6.23/3.97	WORST_CASE(Omega(n^1), ?)
6.23/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
6.23/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.23/3.98	
6.23/3.98	
6.23/3.98	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
6.23/3.98	
6.23/3.98	(0) CpxIntTrs
6.23/3.98	(1) Loat Proof [FINISHED, 323 ms]
6.23/3.98	(2) BOUNDS(n^1, INF)
6.23/3.98	
6.23/3.98	
6.23/3.98	----------------------------------------
6.23/3.98	
6.23/3.98	(0)
6.23/3.98	Obligation:
6.23/3.98	Complexity Int TRS consisting of the following rules:
6.23/3.98	eval(A, B, C) -> Com_1(eval(A - 1, B, C - 1)) :|: A >= 0 && B * B * B >= C
6.23/3.98	eval(A, B, C) -> Com_1(eval(A, B + C, C - 1)) :|: A >= 0 && B * B * B >= C
6.23/3.98	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
6.23/3.98	
6.23/3.98	The start-symbols are:[start_3]
6.23/3.98	
6.23/3.98	
6.23/3.98	----------------------------------------
6.23/3.98	
6.23/3.98	(1) Loat Proof (FINISHED)
6.23/3.98	
6.23/3.98	
6.23/3.98	### Pre-processing the ITS problem ###
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Initial linear ITS problem
6.23/3.98	
6.23/3.98	   Start location: start
6.23/3.98	
6.23/3.98	      0: eval -> eval : A'=-1+A, C'=-1+C, [ A>=0 && B^3>=C ], cost: 1
6.23/3.98	
6.23/3.98	      1: eval -> eval : B'=C+B, C'=-1+C, [ A>=0 && B^3>=C ], cost: 1
6.23/3.98	
6.23/3.98	      2: start -> eval : [], cost: 1
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	### Simplification by acceleration and chaining ###
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Accelerating simple loops of location 0.
6.23/3.98	
6.23/3.98	   Accelerating the following rules:
6.23/3.98	
6.23/3.98	      0: eval -> eval : A'=-1+A, C'=-1+C, [ A>=0 && B^3>=C ], cost: 1
6.23/3.98	
6.23/3.98	      1: eval -> eval : B'=C+B, C'=-1+C, [ A>=0 && B^3>=C ], cost: 1
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	   Accelerated rule 0 with metering function 1+A, yielding the new rule 3.
6.23/3.98	
6.23/3.98	   Found no metering function for rule 1 (rule is too complicated).
6.23/3.98	
6.23/3.98	   Removing the simple loops: 0.
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Accelerated all simple loops using metering functions (where possible):
6.23/3.98	
6.23/3.98	   Start location: start
6.23/3.98	
6.23/3.98	      1: eval -> eval : B'=C+B, C'=-1+C, [ A>=0 && B^3>=C ], cost: 1
6.23/3.98	
6.23/3.98	      3: eval -> eval : A'=-1, C'=-1+C-A, [ A>=0 && B^3>=C ], cost: 1+A
6.23/3.98	
6.23/3.98	      2: start -> eval : [], cost: 1
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Chained accelerated rules (with incoming rules):
6.23/3.98	
6.23/3.98	   Start location: start
6.23/3.98	
6.23/3.98	      2: start -> eval : [], cost: 1
6.23/3.98	
6.23/3.98	      4: start -> eval : B'=C+B, C'=-1+C, [ A>=0 && B^3>=C ], cost: 2
6.23/3.98	
6.23/3.98	      5: start -> eval : A'=-1, C'=-1+C-A, [ A>=0 && B^3>=C ], cost: 2+A
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Removed unreachable locations (and leaf rules with constant cost):
6.23/3.98	
6.23/3.98	   Start location: start
6.23/3.98	
6.23/3.98	      5: start -> eval : A'=-1, C'=-1+C-A, [ A>=0 && B^3>=C ], cost: 2+A
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	### Computing asymptotic complexity ###
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Fully simplified ITS problem
6.23/3.98	
6.23/3.98	   Start location: start
6.23/3.98	
6.23/3.98	      5: start -> eval : A'=-1, C'=-1+C-A, [ A>=0 && B^3>=C ], cost: 2+A
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Computing asymptotic complexity for rule 5
6.23/3.98	
6.23/3.98	   Solved the limit problem by the following transformations:
6.23/3.98	
6.23/3.98	      Created initial limit problem:
6.23/3.98	
6.23/3.98	      1-C+B^3 (+/+!), 2+A (+), 1+A (+/+!) [not solved]
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	      removing all constraints (solved by SMT)
6.23/3.98	
6.23/3.98	      resulting limit problem: [solved]
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	      applying transformation rule (C) using substitution {C==0,A==n,B==n}
6.23/3.98	
6.23/3.98	      resulting limit problem:
6.23/3.98	
6.23/3.98	      [solved]
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	   Solution:
6.23/3.98	
6.23/3.98	      C / 0
6.23/3.98	
6.23/3.98	      A / n
6.23/3.98	
6.23/3.98	      B / n
6.23/3.98	
6.23/3.98	   Resulting cost 2+n has complexity: Poly(n^1)
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Found new complexity Poly(n^1).
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	Obtained the following overall complexity (w.r.t. the length of the input n):
6.23/3.98	
6.23/3.98	   Complexity:  Poly(n^1)
6.23/3.98	
6.23/3.98	   Cpx degree:  1
6.23/3.98	
6.23/3.98	   Solved cost: 2+n
6.23/3.98	
6.23/3.98	   Rule cost:   2+A
6.23/3.98	
6.23/3.98	   Rule guard:  [ A>=0 && B^3>=C ]
6.23/3.98	
6.23/3.98	
6.23/3.98	
6.23/3.98	WORST_CASE(Omega(n^1),?)
6.23/3.98	
6.23/3.98	
6.23/3.98	----------------------------------------
6.23/3.98	
6.23/3.98	(2)
6.23/3.98	BOUNDS(n^1, INF)
6.23/4.00	EOF