6.50/4.06	WORST_CASE(Omega(n^1), ?)
6.61/4.07	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
6.61/4.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.61/4.07	
6.61/4.07	
6.61/4.07	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
6.61/4.07	
6.61/4.07	(0) CpxIntTrs
6.61/4.07	(1) Loat Proof [FINISHED, 265 ms]
6.61/4.07	(2) BOUNDS(n^1, INF)
6.61/4.07	
6.61/4.07	
6.61/4.07	----------------------------------------
6.61/4.07	
6.61/4.07	(0)
6.61/4.07	Obligation:
6.61/4.07	Complexity Int TRS consisting of the following rules:
6.61/4.07	f0(A, B, C, D) -> Com_1(f1(A, B, C, D)) :|: TRUE
6.61/4.07	f1(A, B, C, D) -> Com_1(f1(A + B, B + C, C + D, D - 1)) :|: A >= 1
6.61/4.07	f1(A, B, C, D) -> Com_1(f1(A - 1, B, C, D)) :|: A >= 1
6.61/4.07	
6.61/4.07	The start-symbols are:[f0_4]
6.61/4.07	
6.61/4.07	
6.61/4.07	----------------------------------------
6.61/4.07	
6.61/4.07	(1) Loat Proof (FINISHED)
6.61/4.07	
6.61/4.07	
6.61/4.07	### Pre-processing the ITS problem ###
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Initial linear ITS problem
6.61/4.07	
6.61/4.07	   Start location: f0
6.61/4.07	
6.61/4.07	      0: f0 -> f1 : [], cost: 1
6.61/4.07	
6.61/4.07	      1: f1 -> f1 : A'=A+B, B'=C+B, C'=C+D, D'=-1+D, [ A>=1 ], cost: 1
6.61/4.07	
6.61/4.07	      2: f1 -> f1 : A'=-1+A, [ A>=1 ], cost: 1
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	### Simplification by acceleration and chaining ###
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Accelerating simple loops of location 1.
6.61/4.07	
6.61/4.07	   Accelerating the following rules:
6.61/4.07	
6.61/4.07	      1: f1 -> f1 : A'=A+B, B'=C+B, C'=C+D, D'=-1+D, [ A>=1 ], cost: 1
6.61/4.07	
6.61/4.07	      2: f1 -> f1 : A'=-1+A, [ A>=1 ], cost: 1
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	   Found no metering function for rule 1.
6.61/4.07	
6.61/4.07	   Accelerated rule 2 with metering function A, yielding the new rule 3.
6.61/4.07	
6.61/4.07	   Removing the simple loops: 2.
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Accelerated all simple loops using metering functions (where possible):
6.61/4.07	
6.61/4.07	   Start location: f0
6.61/4.07	
6.61/4.07	      0: f0 -> f1 : [], cost: 1
6.61/4.07	
6.61/4.07	      1: f1 -> f1 : A'=A+B, B'=C+B, C'=C+D, D'=-1+D, [ A>=1 ], cost: 1
6.61/4.07	
6.61/4.07	      3: f1 -> f1 : A'=0, [ A>=1 ], cost: A
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Chained accelerated rules (with incoming rules):
6.61/4.07	
6.61/4.07	   Start location: f0
6.61/4.07	
6.61/4.07	      0: f0 -> f1 : [], cost: 1
6.61/4.07	
6.61/4.07	      4: f0 -> f1 : A'=A+B, B'=C+B, C'=C+D, D'=-1+D, [ A>=1 ], cost: 2
6.61/4.07	
6.61/4.07	      5: f0 -> f1 : A'=0, [ A>=1 ], cost: 1+A
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Removed unreachable locations (and leaf rules with constant cost):
6.61/4.07	
6.61/4.07	   Start location: f0
6.61/4.07	
6.61/4.07	      5: f0 -> f1 : A'=0, [ A>=1 ], cost: 1+A
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	### Computing asymptotic complexity ###
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Fully simplified ITS problem
6.61/4.07	
6.61/4.07	   Start location: f0
6.61/4.07	
6.61/4.07	      5: f0 -> f1 : A'=0, [ A>=1 ], cost: 1+A
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Computing asymptotic complexity for rule 5
6.61/4.07	
6.61/4.07	   Solved the limit problem by the following transformations:
6.61/4.07	
6.61/4.07	      Created initial limit problem:
6.61/4.07	
6.61/4.07	      A (+/+!), 1+A (+) [not solved]
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	      removing all constraints (solved by SMT)
6.61/4.07	
6.61/4.07	      resulting limit problem: [solved]
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	      applying transformation rule (C) using substitution {A==n}
6.61/4.07	
6.61/4.07	      resulting limit problem:
6.61/4.07	
6.61/4.07	      [solved]
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	   Solution:
6.61/4.07	
6.61/4.07	      A / n
6.61/4.07	
6.61/4.07	   Resulting cost 1+n has complexity: Poly(n^1)
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Found new complexity Poly(n^1).
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	Obtained the following overall complexity (w.r.t. the length of the input n):
6.61/4.07	
6.61/4.07	   Complexity:  Poly(n^1)
6.61/4.07	
6.61/4.07	   Cpx degree:  1
6.61/4.07	
6.61/4.07	   Solved cost: 1+n
6.61/4.07	
6.61/4.07	   Rule cost:   1+A
6.61/4.07	
6.61/4.07	   Rule guard:  [ A>=1 ]
6.61/4.07	
6.61/4.07	
6.61/4.07	
6.61/4.07	WORST_CASE(Omega(n^1),?)
6.61/4.07	
6.61/4.07	
6.61/4.07	----------------------------------------
6.61/4.07	
6.61/4.07	(2)
6.61/4.07	BOUNDS(n^1, INF)
6.63/4.11	EOF