6.92/4.40	WORST_CASE(Omega(n^1), ?)
6.92/4.41	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
6.92/4.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.92/4.41	
6.92/4.41	
6.92/4.41	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
6.92/4.41	
6.92/4.41	(0) CpxIntTrs
6.92/4.41	(1) Loat Proof [FINISHED, 417 ms]
6.92/4.41	(2) BOUNDS(n^1, INF)
6.92/4.41	
6.92/4.41	
6.92/4.41	----------------------------------------
6.92/4.41	
6.92/4.41	(0)
6.92/4.41	Obligation:
6.92/4.41	Complexity Int TRS consisting of the following rules:
6.92/4.41	f0(A, B, C) -> Com_1(f2(A, B, C)) :|: A >= 0 && B >= C
6.92/4.41	f2(A, B, C) -> Com_1(f2(A - 1, B, C - 1)) :|: A >= 1 && B + 1 >= C
6.92/4.41	f2(A, B, C) -> Com_1(f2(A, B + C - 1, C - 1)) :|: A >= 0 && B >= 0
6.92/4.41	
6.92/4.41	The start-symbols are:[f0_3]
6.92/4.41	
6.92/4.41	
6.92/4.41	----------------------------------------
6.92/4.41	
6.92/4.41	(1) Loat Proof (FINISHED)
6.92/4.41	
6.92/4.41	
6.92/4.41	### Pre-processing the ITS problem ###
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Initial linear ITS problem
6.92/4.41	
6.92/4.41	   Start location: f0
6.92/4.41	
6.92/4.41	      0: f0 -> f2 : [ A>=0 && B>=C ], cost: 1
6.92/4.41	
6.92/4.41	      1: f2 -> f2 : A'=-1+A, C'=-1+C, [ A>=1 && 1+B>=C ], cost: 1
6.92/4.41	
6.92/4.41	      2: f2 -> f2 : B'=-1+C+B, C'=-1+C, [ A>=0 && B>=0 ], cost: 1
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	### Simplification by acceleration and chaining ###
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Accelerating simple loops of location 1.
6.92/4.41	
6.92/4.41	   Accelerating the following rules:
6.92/4.41	
6.92/4.41	      1: f2 -> f2 : A'=-1+A, C'=-1+C, [ A>=1 && 1+B>=C ], cost: 1
6.92/4.41	
6.92/4.41	      2: f2 -> f2 : B'=-1+C+B, C'=-1+C, [ A>=0 && B>=0 ], cost: 1
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	   Accelerated rule 1 with metering function A, yielding the new rule 3.
6.92/4.41	
6.92/4.41	   Found no metering function for rule 2.
6.92/4.41	
6.92/4.41	   Removing the simple loops: 1.
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Accelerated all simple loops using metering functions (where possible):
6.92/4.41	
6.92/4.41	   Start location: f0
6.92/4.41	
6.92/4.41	      0: f0 -> f2 : [ A>=0 && B>=C ], cost: 1
6.92/4.41	
6.92/4.41	      2: f2 -> f2 : B'=-1+C+B, C'=-1+C, [ A>=0 && B>=0 ], cost: 1
6.92/4.41	
6.92/4.41	      3: f2 -> f2 : A'=0, C'=C-A, [ A>=1 && 1+B>=C ], cost: A
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Chained accelerated rules (with incoming rules):
6.92/4.41	
6.92/4.41	   Start location: f0
6.92/4.41	
6.92/4.41	      0: f0 -> f2 : [ A>=0 && B>=C ], cost: 1
6.92/4.41	
6.92/4.41	      4: f0 -> f2 : B'=-1+C+B, C'=-1+C, [ A>=0 && B>=C && B>=0 ], cost: 2
6.92/4.41	
6.92/4.41	      5: f0 -> f2 : A'=0, C'=C-A, [ B>=C && A>=1 ], cost: 1+A
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Removed unreachable locations (and leaf rules with constant cost):
6.92/4.41	
6.92/4.41	   Start location: f0
6.92/4.41	
6.92/4.41	      5: f0 -> f2 : A'=0, C'=C-A, [ B>=C && A>=1 ], cost: 1+A
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	### Computing asymptotic complexity ###
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Fully simplified ITS problem
6.92/4.41	
6.92/4.41	   Start location: f0
6.92/4.41	
6.92/4.41	      5: f0 -> f2 : A'=0, C'=C-A, [ B>=C && A>=1 ], cost: 1+A
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Computing asymptotic complexity for rule 5
6.92/4.41	
6.92/4.41	   Solved the limit problem by the following transformations:
6.92/4.41	
6.92/4.41	      Created initial limit problem:
6.92/4.41	
6.92/4.41	      1-C+B (+/+!), A (+/+!), 1+A (+) [not solved]
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	      removing all constraints (solved by SMT)
6.92/4.41	
6.92/4.41	      resulting limit problem: [solved]
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	      applying transformation rule (C) using substitution {C==0,A==n,B==0}
6.92/4.41	
6.92/4.41	      resulting limit problem:
6.92/4.41	
6.92/4.41	      [solved]
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	   Solution:
6.92/4.41	
6.92/4.41	      C / 0
6.92/4.41	
6.92/4.41	      A / n
6.92/4.41	
6.92/4.41	      B / 0
6.92/4.41	
6.92/4.41	   Resulting cost 1+n has complexity: Poly(n^1)
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Found new complexity Poly(n^1).
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	Obtained the following overall complexity (w.r.t. the length of the input n):
6.92/4.41	
6.92/4.41	   Complexity:  Poly(n^1)
6.92/4.41	
6.92/4.41	   Cpx degree:  1
6.92/4.41	
6.92/4.41	   Solved cost: 1+n
6.92/4.41	
6.92/4.41	   Rule cost:   1+A
6.92/4.41	
6.92/4.41	   Rule guard:  [ B>=C && A>=1 ]
6.92/4.41	
6.92/4.41	
6.92/4.41	
6.92/4.41	WORST_CASE(Omega(n^1),?)
6.92/4.41	
6.92/4.41	
6.92/4.41	----------------------------------------
6.92/4.41	
6.92/4.41	(2)
6.92/4.41	BOUNDS(n^1, INF)
7.07/4.44	EOF