32.34/9.92	WORST_CASE(Omega(n^1), O(n^1))
32.34/9.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.34/9.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.34/9.94	
32.34/9.94	
32.34/9.94	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.34/9.94	
32.34/9.94	(0) CpxTRS
32.34/9.94	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	(2) CpxTRS
32.34/9.94	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (4) CpxTRS
32.34/9.94	    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (6) CpxWeightedTrs
32.34/9.94	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (8) CpxTypedWeightedTrs
32.34/9.94	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (10) CpxTypedWeightedCompleteTrs
32.34/9.94	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (12) CpxTypedWeightedCompleteTrs
32.34/9.94	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (14) CpxRNTS
32.34/9.94	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (16) CpxRNTS
32.34/9.94	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
32.34/9.94	    (18) CpxRNTS
32.34/9.94	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (20) CpxRNTS
32.34/9.94	    (21) IntTrsBoundProof [UPPER BOUND(ID), 273 ms]
32.34/9.94	    (22) CpxRNTS
32.34/9.94	    (23) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
32.34/9.94	    (24) CpxRNTS
32.34/9.94	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (26) CpxRNTS
32.34/9.94	    (27) IntTrsBoundProof [UPPER BOUND(ID), 264 ms]
32.34/9.94	    (28) CpxRNTS
32.34/9.94	    (29) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
32.34/9.94	    (30) CpxRNTS
32.34/9.94	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (32) CpxRNTS
32.34/9.94	    (33) IntTrsBoundProof [UPPER BOUND(ID), 465 ms]
32.34/9.94	    (34) CpxRNTS
32.34/9.94	    (35) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
32.34/9.94	    (36) CpxRNTS
32.34/9.94	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.34/9.94	    (38) CpxRNTS
32.34/9.94	    (39) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
32.34/9.94	    (40) CpxRNTS
32.34/9.94	    (41) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
32.34/9.94	    (42) CpxRNTS
32.34/9.94	    (43) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
32.34/9.94	    (44) CpxRNTS
32.34/9.94	    (45) IntTrsBoundProof [UPPER BOUND(ID), 361 ms]
32.34/9.94	    (46) CpxRNTS
32.34/9.94	    (47) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
32.34/9.94	    (48) CpxRNTS
32.34/9.94	    (49) FinalProof [FINISHED, 0 ms]
32.34/9.94	    (50) BOUNDS(1, n^1)
32.34/9.94	(51) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
32.34/9.94	(52) TRS for Loop Detection
32.34/9.94	    (53) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
32.34/9.94	    (54) BEST
32.34/9.94	        (55) proven lower bound
32.34/9.94	            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
32.34/9.94	            (57) BOUNDS(n^1, INF)
32.34/9.94	        (58) TRS for Loop Detection
32.34/9.94	
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(0)
32.34/9.94	Obligation:
32.34/9.94	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.34/9.94	
32.34/9.94	
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x)
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.94	   ++(x, nil) -> x
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.94	   null(nil) -> true
32.34/9.94	   null(g(x, y)) -> false
32.34/9.94	   mem(nil, y) -> false
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.94	   mem(x, max(x)) -> not(null(x))
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.94	
32.34/9.94	S is empty.
32.34/9.94	Rewrite Strategy: FULL
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
32.34/9.94	The TRS does not nest defined symbols.
32.34/9.94	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
32.34/9.94	mem(x, max(x)) -> not(null(x))
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(2)
32.34/9.94	Obligation:
32.34/9.94	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
32.34/9.94	
32.34/9.94	
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x)
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.94	   ++(x, nil) -> x
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.94	   null(nil) -> true
32.34/9.94	   null(g(x, y)) -> false
32.34/9.94	   mem(nil, y) -> false
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.94	
32.34/9.94	S is empty.
32.34/9.94	Rewrite Strategy: FULL
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Converted rc-obligation to irc-obligation.
32.34/9.94	
32.34/9.94	As the TRS does not nest defined symbols, we have rc = irc.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(4)
32.34/9.94	Obligation:
32.34/9.94	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
32.34/9.94	
32.34/9.94	
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x)
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.94	   ++(x, nil) -> x
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.94	   null(nil) -> true
32.34/9.94	   null(g(x, y)) -> false
32.34/9.94	   mem(nil, y) -> false
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.94	
32.34/9.94	S is empty.
32.34/9.94	Rewrite Strategy: INNERMOST
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Transformed relative TRS to weighted TRS
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(6)
32.34/9.94	Obligation:
32.34/9.94	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
32.34/9.94	
32.34/9.94	
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x) [1]
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z) [1]
32.34/9.94	   ++(x, nil) -> x [1]
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z) [1]
32.34/9.94	   null(nil) -> true [1]
32.34/9.94	   null(g(x, y)) -> false [1]
32.34/9.94	   mem(nil, y) -> false [1]
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y) [1]
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]
32.34/9.94	
32.34/9.94	Rewrite Strategy: INNERMOST
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Infered types.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(8)
32.34/9.94	Obligation:
32.34/9.94	Runtime Complexity Weighted TRS with Types.
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x) [1]
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z) [1]
32.34/9.94	   ++(x, nil) -> x [1]
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z) [1]
32.34/9.94	   null(nil) -> true [1]
32.34/9.94	   null(g(x, y)) -> false [1]
32.34/9.94	   mem(nil, y) -> false [1]
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y) [1]
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]
32.34/9.94	
32.34/9.94	The TRS has the following type information:
32.34/9.94	f :: max':u -> nil:g -> nil:g
32.34/9.94	nil :: nil:g
32.34/9.94	g :: nil:g -> max':u -> nil:g
32.34/9.94	++ :: nil:g -> nil:g -> nil:g
32.34/9.94	null :: nil:g -> true:false:or
32.34/9.94	true :: true:false:or
32.34/9.94	false :: true:false:or
32.34/9.94	mem :: nil:g -> a -> true:false:or
32.34/9.94	or :: = -> true:false:or -> true:false:or
32.34/9.94	= :: max':u -> a -> =
32.34/9.94	max :: nil:g -> max':u
32.34/9.94	max' :: max':u -> max':u -> max':u
32.34/9.94	u :: max':u
32.34/9.94	
32.34/9.94	Rewrite Strategy: INNERMOST
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(9) CompletionProof (UPPER BOUND(ID))
32.34/9.94	The transformation into a RNTS is sound, since:
32.34/9.94	
32.34/9.94	(a) The obligation is a constructor system where every type has a constant constructor,
32.34/9.94	
32.34/9.94	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
32.34/9.94	
32.34/9.94	   f_2
32.34/9.94	   ++_2
32.34/9.94	   null_1
32.34/9.94	   mem_2
32.34/9.94	   max_1
32.34/9.94	
32.34/9.94	(c) The following functions are completely defined:
32.34/9.94	none
32.34/9.94	
32.34/9.94	Due to the following rules being added:
32.34/9.94	none
32.34/9.94	
32.34/9.94	And the following fresh constants:    const, const1
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(10)
32.34/9.94	Obligation:
32.34/9.94	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
32.34/9.94	
32.34/9.94	Runtime Complexity Weighted TRS with Types.
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x) [1]
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z) [1]
32.34/9.94	   ++(x, nil) -> x [1]
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z) [1]
32.34/9.94	   null(nil) -> true [1]
32.34/9.94	   null(g(x, y)) -> false [1]
32.34/9.94	   mem(nil, y) -> false [1]
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y) [1]
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]
32.34/9.94	
32.34/9.94	The TRS has the following type information:
32.34/9.94	f :: max':u -> nil:g -> nil:g
32.34/9.94	nil :: nil:g
32.34/9.94	g :: nil:g -> max':u -> nil:g
32.34/9.94	++ :: nil:g -> nil:g -> nil:g
32.34/9.94	null :: nil:g -> true:false:or
32.34/9.94	true :: true:false:or
32.34/9.94	false :: true:false:or
32.34/9.94	mem :: nil:g -> a -> true:false:or
32.34/9.94	or :: = -> true:false:or -> true:false:or
32.34/9.94	= :: max':u -> a -> =
32.34/9.94	max :: nil:g -> max':u
32.34/9.94	max' :: max':u -> max':u -> max':u
32.34/9.94	u :: max':u
32.34/9.94	const :: a
32.34/9.94	const1 :: =
32.34/9.94	
32.34/9.94	Rewrite Strategy: INNERMOST
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(12)
32.34/9.94	Obligation:
32.34/9.94	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
32.34/9.94	
32.34/9.94	Runtime Complexity Weighted TRS with Types.
32.34/9.94	The TRS R consists of the following rules:
32.34/9.94	
32.34/9.94	   f(x, nil) -> g(nil, x) [1]
32.34/9.94	   f(x, g(y, z)) -> g(f(x, y), z) [1]
32.34/9.94	   ++(x, nil) -> x [1]
32.34/9.94	   ++(x, g(y, z)) -> g(++(x, y), z) [1]
32.34/9.94	   null(nil) -> true [1]
32.34/9.94	   null(g(x, y)) -> false [1]
32.34/9.94	   mem(nil, y) -> false [1]
32.34/9.94	   mem(g(x, y), z) -> or(=(y, z), mem(x, z)) [1]
32.34/9.94	   max(g(g(nil, x), y)) -> max'(x, y) [1]
32.34/9.94	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u) [1]
32.34/9.94	
32.34/9.94	The TRS has the following type information:
32.34/9.94	f :: max':u -> nil:g -> nil:g
32.34/9.94	nil :: nil:g
32.34/9.94	g :: nil:g -> max':u -> nil:g
32.34/9.94	++ :: nil:g -> nil:g -> nil:g
32.34/9.94	null :: nil:g -> true:false:or
32.34/9.94	true :: true:false:or
32.34/9.94	false :: true:false:or
32.34/9.94	mem :: nil:g -> a -> true:false:or
32.34/9.94	or :: = -> true:false:or -> true:false:or
32.34/9.94	= :: max':u -> a -> =
32.34/9.94	max :: nil:g -> max':u
32.34/9.94	max' :: max':u -> max':u -> max':u
32.34/9.94	u :: max':u
32.34/9.94	const :: a
32.34/9.94	const1 :: =
32.34/9.94	
32.34/9.94	Rewrite Strategy: INNERMOST
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
32.34/9.94	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
32.34/9.94	The constant constructors are abstracted as follows:
32.34/9.94	
32.34/9.94	   nil => 0
32.34/9.94	   true => 1
32.34/9.94	   false => 0
32.34/9.94	   u => 0
32.34/9.94	   const => 0
32.34/9.94	   const1 => 0
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(14)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + x :|: z'' = 0, z' = x, x >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z) + mem(x, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(16)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
32.34/9.94	Found the following analysis order by SCC decomposition:
32.34/9.94	
32.34/9.94	   { mem }
32.34/9.94	   { max }
32.34/9.94	   { f }
32.34/9.94	   { null }
32.34/9.94	   { ++ }
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(18)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(19) ResultPropagationProof (UPPER BOUND(ID))
32.34/9.94	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(20)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(21) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed SIZE bound using CoFloCo for: mem
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^2) with polynomial bound: z' + z'*z'' + z'^2
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(22)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: ?, size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(23) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed RUNTIME bound using CoFloCo for: mem
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^1) with polynomial bound: 1 + z'
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(24)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {max}, {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(25) ResultPropagationProof (UPPER BOUND(ID))
32.34/9.94	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(26)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {max}, {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(27) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed SIZE bound using KoAT for: max
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^1) with polynomial bound: z'
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(28)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {max}, {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	  max: runtime: ?, size: O(n^1) [z']
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(29) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed RUNTIME bound using KoAT for: max
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^1) with polynomial bound: 1 + z'
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(30)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(31) ResultPropagationProof (UPPER BOUND(ID))
32.34/9.94	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(32)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(33) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed SIZE bound using CoFloCo for: f
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^1) with polynomial bound: 1 + z' + z''
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(34)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {f}, {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.94	  f: runtime: ?, size: O(n^1) [1 + z' + z'']
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(35) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.94	
32.34/9.94	Computed RUNTIME bound using CoFloCo for: f
32.34/9.94	after applying outer abstraction to obtain an ITS,
32.34/9.94	resulting in: O(n^1) with polynomial bound: 1 + z''
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(36)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.94	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.94	
32.34/9.94	Function symbols to be analyzed: {null}, {++}
32.34/9.94	Previous analysis results are:
32.34/9.94	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.94	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.94	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.94	
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(37) ResultPropagationProof (UPPER BOUND(ID))
32.34/9.94	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.34/9.94	----------------------------------------
32.34/9.94	
32.34/9.94	(38)
32.34/9.94	Obligation:
32.34/9.94	Complexity RNTS consisting of the following rules:
32.34/9.94	
32.34/9.94	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.94	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.94	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.94	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.94	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.94	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.94	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: {null}, {++}
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(39) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.95	
32.34/9.95	Computed SIZE bound using CoFloCo for: null
32.34/9.95	after applying outer abstraction to obtain an ITS,
32.34/9.95	resulting in: O(1) with polynomial bound: 1
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(40)
32.34/9.95	Obligation:
32.34/9.95	Complexity RNTS consisting of the following rules:
32.34/9.95	
32.34/9.95	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.95	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.95	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.95	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.95	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.95	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: {null}, {++}
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	  null: runtime: ?, size: O(1) [1]
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(41) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.95	
32.34/9.95	Computed RUNTIME bound using CoFloCo for: null
32.34/9.95	after applying outer abstraction to obtain an ITS,
32.34/9.95	resulting in: O(1) with polynomial bound: 1
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(42)
32.34/9.95	Obligation:
32.34/9.95	Complexity RNTS consisting of the following rules:
32.34/9.95	
32.34/9.95	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.95	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.95	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.95	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.95	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.95	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: {++}
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	  null: runtime: O(1) [1], size: O(1) [1]
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(43) ResultPropagationProof (UPPER BOUND(ID))
32.34/9.95	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(44)
32.34/9.95	Obligation:
32.34/9.95	Complexity RNTS consisting of the following rules:
32.34/9.95	
32.34/9.95	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.95	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.95	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.95	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.95	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.95	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: {++}
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	  null: runtime: O(1) [1], size: O(1) [1]
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(45) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.95	
32.34/9.95	Computed SIZE bound using CoFloCo for: ++
32.34/9.95	after applying outer abstraction to obtain an ITS,
32.34/9.95	resulting in: O(n^1) with polynomial bound: z' + z''
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(46)
32.34/9.95	Obligation:
32.34/9.95	Complexity RNTS consisting of the following rules:
32.34/9.95	
32.34/9.95	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.95	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.95	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.95	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.95	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.95	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: {++}
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	  null: runtime: O(1) [1], size: O(1) [1]
32.34/9.95	  ++: runtime: ?, size: O(n^1) [z' + z'']
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(47) IntTrsBoundProof (UPPER BOUND(ID))
32.34/9.95	
32.34/9.95	Computed RUNTIME bound using CoFloCo for: ++
32.34/9.95	after applying outer abstraction to obtain an ITS,
32.34/9.95	resulting in: O(n^1) with polynomial bound: 1 + z''
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(48)
32.34/9.95	Obligation:
32.34/9.95	Complexity RNTS consisting of the following rules:
32.34/9.95	
32.34/9.95	   ++(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
32.34/9.95	   ++(z', z'') -{ 1 }-> 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 2 + y }-> 1 + s'' + z :|: s'' >= 0, s'' <= z' + y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
32.34/9.95	   f(z', z'') -{ 1 }-> 1 + 0 + z' :|: z'' = 0, z' >= 0
32.34/9.95	   max(z') -{ 4 + x + y + z }-> 1 + s' + 0 :|: s' >= 0, s' <= 1 + (1 + x + y) + z, z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
32.34/9.95	   max(z') -{ 1 }-> 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
32.34/9.95	   mem(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
32.34/9.95	   mem(z', z'') -{ 2 + x }-> 1 + (1 + y + z'') + s :|: s >= 0, s <= x * x + x * z'' + x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	   null(z') -{ 1 }-> 1 :|: z' = 0
32.34/9.95	   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
32.34/9.95	
32.34/9.95	Function symbols to be analyzed: 
32.34/9.95	Previous analysis results are:
32.34/9.95	  mem: runtime: O(n^1) [1 + z'], size: O(n^2) [z' + z'*z'' + z'^2]
32.34/9.95	  max: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
32.34/9.95	  f: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z'']
32.34/9.95	  null: runtime: O(1) [1], size: O(1) [1]
32.34/9.95	  ++: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(49) FinalProof (FINISHED)
32.34/9.95	Computed overall runtime complexity
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(50)
32.34/9.95	BOUNDS(1, n^1)
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(51) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
32.34/9.95	Transformed a relative TRS into a decreasing-loop problem.
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(52)
32.34/9.95	Obligation:
32.34/9.95	Analyzing the following TRS for decreasing loops:
32.34/9.95	
32.34/9.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.34/9.95	
32.34/9.95	
32.34/9.95	The TRS R consists of the following rules:
32.34/9.95	
32.34/9.95	   f(x, nil) -> g(nil, x)
32.34/9.95	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.95	   ++(x, nil) -> x
32.34/9.95	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.95	   null(nil) -> true
32.34/9.95	   null(g(x, y)) -> false
32.34/9.95	   mem(nil, y) -> false
32.34/9.95	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.95	   mem(x, max(x)) -> not(null(x))
32.34/9.95	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.95	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.95	
32.34/9.95	S is empty.
32.34/9.95	Rewrite Strategy: FULL
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(53) DecreasingLoopProof (LOWER BOUND(ID))
32.34/9.95	The following loop(s) give(s) rise to the lower bound Omega(n^1):
32.34/9.95	
32.34/9.95	The rewrite sequence
32.34/9.95	
32.34/9.95	++(x, g(y, z)) ->^+ g(++(x, y), z)
32.34/9.95	
32.34/9.95	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
32.34/9.95	
32.34/9.95	The pumping substitution is [y / g(y, z)].
32.34/9.95	
32.34/9.95	The result substitution is [ ].
32.34/9.95	
32.34/9.95	
32.34/9.95	
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(54)
32.34/9.95	Complex Obligation (BEST)
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(55)
32.34/9.95	Obligation:
32.34/9.95	Proved the lower bound n^1 for the following obligation:
32.34/9.95	
32.34/9.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.34/9.95	
32.34/9.95	
32.34/9.95	The TRS R consists of the following rules:
32.34/9.95	
32.34/9.95	   f(x, nil) -> g(nil, x)
32.34/9.95	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.95	   ++(x, nil) -> x
32.34/9.95	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.95	   null(nil) -> true
32.34/9.95	   null(g(x, y)) -> false
32.34/9.95	   mem(nil, y) -> false
32.34/9.95	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.95	   mem(x, max(x)) -> not(null(x))
32.34/9.95	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.95	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.95	
32.34/9.95	S is empty.
32.34/9.95	Rewrite Strategy: FULL
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(56) LowerBoundPropagationProof (FINISHED)
32.34/9.95	Propagated lower bound.
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(57)
32.34/9.95	BOUNDS(n^1, INF)
32.34/9.95	
32.34/9.95	----------------------------------------
32.34/9.95	
32.34/9.95	(58)
32.34/9.95	Obligation:
32.34/9.95	Analyzing the following TRS for decreasing loops:
32.34/9.95	
32.34/9.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.34/9.95	
32.34/9.95	
32.34/9.95	The TRS R consists of the following rules:
32.34/9.95	
32.34/9.95	   f(x, nil) -> g(nil, x)
32.34/9.95	   f(x, g(y, z)) -> g(f(x, y), z)
32.34/9.95	   ++(x, nil) -> x
32.34/9.95	   ++(x, g(y, z)) -> g(++(x, y), z)
32.34/9.95	   null(nil) -> true
32.34/9.95	   null(g(x, y)) -> false
32.34/9.95	   mem(nil, y) -> false
32.34/9.95	   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
32.34/9.95	   mem(x, max(x)) -> not(null(x))
32.34/9.95	   max(g(g(nil, x), y)) -> max'(x, y)
32.34/9.95	   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)
32.34/9.95	
32.34/9.95	S is empty.
32.34/9.95	Rewrite Strategy: FULL
32.34/9.99	EOF