5.22/3.07	WORST_CASE(Omega(n^1), ?)
5.22/3.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.22/3.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.22/3.07	
5.22/3.07	
5.22/3.07	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
5.22/3.07	
5.22/3.07	(0) CpxIntTrs
5.22/3.07	(1) Loat Proof [FINISHED, 316 ms]
5.22/3.07	(2) BOUNDS(n^1, INF)
5.22/3.07	
5.22/3.07	
5.22/3.07	----------------------------------------
5.22/3.07	
5.22/3.07	(0)
5.22/3.07	Obligation:
5.22/3.07	Complexity Int TRS consisting of the following rules:
5.22/3.07	eval(A, B) -> Com_1(eval(A - 1, A)) :|: A >= 1 && B >= 1
5.22/3.07	eval(A, B) -> Com_1(eval(B - 2, A + 1)) :|: A >= 1 && B >= 1
5.22/3.07	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
5.22/3.07	
5.22/3.07	The start-symbols are:[start_2]
5.22/3.07	
5.22/3.07	
5.22/3.07	----------------------------------------
5.22/3.07	
5.22/3.07	(1) Loat Proof (FINISHED)
5.22/3.07	
5.22/3.07	
5.22/3.07	### Pre-processing the ITS problem ###
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Initial linear ITS problem
5.22/3.07	
5.22/3.07	   Start location: start
5.22/3.07	
5.22/3.07	      0: eval -> eval : A'=-1+A, B'=A, [ A>=1 && B>=1 ], cost: 1
5.22/3.07	
5.22/3.07	      1: eval -> eval : A'=-2+B, B'=1+A, [ A>=1 && B>=1 ], cost: 1
5.22/3.07	
5.22/3.07	      2: start -> eval : [], cost: 1
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	### Simplification by acceleration and chaining ###
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Accelerating simple loops of location 0.
5.22/3.07	
5.22/3.07	   Accelerating the following rules:
5.22/3.07	
5.22/3.07	      0: eval -> eval : A'=-1+A, B'=A, [ A>=1 && B>=1 ], cost: 1
5.22/3.07	
5.22/3.07	      1: eval -> eval : A'=-2+B, B'=1+A, [ A>=1 && B>=1 ], cost: 1
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	   Accelerated rule 0 with metering function A, yielding the new rule 3.
5.22/3.07	
5.22/3.07	   Found no metering function for rule 1.
5.22/3.07	
5.22/3.07	   Removing the simple loops: 0.
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Accelerated all simple loops using metering functions (where possible):
5.22/3.07	
5.22/3.07	   Start location: start
5.22/3.07	
5.22/3.07	      1: eval -> eval : A'=-2+B, B'=1+A, [ A>=1 && B>=1 ], cost: 1
5.22/3.07	
5.22/3.07	      3: eval -> eval : A'=0, B'=1, [ A>=1 && B>=1 ], cost: A
5.22/3.07	
5.22/3.07	      2: start -> eval : [], cost: 1
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Chained accelerated rules (with incoming rules):
5.22/3.07	
5.22/3.07	   Start location: start
5.22/3.07	
5.22/3.07	      2: start -> eval : [], cost: 1
5.22/3.07	
5.22/3.07	      4: start -> eval : A'=-2+B, B'=1+A, [ A>=1 && B>=1 ], cost: 2
5.22/3.07	
5.22/3.07	      5: start -> eval : A'=0, B'=1, [ A>=1 && B>=1 ], cost: 1+A
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Removed unreachable locations (and leaf rules with constant cost):
5.22/3.07	
5.22/3.07	   Start location: start
5.22/3.07	
5.22/3.07	      5: start -> eval : A'=0, B'=1, [ A>=1 && B>=1 ], cost: 1+A
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	### Computing asymptotic complexity ###
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Fully simplified ITS problem
5.22/3.07	
5.22/3.07	   Start location: start
5.22/3.07	
5.22/3.07	      5: start -> eval : A'=0, B'=1, [ A>=1 && B>=1 ], cost: 1+A
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Computing asymptotic complexity for rule 5
5.22/3.07	
5.22/3.07	   Solved the limit problem by the following transformations:
5.22/3.07	
5.22/3.07	      Created initial limit problem:
5.22/3.07	
5.22/3.07	      A (+/+!), B (+/+!), 1+A (+) [not solved]
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	      removing all constraints (solved by SMT)
5.22/3.07	
5.22/3.07	      resulting limit problem: [solved]
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	      applying transformation rule (C) using substitution {A==n,B==1}
5.22/3.07	
5.22/3.07	      resulting limit problem:
5.22/3.07	
5.22/3.07	      [solved]
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	   Solution:
5.22/3.07	
5.22/3.07	      A / n
5.22/3.07	
5.22/3.07	      B / 1
5.22/3.07	
5.22/3.07	   Resulting cost 1+n has complexity: Poly(n^1)
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Found new complexity Poly(n^1).
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	Obtained the following overall complexity (w.r.t. the length of the input n):
5.22/3.07	
5.22/3.07	   Complexity:  Poly(n^1)
5.22/3.07	
5.22/3.07	   Cpx degree:  1
5.22/3.07	
5.22/3.07	   Solved cost: 1+n
5.22/3.07	
5.22/3.07	   Rule cost:   1+A
5.22/3.07	
5.22/3.07	   Rule guard:  [ A>=1 && B>=1 ]
5.22/3.07	
5.22/3.07	
5.22/3.07	
5.22/3.07	WORST_CASE(Omega(n^1),?)
5.22/3.07	
5.22/3.07	
5.22/3.07	----------------------------------------
5.22/3.07	
5.22/3.07	(2)
5.22/3.07	BOUNDS(n^1, INF)
5.34/3.11	EOF