19.78/6.27	WORST_CASE(Omega(n^1), O(n^1))
20.03/6.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.03/6.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.03/6.29	
20.03/6.29	
20.03/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.03/6.29	
20.03/6.29	(0) CpxTRS
20.03/6.29	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	(2) CpxTRS
20.03/6.29	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	    (4) CpxWeightedTrs
20.03/6.29	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	    (6) CpxTypedWeightedTrs
20.03/6.29	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
20.03/6.29	    (8) CpxTypedWeightedCompleteTrs
20.03/6.29	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	    (10) CpxTypedWeightedCompleteTrs
20.03/6.29	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.03/6.29	    (12) CpxRNTS
20.03/6.29	    (13) InliningProof [UPPER BOUND(ID), 84 ms]
20.03/6.29	    (14) CpxRNTS
20.03/6.29	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	    (16) CpxRNTS
20.03/6.29	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.03/6.29	    (18) CpxRNTS
20.03/6.29	    (19) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
20.03/6.29	    (20) CpxRNTS
20.03/6.29	    (21) IntTrsBoundProof [UPPER BOUND(ID), 210 ms]
20.03/6.29	    (22) CpxRNTS
20.03/6.29	    (23) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
20.03/6.29	    (24) CpxRNTS
20.03/6.29	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.03/6.29	    (26) CpxRNTS
20.03/6.29	    (27) IntTrsBoundProof [UPPER BOUND(ID), 757 ms]
20.03/6.29	    (28) CpxRNTS
20.03/6.29	    (29) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
20.03/6.29	    (30) CpxRNTS
20.03/6.29	    (31) FinalProof [FINISHED, 0 ms]
20.03/6.29	    (32) BOUNDS(1, n^1)
20.03/6.29	(33) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
20.03/6.29	(34) TRS for Loop Detection
20.03/6.29	    (35) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
20.03/6.29	    (36) BEST
20.03/6.29	        (37) proven lower bound
20.03/6.29	            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
20.03/6.29	            (39) BOUNDS(n^1, INF)
20.03/6.29	        (40) TRS for Loop Detection
20.03/6.29	
20.03/6.29	
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(0)
20.03/6.29	Obligation:
20.03/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.03/6.29	
20.03/6.29	
20.03/6.29	The TRS R consists of the following rules:
20.03/6.29	
20.03/6.29	   first(0, X) -> nil
20.03/6.29	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
20.03/6.29	   from(X) -> cons(X, n__from(s(X)))
20.03/6.29	   first(X1, X2) -> n__first(X1, X2)
20.03/6.29	   from(X) -> n__from(X)
20.03/6.29	   activate(n__first(X1, X2)) -> first(X1, X2)
20.03/6.29	   activate(n__from(X)) -> from(X)
20.03/6.29	   activate(X) -> X
20.03/6.29	
20.03/6.29	S is empty.
20.03/6.29	Rewrite Strategy: FULL
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.03/6.29	Converted rc-obligation to irc-obligation.
20.03/6.29	
20.03/6.29	As the TRS does not nest defined symbols, we have rc = irc.
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(2)
20.03/6.29	Obligation:
20.03/6.29	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.03/6.29	
20.03/6.29	
20.03/6.29	The TRS R consists of the following rules:
20.03/6.29	
20.03/6.29	   first(0, X) -> nil
20.03/6.29	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
20.03/6.29	   from(X) -> cons(X, n__from(s(X)))
20.03/6.29	   first(X1, X2) -> n__first(X1, X2)
20.03/6.29	   from(X) -> n__from(X)
20.03/6.29	   activate(n__first(X1, X2)) -> first(X1, X2)
20.03/6.29	   activate(n__from(X)) -> from(X)
20.03/6.29	   activate(X) -> X
20.03/6.29	
20.03/6.29	S is empty.
20.03/6.29	Rewrite Strategy: INNERMOST
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.03/6.29	Transformed relative TRS to weighted TRS
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(4)
20.03/6.29	Obligation:
20.03/6.29	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.03/6.29	
20.03/6.29	
20.03/6.29	The TRS R consists of the following rules:
20.03/6.29	
20.03/6.29	   first(0, X) -> nil [1]
20.03/6.29	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
20.03/6.29	   from(X) -> cons(X, n__from(s(X))) [1]
20.03/6.29	   first(X1, X2) -> n__first(X1, X2) [1]
20.03/6.29	   from(X) -> n__from(X) [1]
20.03/6.29	   activate(n__first(X1, X2)) -> first(X1, X2) [1]
20.03/6.29	   activate(n__from(X)) -> from(X) [1]
20.03/6.29	   activate(X) -> X [1]
20.03/6.29	
20.03/6.29	Rewrite Strategy: INNERMOST
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.03/6.29	Infered types.
20.03/6.29	----------------------------------------
20.03/6.29	
20.03/6.29	(6)
20.03/6.29	Obligation:
20.03/6.29	Runtime Complexity Weighted TRS with Types.
20.03/6.29	The TRS R consists of the following rules:
20.03/6.29	
20.03/6.29	   first(0, X) -> nil [1]
20.03/6.29	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
20.03/6.29	   from(X) -> cons(X, n__from(s(X))) [1]
20.03/6.29	   first(X1, X2) -> n__first(X1, X2) [1]
20.03/6.30	   from(X) -> n__from(X) [1]
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2) [1]
20.03/6.30	   activate(n__from(X)) -> from(X) [1]
20.03/6.30	   activate(X) -> X [1]
20.03/6.30	
20.03/6.30	The TRS has the following type information:
20.03/6.30	first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	0 :: 0:s
20.03/6.30	nil :: nil:cons:n__first:n__from
20.03/6.30	s :: 0:s -> 0:s
20.03/6.30	cons :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	n__first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	activate :: nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	n__from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	
20.03/6.30	Rewrite Strategy: INNERMOST
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(7) CompletionProof (UPPER BOUND(ID))
20.03/6.30	The transformation into a RNTS is sound, since:
20.03/6.30	
20.03/6.30	(a) The obligation is a constructor system where every type has a constant constructor,
20.03/6.30	
20.03/6.30	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.03/6.30	
20.03/6.30	   first_2
20.03/6.30	   from_1
20.03/6.30	   activate_1
20.03/6.30	
20.03/6.30	(c) The following functions are completely defined:
20.03/6.30	none
20.03/6.30	
20.03/6.30	Due to the following rules being added:
20.03/6.30	none
20.03/6.30	
20.03/6.30	And the following fresh constants: none
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(8)
20.03/6.30	Obligation:
20.03/6.30	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.03/6.30	
20.03/6.30	Runtime Complexity Weighted TRS with Types.
20.03/6.30	The TRS R consists of the following rules:
20.03/6.30	
20.03/6.30	   first(0, X) -> nil [1]
20.03/6.30	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
20.03/6.30	   from(X) -> cons(X, n__from(s(X))) [1]
20.03/6.30	   first(X1, X2) -> n__first(X1, X2) [1]
20.03/6.30	   from(X) -> n__from(X) [1]
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2) [1]
20.03/6.30	   activate(n__from(X)) -> from(X) [1]
20.03/6.30	   activate(X) -> X [1]
20.03/6.30	
20.03/6.30	The TRS has the following type information:
20.03/6.30	first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	0 :: 0:s
20.03/6.30	nil :: nil:cons:n__first:n__from
20.03/6.30	s :: 0:s -> 0:s
20.03/6.30	cons :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	n__first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	activate :: nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	n__from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	
20.03/6.30	Rewrite Strategy: INNERMOST
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
20.03/6.30	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(10)
20.03/6.30	Obligation:
20.03/6.30	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.03/6.30	
20.03/6.30	Runtime Complexity Weighted TRS with Types.
20.03/6.30	The TRS R consists of the following rules:
20.03/6.30	
20.03/6.30	   first(0, X) -> nil [1]
20.03/6.30	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
20.03/6.30	   from(X) -> cons(X, n__from(s(X))) [1]
20.03/6.30	   first(X1, X2) -> n__first(X1, X2) [1]
20.03/6.30	   from(X) -> n__from(X) [1]
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2) [1]
20.03/6.30	   activate(n__from(X)) -> from(X) [1]
20.03/6.30	   activate(X) -> X [1]
20.03/6.30	
20.03/6.30	The TRS has the following type information:
20.03/6.30	first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	0 :: 0:s
20.03/6.30	nil :: nil:cons:n__first:n__from
20.03/6.30	s :: 0:s -> 0:s
20.03/6.30	cons :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	n__first :: 0:s -> nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	activate :: nil:cons:n__first:n__from -> nil:cons:n__first:n__from
20.03/6.30	from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	n__from :: 0:s -> nil:cons:n__first:n__from
20.03/6.30	
20.03/6.30	Rewrite Strategy: INNERMOST
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.03/6.30	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.03/6.30	The constant constructors are abstracted as follows:
20.03/6.30	
20.03/6.30	   0 => 0
20.03/6.30	   nil => 0
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(12)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> X :|: X >= 0, z = X
20.03/6.30	   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
20.03/6.30	   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.03/6.30	   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
20.03/6.30	
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(13) InliningProof (UPPER BOUND(ID))
20.03/6.30	Inlined the following terminating rules on right-hand sides where appropriate:
20.03/6.30	
20.03/6.30	   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.03/6.30	   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(14)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> X :|: X >= 0, z = X
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
20.03/6.30	   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.03/6.30	   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
20.03/6.30	
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
20.03/6.30	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(16)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.03/6.30	Found the following analysis order by SCC decomposition:
20.03/6.30	
20.03/6.30	   { from }
20.03/6.30	   { first, activate }
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(18)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {from}, {first,activate}
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(19) ResultPropagationProof (UPPER BOUND(ID))
20.03/6.30	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(20)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {from}, {first,activate}
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.03/6.30	
20.03/6.30	Computed SIZE bound using CoFloCo for: from
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(n^1) with polynomial bound: 3 + 2*z
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(22)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {from}, {first,activate}
20.03/6.30	Previous analysis results are:
20.03/6.30	  from: runtime: ?, size: O(n^1) [3 + 2*z]
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(23) IntTrsBoundProof (UPPER BOUND(ID))
20.03/6.30	
20.03/6.30	Computed RUNTIME bound using CoFloCo for: from
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(1) with polynomial bound: 1
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(24)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {first,activate}
20.03/6.30	Previous analysis results are:
20.03/6.30	  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(25) ResultPropagationProof (UPPER BOUND(ID))
20.03/6.30	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(26)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {first,activate}
20.03/6.30	Previous analysis results are:
20.03/6.30	  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(27) IntTrsBoundProof (UPPER BOUND(ID))
20.03/6.30	
20.03/6.30	Computed SIZE bound using KoAT for: first
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(n^1) with polynomial bound: 1 + z + 2*z'
20.03/6.30	
20.03/6.30	Computed SIZE bound using KoAT for: activate
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(n^1) with polynomial bound: 1 + 2*z
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(28)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: {first,activate}
20.03/6.30	Previous analysis results are:
20.03/6.30	  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]
20.03/6.30	  first: runtime: ?, size: O(n^1) [1 + z + 2*z']
20.03/6.30	  activate: runtime: ?, size: O(n^1) [1 + 2*z]
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(29) IntTrsBoundProof (UPPER BOUND(ID))
20.03/6.30	
20.03/6.30	Computed RUNTIME bound using KoAT for: first
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(n^1) with polynomial bound: 5 + 2*z'
20.03/6.30	
20.03/6.30	Computed RUNTIME bound using KoAT for: activate
20.03/6.30	after applying outer abstraction to obtain an ITS,
20.03/6.30	resulting in: O(n^1) with polynomial bound: 9 + 2*z
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(30)
20.03/6.30	Obligation:
20.03/6.30	Complexity RNTS consisting of the following rules:
20.03/6.30	
20.03/6.30	   activate(z) -{ 1 }-> z :|: z >= 0
20.03/6.30	   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
20.03/6.30	   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.03/6.30	   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
20.03/6.30	   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z :|: z >= 0
20.03/6.30	   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
20.03/6.30	
20.03/6.30	Function symbols to be analyzed: 
20.03/6.30	Previous analysis results are:
20.03/6.30	  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]
20.03/6.30	  first: runtime: O(n^1) [5 + 2*z'], size: O(n^1) [1 + z + 2*z']
20.03/6.30	  activate: runtime: O(n^1) [9 + 2*z], size: O(n^1) [1 + 2*z]
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(31) FinalProof (FINISHED)
20.03/6.30	Computed overall runtime complexity
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(32)
20.03/6.30	BOUNDS(1, n^1)
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(33) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
20.03/6.30	Transformed a relative TRS into a decreasing-loop problem.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(34)
20.03/6.30	Obligation:
20.03/6.30	Analyzing the following TRS for decreasing loops:
20.03/6.30	
20.03/6.30	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.03/6.30	
20.03/6.30	
20.03/6.30	The TRS R consists of the following rules:
20.03/6.30	
20.03/6.30	   first(0, X) -> nil
20.03/6.30	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
20.03/6.30	   from(X) -> cons(X, n__from(s(X)))
20.03/6.30	   first(X1, X2) -> n__first(X1, X2)
20.03/6.30	   from(X) -> n__from(X)
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2)
20.03/6.30	   activate(n__from(X)) -> from(X)
20.03/6.30	   activate(X) -> X
20.03/6.30	
20.03/6.30	S is empty.
20.03/6.30	Rewrite Strategy: FULL
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(35) DecreasingLoopProof (LOWER BOUND(ID))
20.03/6.30	The following loop(s) give(s) rise to the lower bound Omega(n^1):
20.03/6.30	
20.03/6.30	The rewrite sequence
20.03/6.30	
20.03/6.30	activate(n__first(s(X1_0), cons(Y2_0, Z3_0))) ->^+ cons(Y2_0, n__first(X1_0, activate(Z3_0)))
20.03/6.30	
20.03/6.30	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
20.03/6.30	
20.03/6.30	The pumping substitution is [Z3_0 / n__first(s(X1_0), cons(Y2_0, Z3_0))].
20.03/6.30	
20.03/6.30	The result substitution is [ ].
20.03/6.30	
20.03/6.30	
20.03/6.30	
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(36)
20.03/6.30	Complex Obligation (BEST)
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(37)
20.03/6.30	Obligation:
20.03/6.30	Proved the lower bound n^1 for the following obligation:
20.03/6.30	
20.03/6.30	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.03/6.30	
20.03/6.30	
20.03/6.30	The TRS R consists of the following rules:
20.03/6.30	
20.03/6.30	   first(0, X) -> nil
20.03/6.30	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
20.03/6.30	   from(X) -> cons(X, n__from(s(X)))
20.03/6.30	   first(X1, X2) -> n__first(X1, X2)
20.03/6.30	   from(X) -> n__from(X)
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2)
20.03/6.30	   activate(n__from(X)) -> from(X)
20.03/6.30	   activate(X) -> X
20.03/6.30	
20.03/6.30	S is empty.
20.03/6.30	Rewrite Strategy: FULL
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(38) LowerBoundPropagationProof (FINISHED)
20.03/6.30	Propagated lower bound.
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(39)
20.03/6.30	BOUNDS(n^1, INF)
20.03/6.30	
20.03/6.30	----------------------------------------
20.03/6.30	
20.03/6.30	(40)
20.03/6.30	Obligation:
20.03/6.30	Analyzing the following TRS for decreasing loops:
20.03/6.30	
20.03/6.30	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.03/6.30	
20.03/6.30	
20.03/6.30	The TRS R consists of the following rules:
20.03/6.30	
20.03/6.30	   first(0, X) -> nil
20.03/6.30	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
20.03/6.30	   from(X) -> cons(X, n__from(s(X)))
20.03/6.30	   first(X1, X2) -> n__first(X1, X2)
20.03/6.30	   from(X) -> n__from(X)
20.03/6.30	   activate(n__first(X1, X2)) -> first(X1, X2)
20.03/6.30	   activate(n__from(X)) -> from(X)
20.03/6.30	   activate(X) -> X
20.03/6.30	
20.03/6.30	S is empty.
20.03/6.30	Rewrite Strategy: FULL
20.17/9.97	EOF