflicts with arithmetic symbols:
20.62/7.02	
20.62/7.02	- => minus
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(6)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   minus(0, y) -> 0 [1]
20.62/7.02	   minus(x, 0) -> x [1]
20.62/7.02	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
20.62/7.02	   p(0) -> 0 [1]
20.62/7.02	   p(s(x)) -> x [1]
20.62/7.02	
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Infered types.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(8)
20.62/7.02	Obligation:
20.62/7.02	Runtime Complexity Weighted TRS with Types.
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   minus(0, y) -> 0 [1]
20.62/7.02	   minus(x, 0) -> x [1]
20.62/7.02	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
20.62/7.02	   p(0) -> 0 [1]
20.62/7.02	   p(s(x)) -> x [1]
20.62/7.02	
20.62/7.02	The TRS has the following type information:
20.62/7.02	minus :: 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	0 :: 0:s:if
20.62/7.02	s :: 0:s:if -> 0:s:if
20.62/7.02	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	greater :: 0:s:if -> 0:s:if -> greater
20.62/7.02	p :: 0:s:if -> 0:s:if
20.62/7.02	
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(9) CompletionProof (UPPER BOUND(ID))
20.62/7.02	The transformation into a RNTS is sound, since:
20.62/7.02	
20.62/7.02	(a) The obligation is a constructor system where every type has a constant constructor,
20.62/7.02	
20.62/7.02	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.62/7.02	
20.62/7.02	   minus_2
20.62/7.02	
20.62/7.02	(c) The following functions are completely defined:
20.62/7.02	
20.62/7.02	   p_1
20.62/7.02	
20.62/7.02	Due to the following rules being added:
20.62/7.02	
20.62/7.02	   p(v0) -> 0 [0]
20.62/7.02	
20.62/7.02	And the following fresh constants:    const
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(10)
20.62/7.02	Obligation:
20.62/7.02	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.62/7.02	
20.62/7.02	Runtime Complexity Weighted TRS with Types.
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   minus(0, y) -> 0 [1]
20.62/7.02	   minus(x, 0) -> x [1]
20.62/7.02	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
20.62/7.02	   p(0) -> 0 [1]
20.62/7.02	   p(s(x)) -> x [1]
20.62/7.02	   p(v0) -> 0 [0]
20.62/7.02	
20.62/7.02	The TRS has the following type information:
20.62/7.02	minus :: 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	0 :: 0:s:if
20.62/7.02	s :: 0:s:if -> 0:s:if
20.62/7.02	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	greater :: 0:s:if -> 0:s:if -> greater
20.62/7.02	p :: 0:s:if -> 0:s:if
20.62/7.02	const :: greater
20.62/7.02	
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(12)
20.62/7.02	Obligation:
20.62/7.02	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.62/7.02	
20.62/7.02	Runtime Complexity Weighted TRS with Types.
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   minus(0, y) -> 0 [1]
20.62/7.02	   minus(x, 0) -> x [1]
20.62/7.02	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, y)), 0) [2]
20.62/7.02	   minus(x, s(y)) -> if(greater(x, s(y)), s(minus(x, 0)), 0) [1]
20.62/7.02	   p(0) -> 0 [1]
20.62/7.02	   p(s(x)) -> x [1]
20.62/7.02	   p(v0) -> 0 [0]
20.62/7.02	
20.62/7.02	The TRS has the following type information:
20.62/7.02	minus :: 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	0 :: 0:s:if
20.62/7.02	s :: 0:s:if -> 0:s:if
20.62/7.02	if :: greater -> 0:s:if -> 0:s:if -> 0:s:if
20.62/7.02	greater :: 0:s:if -> 0:s:if -> greater
20.62/7.02	p :: 0:s:if -> 0:s:if
20.62/7.02	const :: greater
20.62/7.02	
20.62/7.02	Rewrite Strategy: INNERMOST
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.62/7.02	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.62/7.02	The constant constructors are abstracted as follows:
20.62/7.02	
20.62/7.02	   0 => 0
20.62/7.02	   const => 0
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(14)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, y)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + x + (1 + y)) + (1 + minus(x, 0)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
20.62/7.02	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
20.62/7.02	
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(16)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Found the following analysis order by SCC decomposition:
20.62/7.02	
20.62/7.02	   { minus }
20.62/7.02	   { p }
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(18)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {minus}, {p}
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(19) ResultPropagationProof (UPPER BOUND(ID))
20.62/7.02	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(20)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {minus}, {p}
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.62/7.02	
20.62/7.02	Computed SIZE bound using CoFloCo for: minus
20.62/7.02	after applying outer abstraction to obtain an ITS,
20.62/7.02	resulting in: O(n^2) with polynomial bound: 3 + 2*z + z*z' + 3*z' + z'^2
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(22)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {minus}, {p}
20.62/7.02	Previous analysis results are:
20.62/7.02	  minus: runtime: ?, size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(23) IntTrsBoundProof (UPPER BOUND(ID))
20.62/7.02	
20.62/7.02	Computed RUNTIME bound using CoFloCo for: minus
20.62/7.02	after applying outer abstraction to obtain an ITS,
20.62/7.02	resulting in: O(n^1) with polynomial bound: 4 + 2*z'
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(24)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, 0)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   minus(z, z') -{ 2 }-> 1 + (1 + z + (1 + (z' - 1))) + (1 + minus(z, z' - 1)) + 0 :|: z >= 0, z' - 1 >= 0
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {p}
20.62/7.02	Previous analysis results are:
20.62/7.02	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(25) ResultPropagationProof (UPPER BOUND(ID))
20.62/7.02	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(26)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   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
20.62/7.02	   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
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {p}
20.62/7.02	Previous analysis results are:
20.62/7.02	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(27) IntTrsBoundProof (UPPER BOUND(ID))
20.62/7.02	
20.62/7.02	Computed SIZE bound using KoAT for: p
20.62/7.02	after applying outer abstraction to obtain an ITS,
20.62/7.02	resulting in: O(n^1) with polynomial bound: z
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(28)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   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
20.62/7.02	   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
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: {p}
20.62/7.02	Previous analysis results are:
20.62/7.02	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
20.62/7.02	  p: runtime: ?, size: O(n^1) [z]
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(29) IntTrsBoundProof (UPPER BOUND(ID))
20.62/7.02	
20.62/7.02	Computed RUNTIME bound using CoFloCo for: p
20.62/7.02	after applying outer abstraction to obtain an ITS,
20.62/7.02	resulting in: O(1) with polynomial bound: 1
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(30)
20.62/7.02	Obligation:
20.62/7.02	Complexity RNTS consisting of the following rules:
20.62/7.02	
20.62/7.02	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
20.62/7.02	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.62/7.02	   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
20.62/7.02	   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
20.62/7.02	   p(z) -{ 1 }-> 0 :|: z = 0
20.62/7.02	   p(z) -{ 0 }-> 0 :|: z >= 0
20.62/7.02	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
20.62/7.02	
20.62/7.02	Function symbols to be analyzed: 
20.62/7.02	Previous analysis results are:
20.62/7.02	  minus: runtime: O(n^1) [4 + 2*z'], size: O(n^2) [3 + 2*z + z*z' + 3*z' + z'^2]
20.62/7.02	  p: runtime: O(1) [1], size: O(n^1) [z]
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(31) FinalProof (FINISHED)
20.62/7.02	Computed overall runtime complexity
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(32)
20.62/7.02	BOUNDS(1, n^1)
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(33) RenamingProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Renamed function symbols to avoid clashes with predefined symbol.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(34)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   -(0', y) -> 0'
20.62/7.02	   -(x, 0') -> x
20.62/7.02	   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
20.62/7.02	   p(0') -> 0'
20.62/7.02	   p(s(x)) -> x
20.62/7.02	
20.62/7.02	S is empty.
20.62/7.02	Rewrite Strategy: FULL
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(35) SlicingProof (LOWER BOUND(ID))
20.62/7.02	Sliced the following arguments:
20.62/7.02	if/0
20.62/7.02	if/2
20.62/7.02	greater/0
20.62/7.02	greater/1
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(36)
20.62/7.02	Obligation:
20.62/7.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
20.62/7.02	
20.62/7.02	
20.62/7.02	The TRS R consists of the following rules:
20.62/7.02	
20.62/7.02	   -(0', y) -> 0'
20.62/7.02	   -(x, 0') -> x
20.62/7.02	   -(x, s(y)) -> if(s(-(x, p(s(y)))))
20.62/7.02	   p(0') -> 0'
20.62/7.02	   p(s(x)) -> x
20.62/7.02	
20.62/7.02	S is empty.
20.62/7.02	Rewrite Strategy: FULL
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(37) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.62/7.02	Infered types.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(38)
20.62/7.02	Obligation:
20.62/7.02	TRS:
20.62/7.02	Rules:
20.62/7.02	-(0', y) -> 0'
20.62/7.02	-(x, 0') -> x
20.62/7.02	-(x, s(y)) -> if(s(-(x, p(s(y)))))
20.62/7.02	p(0') -> 0'
20.62/7.02	p(s(x)) -> x
20.62/7.02	
20.62/7.02	Types:
20.62/7.02	- :: 0':s:if -> 0':s:if -> 0':s:if
20.62/7.02	0' :: 0':s:if
20.62/7.02	s :: 0':s:if -> 0':s:if
20.62/7.02	if :: 0':s:if -> 0':s:if
20.62/7.02	p :: 0':s:if -> 0':s:if
20.62/7.02	hole_0':s:if1_0 :: 0':s:if
20.62/7.02	gen_0':s:if2_0 :: Nat -> 0':s:if
20.62/7.02	
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(39) OrderProof (LOWER BOUND(ID))
20.62/7.02	Heuristically decided to analyse the following defined symbols:
20.62/7.02	-
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(40)
20.62/7.02	Obligation:
20.62/7.02	TRS:
20.62/7.02	Rules:
20.62/7.02	-(0', y) -> 0'
20.62/7.02	-(x, 0') -> x
20.62/7.02	-(x, s(y)) -> if(s(-(x, p(s(y)))))
20.62/7.02	p(0') -> 0'
20.62/7.02	p(s(x)) -> x
20.62/7.02	
20.62/7.02	Types:
20.62/7.02	- :: 0':s:if -> 0':s:if -> 0':s:if
20.62/7.02	0' :: 0':s:if
20.62/7.02	s :: 0':s:if -> 0':s:if
20.62/7.02	if :: 0':s:if -> 0':s:if
20.62/7.02	p :: 0':s:if -> 0':s:if
20.62/7.02	hole_0':s:if1_0 :: 0':s:if
20.62/7.02	gen_0':s:if2_0 :: Nat -> 0':s:if
20.62/7.02	
20.62/7.02	
20.62/7.02	Generator Equations:
20.62/7.02	gen_0':s:if2_0(0) <=> 0'
20.62/7.02	gen_0':s:if2_0(+(x, 1)) <=> s(gen_0':s:if2_0(x))
20.62/7.02	
20.62/7.02	
20.62/7.02	The following defined symbols remain to be analysed:
20.62/7.02	-
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(41) RewriteLemmaProof (LOWER BOUND(ID))
20.62/7.02	Proved the following rewrite lemma:
20.62/7.02	-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
20.62/7.02	
20.62/7.02	Induction Base:
20.62/7.02	-(gen_0':s:if2_0(a), gen_0':s:if2_0(0))
20.62/7.02	
20.62/7.02	Induction Step:
20.62/7.02	-(gen_0':s:if2_0(a), gen_0':s:if2_0(+(n4_0, 1))) ->_R^Omega(1)
20.62/7.02	if(s(-(gen_0':s:if2_0(a), p(s(gen_0':s:if2_0(n4_0)))))) ->_R^Omega(1)
20.62/7.02	if(s(-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)))) ->_IH
20.62/7.02	if(s(*3_0))
20.62/7.02	
20.62/7.02	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(42)
20.62/7.02	Obligation:
20.62/7.02	Proved the lower bound n^1 for the following obligation:
20.62/7.02	
20.62/7.02	TRS:
20.62/7.02	Rules:
20.62/7.02	-(0', y) -> 0'
20.62/7.02	-(x, 0') -> x
20.62/7.02	-(x, s(y)) -> if(s(-(x, p(s(y)))))
20.62/7.02	p(0') -> 0'
20.62/7.02	p(s(x)) -> x
20.62/7.02	
20.62/7.02	Types:
20.62/7.02	- :: 0':s:if -> 0':s:if -> 0':s:if
20.62/7.02	0' :: 0':s:if
20.62/7.02	s :: 0':s:if -> 0':s:if
20.62/7.02	if :: 0':s:if -> 0':s:if
20.62/7.02	p :: 0':s:if -> 0':s:if
20.62/7.02	hole_0':s:if1_0 :: 0':s:if
20.62/7.02	gen_0':s:if2_0 :: Nat -> 0':s:if
20.62/7.02	
20.62/7.02	
20.62/7.02	Generator Equations:
20.62/7.02	gen_0':s:if2_0(0) <=> 0'
20.62/7.02	gen_0':s:if2_0(+(x, 1)) <=> s(gen_0':s:if2_0(x))
20.62/7.02	
20.62/7.02	
20.62/7.02	The following defined symbols remain to be analysed:
20.62/7.02	-
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(43) LowerBoundPropagationProof (FINISHED)
20.62/7.02	Propagated lower bound.
20.62/7.02	----------------------------------------
20.62/7.02	
20.62/7.02	(44)
20.62/7.02	BOUNDS(n^1, INF)
20.62/7.05	EOF