323.64/291.49	WORST_CASE(Omega(n^2), O(n^3))
323.73/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
323.73/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
323.73/291.51	
323.73/291.51	
323.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^3).
323.73/291.51	
323.73/291.51	(0) CpxTRS
323.73/291.51	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	(2) CpxTRS
323.73/291.51	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (4) CpxWeightedTrs
323.73/291.51	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (6) CpxTypedWeightedTrs
323.73/291.51	    (7) CompletionProof [UPPER BOUND(ID), 3 ms]
323.73/291.51	    (8) CpxTypedWeightedCompleteTrs
323.73/291.51	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (10) CpxTypedWeightedCompleteTrs
323.73/291.51	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
323.73/291.51	    (12) CpxRNTS
323.73/291.51	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (14) CpxRNTS
323.73/291.51	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (16) CpxRNTS
323.73/291.51	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
323.73/291.51	    (18) CpxRNTS
323.73/291.51	    (19) IntTrsBoundProof [UPPER BOUND(ID), 611 ms]
323.73/291.51	    (20) CpxRNTS
323.73/291.51	    (21) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
323.73/291.51	    (22) CpxRNTS
323.73/291.51	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
323.73/291.51	    (24) CpxRNTS
323.73/291.51	    (25) IntTrsBoundProof [UPPER BOUND(ID), 606 ms]
323.73/291.51	    (26) CpxRNTS
323.73/291.51	    (27) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
323.73/291.51	    (28) CpxRNTS
323.73/291.51	    (29) FinalProof [FINISHED, 0 ms]
323.73/291.51	    (30) BOUNDS(1, n^3)
323.73/291.51	(31) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	(32) CpxTRS
323.73/291.51	    (33) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
323.73/291.51	    (34) typed CpxTrs
323.73/291.51	    (35) OrderProof [LOWER BOUND(ID), 0 ms]
323.73/291.51	    (36) typed CpxTrs
323.73/291.51	    (37) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
323.73/291.51	    (38) BEST
323.73/291.51	        (39) proven lower bound
323.73/291.51	            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
323.73/291.51	            (41) BOUNDS(n^1, INF)
323.73/291.51	        (42) typed CpxTrs
323.73/291.51	            (43) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
323.73/291.51	            (44) proven lower bound
323.73/291.51	            (45) LowerBoundPropagationProof [FINISHED, 0 ms]
323.73/291.51	            (46) BOUNDS(n^2, INF)
323.73/291.51	
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(0)
323.73/291.51	Obligation:
323.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^3).
323.73/291.51	
323.73/291.51	
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	   plus(x, 0) -> x
323.73/291.51	   plus(0, x) -> x
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	   plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	S is empty.
323.73/291.51	Rewrite Strategy: FULL
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Converted rc-obligation to irc-obligation.
323.73/291.51	
323.73/291.51	The duplicating contexts are:
323.73/291.51	times([], s(y))
323.73/291.51	
323.73/291.51	
323.73/291.51	The defined contexts are:
323.73/291.51	plus([], x1)
323.73/291.51	
323.73/291.51	
323.73/291.51	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(2)
323.73/291.51	Obligation:
323.73/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).
323.73/291.51	
323.73/291.51	
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	   plus(x, 0) -> x
323.73/291.51	   plus(0, x) -> x
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	   plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	S is empty.
323.73/291.51	Rewrite Strategy: INNERMOST
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Transformed relative TRS to weighted TRS
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(4)
323.73/291.51	Obligation:
323.73/291.51	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
323.73/291.51	
323.73/291.51	
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0 [1]
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x) [1]
323.73/291.51	   plus(x, 0) -> x [1]
323.73/291.51	   plus(0, x) -> x [1]
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y)) [1]
323.73/291.51	   plus(s(x), y) -> s(plus(x, y)) [1]
323.73/291.51	
323.73/291.51	Rewrite Strategy: INNERMOST
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Infered types.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(6)
323.73/291.51	Obligation:
323.73/291.51	Runtime Complexity Weighted TRS with Types.
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0 [1]
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x) [1]
323.73/291.51	   plus(x, 0) -> x [1]
323.73/291.51	   plus(0, x) -> x [1]
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y)) [1]
323.73/291.51	   plus(s(x), y) -> s(plus(x, y)) [1]
323.73/291.51	
323.73/291.51	The TRS has the following type information:
323.73/291.51	times :: 0:s -> 0:s -> 0:s
323.73/291.51	0 :: 0:s
323.73/291.51	s :: 0:s -> 0:s
323.73/291.51	plus :: 0:s -> 0:s -> 0:s
323.73/291.51	
323.73/291.51	Rewrite Strategy: INNERMOST
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(7) CompletionProof (UPPER BOUND(ID))
323.73/291.51	The transformation into a RNTS is sound, since:
323.73/291.51	
323.73/291.51	(a) The obligation is a constructor system where every type has a constant constructor,
323.73/291.51	
323.73/291.51	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
323.73/291.51	none
323.73/291.51	
323.73/291.51	(c) The following functions are completely defined:
323.73/291.51	
323.73/291.51	   times_2
323.73/291.51	   plus_2
323.73/291.51	
323.73/291.51	Due to the following rules being added:
323.73/291.51	none
323.73/291.51	
323.73/291.51	And the following fresh constants: none
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(8)
323.73/291.51	Obligation:
323.73/291.51	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
323.73/291.51	
323.73/291.51	Runtime Complexity Weighted TRS with Types.
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0 [1]
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x) [1]
323.73/291.51	   plus(x, 0) -> x [1]
323.73/291.51	   plus(0, x) -> x [1]
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y)) [1]
323.73/291.51	   plus(s(x), y) -> s(plus(x, y)) [1]
323.73/291.51	
323.73/291.51	The TRS has the following type information:
323.73/291.51	times :: 0:s -> 0:s -> 0:s
323.73/291.51	0 :: 0:s
323.73/291.51	s :: 0:s -> 0:s
323.73/291.51	plus :: 0:s -> 0:s -> 0:s
323.73/291.51	
323.73/291.51	Rewrite Strategy: INNERMOST
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(10)
323.73/291.51	Obligation:
323.73/291.51	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
323.73/291.51	
323.73/291.51	Runtime Complexity Weighted TRS with Types.
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0) -> 0 [1]
323.73/291.51	   times(x, s(0)) -> plus(0, x) [2]
323.73/291.51	   times(x, s(s(y'))) -> plus(plus(times(x, y'), x), x) [2]
323.73/291.51	   plus(x, 0) -> x [1]
323.73/291.51	   plus(0, x) -> x [1]
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y)) [1]
323.73/291.51	   plus(s(x), y) -> s(plus(x, y)) [1]
323.73/291.51	
323.73/291.51	The TRS has the following type information:
323.73/291.51	times :: 0:s -> 0:s -> 0:s
323.73/291.51	0 :: 0:s
323.73/291.51	s :: 0:s -> 0:s
323.73/291.51	plus :: 0:s -> 0:s -> 0:s
323.73/291.51	
323.73/291.51	Rewrite Strategy: INNERMOST
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
323.73/291.51	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
323.73/291.51	The constant constructors are abstracted as follows:
323.73/291.51	
323.73/291.51	   0 => 0
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(12)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(x, y'), x), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
323.73/291.51	
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(14)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Found the following analysis order by SCC decomposition:
323.73/291.51	
323.73/291.51	   { plus }
323.73/291.51	   { times }
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(16)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {plus}, {times}
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(17) ResultPropagationProof (UPPER BOUND(ID))
323.73/291.51	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(18)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {plus}, {times}
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(19) IntTrsBoundProof (UPPER BOUND(ID))
323.73/291.51	
323.73/291.51	Computed SIZE bound using CoFloCo for: plus
323.73/291.51	after applying outer abstraction to obtain an ITS,
323.73/291.51	resulting in: O(n^1) with polynomial bound: z + z'
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(20)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {plus}, {times}
323.73/291.51	Previous analysis results are:
323.73/291.51	  plus: runtime: ?, size: O(n^1) [z + z']
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(21) IntTrsBoundProof (UPPER BOUND(ID))
323.73/291.51	
323.73/291.51	Computed RUNTIME bound using CoFloCo for: plus
323.73/291.51	after applying outer abstraction to obtain an ITS,
323.73/291.51	resulting in: O(n^1) with polynomial bound: 1 + z + z'
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(22)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {times}
323.73/291.51	Previous analysis results are:
323.73/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(23) ResultPropagationProof (UPPER BOUND(ID))
323.73/291.51	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(24)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {times}
323.73/291.51	Previous analysis results are:
323.73/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(25) IntTrsBoundProof (UPPER BOUND(ID))
323.73/291.51	
323.73/291.51	Computed SIZE bound using KoAT for: times
323.73/291.51	after applying outer abstraction to obtain an ITS,
323.73/291.51	resulting in: O(n^2) with polynomial bound: z + 2*z*z'
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(26)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: {times}
323.73/291.51	Previous analysis results are:
323.73/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
323.73/291.51	  times: runtime: ?, size: O(n^2) [z + 2*z*z']
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(27) IntTrsBoundProof (UPPER BOUND(ID))
323.73/291.51	
323.73/291.51	Computed RUNTIME bound using KoAT for: times
323.73/291.51	after applying outer abstraction to obtain an ITS,
323.73/291.51	resulting in: O(n^3) with polynomial bound: 4 + z + 4*z*z'^2 + 4*z'
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(28)
323.73/291.51	Obligation:
323.73/291.51	Complexity RNTS consisting of the following rules:
323.73/291.51	
323.73/291.51	   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
323.73/291.51	   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
323.73/291.51	   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
323.73/291.51	   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
323.73/291.51	   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
323.73/291.51	   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
323.73/291.51	
323.73/291.51	Function symbols to be analyzed: 
323.73/291.51	Previous analysis results are:
323.73/291.51	  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
323.73/291.51	  times: runtime: O(n^3) [4 + z + 4*z*z'^2 + 4*z'], size: O(n^2) [z + 2*z*z']
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(29) FinalProof (FINISHED)
323.73/291.51	Computed overall runtime complexity
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(30)
323.73/291.51	BOUNDS(1, n^3)
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(31) RenamingProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Renamed function symbols to avoid clashes with predefined symbol.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(32)
323.73/291.51	Obligation:
323.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
323.73/291.51	
323.73/291.51	
323.73/291.51	The TRS R consists of the following rules:
323.73/291.51	
323.73/291.51	   times(x, 0') -> 0'
323.73/291.51	   times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	   plus(x, 0') -> x
323.73/291.51	   plus(0', x) -> x
323.73/291.51	   plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	   plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	S is empty.
323.73/291.51	Rewrite Strategy: FULL
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(33) TypeInferenceProof (BOTH BOUNDS(ID, ID))
323.73/291.51	Infered types.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(34)
323.73/291.51	Obligation:
323.73/291.51	TRS:
323.73/291.51	Rules:
323.73/291.51	times(x, 0') -> 0'
323.73/291.51	times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	plus(x, 0') -> x
323.73/291.51	plus(0', x) -> x
323.73/291.51	plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	Types:
323.73/291.51	times :: 0':s -> 0':s -> 0':s
323.73/291.51	0' :: 0':s
323.73/291.51	s :: 0':s -> 0':s
323.73/291.51	plus :: 0':s -> 0':s -> 0':s
323.73/291.51	hole_0':s1_0 :: 0':s
323.73/291.51	gen_0':s2_0 :: Nat -> 0':s
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(35) OrderProof (LOWER BOUND(ID))
323.73/291.51	Heuristically decided to analyse the following defined symbols:
323.73/291.51	times, plus
323.73/291.51	
323.73/291.51	They will be analysed ascendingly in the following order:
323.73/291.51	plus < times
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(36)
323.73/291.51	Obligation:
323.73/291.51	TRS:
323.73/291.51	Rules:
323.73/291.51	times(x, 0') -> 0'
323.73/291.51	times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	plus(x, 0') -> x
323.73/291.51	plus(0', x) -> x
323.73/291.51	plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	Types:
323.73/291.51	times :: 0':s -> 0':s -> 0':s
323.73/291.51	0' :: 0':s
323.73/291.51	s :: 0':s -> 0':s
323.73/291.51	plus :: 0':s -> 0':s -> 0':s
323.73/291.51	hole_0':s1_0 :: 0':s
323.73/291.51	gen_0':s2_0 :: Nat -> 0':s
323.73/291.51	
323.73/291.51	
323.73/291.51	Generator Equations:
323.73/291.51	gen_0':s2_0(0) <=> 0'
323.73/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
323.73/291.51	
323.73/291.51	
323.73/291.51	The following defined symbols remain to be analysed:
323.73/291.51	plus, times
323.73/291.51	
323.73/291.51	They will be analysed ascendingly in the following order:
323.73/291.51	plus < times
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(37) RewriteLemmaProof (LOWER BOUND(ID))
323.73/291.51	Proved the following rewrite lemma:
323.73/291.51	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)
323.73/291.51	
323.73/291.51	Induction Base:
323.73/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
323.73/291.51	gen_0':s2_0(a)
323.73/291.51	
323.73/291.51	Induction Step:
323.73/291.51	plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
323.73/291.51	s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
323.73/291.51	s(gen_0':s2_0(+(a, c5_0)))
323.73/291.51	
323.73/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(38)
323.73/291.51	Complex Obligation (BEST)
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(39)
323.73/291.51	Obligation:
323.73/291.51	Proved the lower bound n^1 for the following obligation:
323.73/291.51	
323.73/291.51	TRS:
323.73/291.51	Rules:
323.73/291.51	times(x, 0') -> 0'
323.73/291.51	times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	plus(x, 0') -> x
323.73/291.51	plus(0', x) -> x
323.73/291.51	plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	Types:
323.73/291.51	times :: 0':s -> 0':s -> 0':s
323.73/291.51	0' :: 0':s
323.73/291.51	s :: 0':s -> 0':s
323.73/291.51	plus :: 0':s -> 0':s -> 0':s
323.73/291.51	hole_0':s1_0 :: 0':s
323.73/291.51	gen_0':s2_0 :: Nat -> 0':s
323.73/291.51	
323.73/291.51	
323.73/291.51	Generator Equations:
323.73/291.51	gen_0':s2_0(0) <=> 0'
323.73/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
323.73/291.51	
323.73/291.51	
323.73/291.51	The following defined symbols remain to be analysed:
323.73/291.51	plus, times
323.73/291.51	
323.73/291.51	They will be analysed ascendingly in the following order:
323.73/291.51	plus < times
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(40) LowerBoundPropagationProof (FINISHED)
323.73/291.51	Propagated lower bound.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(41)
323.73/291.51	BOUNDS(n^1, INF)
323.73/291.51	
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(42)
323.73/291.51	Obligation:
323.73/291.51	TRS:
323.73/291.51	Rules:
323.73/291.51	times(x, 0') -> 0'
323.73/291.51	times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	plus(x, 0') -> x
323.73/291.51	plus(0', x) -> x
323.73/291.51	plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	Types:
323.73/291.51	times :: 0':s -> 0':s -> 0':s
323.73/291.51	0' :: 0':s
323.73/291.51	s :: 0':s -> 0':s
323.73/291.51	plus :: 0':s -> 0':s -> 0':s
323.73/291.51	hole_0':s1_0 :: 0':s
323.73/291.51	gen_0':s2_0 :: Nat -> 0':s
323.73/291.51	
323.73/291.51	
323.73/291.51	Lemmas:
323.73/291.51	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)
323.73/291.51	
323.73/291.51	
323.73/291.51	Generator Equations:
323.73/291.51	gen_0':s2_0(0) <=> 0'
323.73/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
323.73/291.51	
323.73/291.51	
323.73/291.51	The following defined symbols remain to be analysed:
323.73/291.51	times
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(43) RewriteLemmaProof (LOWER BOUND(ID))
323.73/291.51	Proved the following rewrite lemma:
323.73/291.51	times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) -> gen_0':s2_0(*(n544_0, a)), rt in Omega(1 + a*n544_0 + n544_0)
323.73/291.51	
323.73/291.51	Induction Base:
323.73/291.51	times(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
323.73/291.51	0'
323.73/291.51	
323.73/291.51	Induction Step:
323.73/291.51	times(gen_0':s2_0(a), gen_0':s2_0(+(n544_0, 1))) ->_R^Omega(1)
323.73/291.51	plus(times(gen_0':s2_0(a), gen_0':s2_0(n544_0)), gen_0':s2_0(a)) ->_IH
323.73/291.51	plus(gen_0':s2_0(*(c545_0, a)), gen_0':s2_0(a)) ->_L^Omega(1 + a)
323.73/291.51	gen_0':s2_0(+(a, *(n544_0, a)))
323.73/291.51	
323.73/291.51	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(44)
323.73/291.51	Obligation:
323.73/291.51	Proved the lower bound n^2 for the following obligation:
323.73/291.51	
323.73/291.51	TRS:
323.73/291.51	Rules:
323.73/291.51	times(x, 0') -> 0'
323.73/291.51	times(x, s(y)) -> plus(times(x, y), x)
323.73/291.51	plus(x, 0') -> x
323.73/291.51	plus(0', x) -> x
323.73/291.51	plus(x, s(y)) -> s(plus(x, y))
323.73/291.51	plus(s(x), y) -> s(plus(x, y))
323.73/291.51	
323.73/291.51	Types:
323.73/291.51	times :: 0':s -> 0':s -> 0':s
323.73/291.51	0' :: 0':s
323.73/291.51	s :: 0':s -> 0':s
323.73/291.51	plus :: 0':s -> 0':s -> 0':s
323.73/291.51	hole_0':s1_0 :: 0':s
323.73/291.51	gen_0':s2_0 :: Nat -> 0':s
323.73/291.51	
323.73/291.51	
323.73/291.51	Lemmas:
323.73/291.51	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)
323.73/291.51	
323.73/291.51	
323.73/291.51	Generator Equations:
323.73/291.51	gen_0':s2_0(0) <=> 0'
323.73/291.51	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
323.73/291.51	
323.73/291.51	
323.73/291.51	The following defined symbols remain to be analysed:
323.73/291.51	times
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(45) LowerBoundPropagationProof (FINISHED)
323.73/291.51	Propagated lower bound.
323.73/291.51	----------------------------------------
323.73/291.51	
323.73/291.51	(46)
323.73/291.51	BOUNDS(n^2, INF)
323.73/291.55	EOF