316.14/291.49	WORST_CASE(Omega(n^2), O(n^3))
316.14/291.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
316.14/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
316.14/291.50	
316.14/291.50	
316.14/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^3).
316.14/291.50	
316.14/291.50	(0) CpxTRS
316.14/291.50	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	(2) CpxWeightedTrs
316.14/291.50	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	    (4) CpxTypedWeightedTrs
316.14/291.50	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
316.14/291.50	    (6) CpxTypedWeightedCompleteTrs
316.14/291.50	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	    (8) CpxTypedWeightedCompleteTrs
316.14/291.50	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
316.14/291.50	    (10) CpxRNTS
316.14/291.50	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	    (12) CpxRNTS
316.14/291.50	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	    (14) CpxRNTS
316.14/291.50	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
316.14/291.50	    (16) CpxRNTS
316.14/291.50	    (17) IntTrsBoundProof [UPPER BOUND(ID), 626 ms]
316.14/291.50	    (18) CpxRNTS
316.14/291.50	    (19) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
316.14/291.50	    (20) CpxRNTS
316.14/291.50	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
316.14/291.50	    (22) CpxRNTS
316.14/291.50	    (23) IntTrsBoundProof [UPPER BOUND(ID), 557 ms]
316.14/291.50	    (24) CpxRNTS
316.14/291.50	    (25) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
316.14/291.50	    (26) CpxRNTS
316.14/291.50	    (27) FinalProof [FINISHED, 0 ms]
316.14/291.50	    (28) BOUNDS(1, n^3)
316.14/291.50	(29) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	(30) CpxTRS
316.14/291.50	    (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
316.14/291.50	    (32) typed CpxTrs
316.14/291.50	    (33) OrderProof [LOWER BOUND(ID), 0 ms]
316.14/291.50	    (34) typed CpxTrs
316.14/291.50	    (35) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
316.14/291.50	    (36) BEST
316.14/291.50	        (37) proven lower bound
316.14/291.50	            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
316.14/291.50	            (39) BOUNDS(n^1, INF)
316.14/291.50	        (40) typed CpxTrs
316.14/291.50	            (41) RewriteLemmaProof [LOWER BOUND(ID), 60 ms]
316.14/291.50	            (42) proven lower bound
316.14/291.50	            (43) LowerBoundPropagationProof [FINISHED, 0 ms]
316.14/291.50	            (44) BOUNDS(n^2, INF)
316.14/291.50	
316.14/291.50	
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(0)
316.14/291.50	Obligation:
316.14/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^3).
316.14/291.50	
316.14/291.50	
316.14/291.50	The TRS R consists of the following rules:
316.14/291.50	
316.14/291.50	   times(x, 0) -> 0
316.14/291.50	   times(x, s(y)) -> plus(times(x, y), x)
316.14/291.50	   plus(x, 0) -> x
316.14/291.50	   plus(0, x) -> x
316.14/291.50	   plus(x, s(y)) -> s(plus(x, y))
316.14/291.50	   plus(s(x), y) -> s(plus(x, y))
316.14/291.50	
316.14/291.50	S is empty.
316.14/291.50	Rewrite Strategy: INNERMOST
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
316.14/291.50	Transformed relative TRS to weighted TRS
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(2)
316.14/291.50	Obligation:
316.14/291.50	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
316.14/291.50	
316.14/291.50	
316.14/291.50	The TRS R consists of the following rules:
316.14/291.50	
316.14/291.50	   times(x, 0) -> 0 [1]
316.14/291.50	   times(x, s(y)) -> plus(times(x, y), x) [1]
316.14/291.50	   plus(x, 0) -> x [1]
316.14/291.50	   plus(0, x) -> x [1]
316.14/291.50	   plus(x, s(y)) -> s(plus(x, y)) [1]
316.14/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
316.14/291.50	
316.14/291.50	Rewrite Strategy: INNERMOST
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
316.14/291.50	Infered types.
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(4)
316.14/291.50	Obligation:
316.14/291.50	Runtime Complexity Weighted TRS with Types.
316.14/291.50	The TRS R consists of the following rules:
316.14/291.50	
316.14/291.50	   times(x, 0) -> 0 [1]
316.14/291.50	   times(x, s(y)) -> plus(times(x, y), x) [1]
316.14/291.50	   plus(x, 0) -> x [1]
316.14/291.50	   plus(0, x) -> x [1]
316.14/291.50	   plus(x, s(y)) -> s(plus(x, y)) [1]
316.14/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
316.14/291.50	
316.14/291.50	The TRS has the following type information:
316.14/291.50	times :: 0:s -> 0:s -> 0:s
316.14/291.50	0 :: 0:s
316.14/291.50	s :: 0:s -> 0:s
316.14/291.50	plus :: 0:s -> 0:s -> 0:s
316.14/291.50	
316.14/291.50	Rewrite Strategy: INNERMOST
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(5) CompletionProof (UPPER BOUND(ID))
316.14/291.50	The transformation into a RNTS is sound, since:
316.14/291.50	
316.14/291.50	(a) The obligation is a constructor system where every type has a constant constructor,
316.14/291.50	
316.14/291.50	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
316.14/291.50	none
316.14/291.50	
316.14/291.50	(c) The following functions are completely defined:
316.14/291.50	
316.14/291.50	   times_2
316.14/291.50	   plus_2
316.14/291.50	
316.14/291.50	Due to the following rules being added:
316.14/291.50	none
316.14/291.50	
316.14/291.50	And the following fresh constants: none
316.14/291.50	
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(6)
316.14/291.50	Obligation:
316.14/291.50	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
316.14/291.50	
316.14/291.50	Runtime Complexity Weighted TRS with Types.
316.14/291.50	The TRS R consists of the following rules:
316.14/291.50	
316.14/291.50	   times(x, 0) -> 0 [1]
316.14/291.50	   times(x, s(y)) -> plus(times(x, y), x) [1]
316.14/291.50	   plus(x, 0) -> x [1]
316.14/291.50	   plus(0, x) -> x [1]
316.14/291.50	   plus(x, s(y)) -> s(plus(x, y)) [1]
316.14/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
316.14/291.50	
316.14/291.50	The TRS has the following type information:
316.14/291.50	times :: 0:s -> 0:s -> 0:s
316.14/291.50	0 :: 0:s
316.14/291.50	s :: 0:s -> 0:s
316.14/291.50	plus :: 0:s -> 0:s -> 0:s
316.14/291.50	
316.14/291.50	Rewrite Strategy: INNERMOST
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
316.14/291.50	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
316.14/291.50	----------------------------------------
316.14/291.50	
316.14/291.50	(8)
316.14/291.50	Obligation:
316.14/291.50	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
316.14/291.50	
316.14/291.50	Runtime Complexity Weighted TRS with Types.
316.14/291.50	The TRS R consists of the following rules:
316.14/291.50	
316.14/291.50	   times(x, 0) -> 0 [1]
316.14/291.50	   times(x, s(0)) -> plus(0, x) [2]
316.14/291.50	   times(x, s(s(y'))) -> plus(plus(times(x, y'), x), x) [2]
316.14/291.51	   plus(x, 0) -> x [1]
316.14/291.51	   plus(0, x) -> x [1]
316.14/291.51	   plus(x, s(y)) -> s(plus(x, y)) [1]
316.14/291.51	   plus(s(x), y) -> s(plus(x, y)) [1]
316.14/291.51	
316.14/291.51	The TRS has the following type information:
316.14/291.51	times :: 0:s -> 0:s -> 0:s
316.14/291.51	0 :: 0:s
316.14/291.51	s :: 0:s -> 0:s
316.14/291.51	plus :: 0:s -> 0:s -> 0:s
316.14/291.51	
316.14/291.51	Rewrite Strategy: INNERMOST
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
316.14/291.51	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
316.14/291.51	The constant constructors are abstracted as follows:
316.14/291.51	
316.14/291.51	   0 => 0
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(10)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(x, y'), x), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
316.14/291.51	
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
316.14/291.51	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(12)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
316.14/291.51	Found the following analysis order by SCC decomposition:
316.14/291.51	
316.14/291.51	   { plus }
316.14/291.51	   { times }
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(14)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {plus}, {times}
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(15) ResultPropagationProof (UPPER BOUND(ID))
316.14/291.51	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(16)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {plus}, {times}
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(17) IntTrsBoundProof (UPPER BOUND(ID))
316.14/291.51	
316.14/291.51	Computed SIZE bound using CoFloCo for: plus
316.14/291.51	after applying outer abstraction to obtain an ITS,
316.14/291.51	resulting in: O(n^1) with polynomial bound: z + z'
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(18)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {plus}, {times}
316.14/291.51	Previous analysis results are:
316.14/291.51	  plus: runtime: ?, size: O(n^1) [z + z']
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(19) IntTrsBoundProof (UPPER BOUND(ID))
316.14/291.51	
316.14/291.51	Computed RUNTIME bound using CoFloCo for: plus
316.14/291.51	after applying outer abstraction to obtain an ITS,
316.14/291.51	resulting in: O(n^1) with polynomial bound: 1 + z + z'
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(20)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {times}
316.14/291.51	Previous analysis results are:
316.14/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(21) ResultPropagationProof (UPPER BOUND(ID))
316.14/291.51	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(22)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {times}
316.14/291.51	Previous analysis results are:
316.14/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(23) IntTrsBoundProof (UPPER BOUND(ID))
316.14/291.51	
316.14/291.51	Computed SIZE bound using KoAT for: times
316.14/291.51	after applying outer abstraction to obtain an ITS,
316.14/291.51	resulting in: O(n^2) with polynomial bound: z + 2*z*z'
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(24)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: {times}
316.14/291.51	Previous analysis results are:
316.14/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
316.14/291.51	  times: runtime: ?, size: O(n^2) [z + 2*z*z']
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(25) IntTrsBoundProof (UPPER BOUND(ID))
316.14/291.51	
316.14/291.51	Computed RUNTIME bound using KoAT for: times
316.14/291.51	after applying outer abstraction to obtain an ITS,
316.14/291.51	resulting in: O(n^3) with polynomial bound: 4 + z + 4*z*z'^2 + 4*z'
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(26)
316.14/291.51	Obligation:
316.14/291.51	Complexity RNTS consisting of the following rules:
316.14/291.51	
316.14/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
316.14/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
316.14/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
316.14/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
316.14/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
316.14/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
316.14/291.51	
316.14/291.51	Function symbols to be analyzed: 
316.14/291.51	Previous analysis results are:
316.14/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
316.14/291.51	  times: runtime: O(n^3) [4 + z + 4*z*z'^2 + 4*z'], size: O(n^2) [z + 2*z*z']
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(27) FinalProof (FINISHED)
316.14/291.51	Computed overall runtime complexity
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(28)
316.14/291.51	BOUNDS(1, n^3)
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(29) RenamingProof (BOTH BOUNDS(ID, ID))
316.14/291.51	Renamed function symbols to avoid clashes with predefined symbol.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(30)
316.14/291.51	Obligation:
316.14/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
316.14/291.51	
316.14/291.51	
316.14/291.51	The TRS R consists of the following rules:
316.14/291.51	
316.14/291.51	   times(x, 0') -> 0'
316.14/291.51	   times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	   plus(x, 0') -> x
316.14/291.51	   plus(0', x) -> x
316.14/291.51	   plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	   plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	S is empty.
316.14/291.51	Rewrite Strategy: INNERMOST
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(31) TypeInferenceProof (BOTH BOUNDS(ID, ID))
316.14/291.51	Infered types.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(32)
316.14/291.51	Obligation:
316.14/291.51	Innermost TRS:
316.14/291.51	Rules:
316.14/291.51	times(x, 0') -> 0'
316.14/291.51	times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	plus(x, 0') -> x
316.14/291.51	plus(0', x) -> x
316.14/291.51	plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	Types:
316.14/291.51	times :: 0':s -> 0':s -> 0':s
316.14/291.51	0' :: 0':s
316.14/291.51	s :: 0':s -> 0':s
316.14/291.51	plus :: 0':s -> 0':s -> 0':s
316.14/291.51	hole_0':s1_0 :: 0':s
316.14/291.51	gen_0':s2_0 :: Nat -> 0':s
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(33) OrderProof (LOWER BOUND(ID))
316.14/291.51	Heuristically decided to analyse the following defined symbols:
316.14/291.51	times, plus
316.14/291.51	
316.14/291.51	They will be analysed ascendingly in the following order:
316.14/291.51	plus < times
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(34)
316.14/291.51	Obligation:
316.14/291.51	Innermost TRS:
316.14/291.51	Rules:
316.14/291.51	times(x, 0') -> 0'
316.14/291.51	times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	plus(x, 0') -> x
316.14/291.51	plus(0', x) -> x
316.14/291.51	plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	Types:
316.14/291.51	times :: 0':s -> 0':s -> 0':s
316.14/291.51	0' :: 0':s
316.14/291.51	s :: 0':s -> 0':s
316.14/291.51	plus :: 0':s -> 0':s -> 0':s
316.14/291.51	hole_0':s1_0 :: 0':s
316.14/291.51	gen_0':s2_0 :: Nat -> 0':s
316.14/291.51	
316.14/291.51	
316.14/291.51	Generator Equations:
316.14/291.51	gen_0':s2_0(0) <=> 0'
316.14/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
316.14/291.51	
316.14/291.51	
316.14/291.51	The following defined symbols remain to be analysed:
316.14/291.51	plus, times
316.14/291.51	
316.14/291.51	They will be analysed ascendingly in the following order:
316.14/291.51	plus < times
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(35) RewriteLemmaProof (LOWER BOUND(ID))
316.14/291.51	Proved the following rewrite lemma:
316.14/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
316.14/291.51	
316.14/291.51	Induction Base:
316.14/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
316.14/291.51	gen_0':s2_0(a)
316.14/291.51	
316.14/291.51	Induction Step:
316.14/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
316.14/291.51	s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
316.14/291.51	s(gen_0':s2_0(+(a, c5_0)))
316.14/291.51	
316.14/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(36)
316.14/291.51	Complex Obligation (BEST)
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(37)
316.14/291.51	Obligation:
316.14/291.51	Proved the lower bound n^1 for the following obligation:
316.14/291.51	
316.14/291.51	Innermost TRS:
316.14/291.51	Rules:
316.14/291.51	times(x, 0') -> 0'
316.14/291.51	times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	plus(x, 0') -> x
316.14/291.51	plus(0', x) -> x
316.14/291.51	plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	Types:
316.14/291.51	times :: 0':s -> 0':s -> 0':s
316.14/291.51	0' :: 0':s
316.14/291.51	s :: 0':s -> 0':s
316.14/291.51	plus :: 0':s -> 0':s -> 0':s
316.14/291.51	hole_0':s1_0 :: 0':s
316.14/291.51	gen_0':s2_0 :: Nat -> 0':s
316.14/291.51	
316.14/291.51	
316.14/291.51	Generator Equations:
316.14/291.51	gen_0':s2_0(0) <=> 0'
316.14/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
316.14/291.51	
316.14/291.51	
316.14/291.51	The following defined symbols remain to be analysed:
316.14/291.51	plus, times
316.14/291.51	
316.14/291.51	They will be analysed ascendingly in the following order:
316.14/291.51	plus < times
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(38) LowerBoundPropagationProof (FINISHED)
316.14/291.51	Propagated lower bound.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(39)
316.14/291.51	BOUNDS(n^1, INF)
316.14/291.51	
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(40)
316.14/291.51	Obligation:
316.14/291.51	Innermost TRS:
316.14/291.51	Rules:
316.14/291.51	times(x, 0') -> 0'
316.14/291.51	times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	plus(x, 0') -> x
316.14/291.51	plus(0', x) -> x
316.14/291.51	plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	Types:
316.14/291.51	times :: 0':s -> 0':s -> 0':s
316.14/291.51	0' :: 0':s
316.14/291.51	s :: 0':s -> 0':s
316.14/291.51	plus :: 0':s -> 0':s -> 0':s
316.14/291.51	hole_0':s1_0 :: 0':s
316.14/291.51	gen_0':s2_0 :: Nat -> 0':s
316.14/291.51	
316.14/291.51	
316.14/291.51	Lemmas:
316.14/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
316.14/291.51	
316.14/291.51	
316.14/291.51	Generator Equations:
316.14/291.51	gen_0':s2_0(0) <=> 0'
316.14/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
316.14/291.51	
316.14/291.51	
316.14/291.51	The following defined symbols remain to be analysed:
316.14/291.51	times
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(41) RewriteLemmaProof (LOWER BOUND(ID))
316.14/291.51	Proved the following rewrite lemma:
316.14/291.51	times(gen_0':s2_0(a), gen_0':s2_0(n592_0)) -> gen_0':s2_0(*(n592_0, a)), rt in Omega(1 + a*n592_0 + n592_0)
316.14/291.51	
316.14/291.51	Induction Base:
316.14/291.51	times(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
316.14/291.51	0'
316.14/291.51	
316.14/291.51	Induction Step:
316.14/291.51	times(gen_0':s2_0(a), gen_0':s2_0(+(n592_0, 1))) ->_R^Omega(1)
316.14/291.51	plus(times(gen_0':s2_0(a), gen_0':s2_0(n592_0)), gen_0':s2_0(a)) ->_IH
316.14/291.51	plus(gen_0':s2_0(*(c593_0, a)), gen_0':s2_0(a)) ->_L^Omega(1 + a)
316.14/291.51	gen_0':s2_0(+(a, *(n592_0, a)))
316.14/291.51	
316.14/291.51	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(42)
316.14/291.51	Obligation:
316.14/291.51	Proved the lower bound n^2 for the following obligation:
316.14/291.51	
316.14/291.51	Innermost TRS:
316.14/291.51	Rules:
316.14/291.51	times(x, 0') -> 0'
316.14/291.51	times(x, s(y)) -> plus(times(x, y), x)
316.14/291.51	plus(x, 0') -> x
316.14/291.51	plus(0', x) -> x
316.14/291.51	plus(x, s(y)) -> s(plus(x, y))
316.14/291.51	plus(s(x), y) -> s(plus(x, y))
316.14/291.51	
316.14/291.51	Types:
316.14/291.51	times :: 0':s -> 0':s -> 0':s
316.14/291.51	0' :: 0':s
316.14/291.51	s :: 0':s -> 0':s
316.14/291.51	plus :: 0':s -> 0':s -> 0':s
316.14/291.51	hole_0':s1_0 :: 0':s
316.14/291.51	gen_0':s2_0 :: Nat -> 0':s
316.14/291.51	
316.14/291.51	
316.14/291.51	Lemmas:
316.14/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
316.14/291.51	
316.14/291.51	
316.14/291.51	Generator Equations:
316.14/291.51	gen_0':s2_0(0) <=> 0'
316.14/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
316.14/291.51	
316.14/291.51	
316.14/291.51	The following defined symbols remain to be analysed:
316.14/291.51	times
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(43) LowerBoundPropagationProof (FINISHED)
316.14/291.51	Propagated lower bound.
316.14/291.51	----------------------------------------
316.14/291.51	
316.14/291.51	(44)
316.14/291.51	BOUNDS(n^2, INF)
316.25/291.54	EOF