5.21/2.59	WORST_CASE(Omega(n^1), ?)
5.21/2.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.21/2.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.21/2.60	
5.21/2.60	
5.21/2.60	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
5.21/2.60	
5.21/2.60	(0) CpxIntTrs
5.21/2.60	(1) Loat Proof [FINISHED, 526 ms]
5.21/2.60	(2) BOUNDS(n^1, INF)
5.21/2.60	
5.21/2.60	
5.21/2.60	----------------------------------------
5.21/2.60	
5.21/2.60	(0)
5.21/2.60	Obligation:
5.21/2.60	Complexity Int TRS consisting of the following rules:
5.21/2.60	eval(A, B) -> Com_1(eval(A - 1, C)) :|: A >= 1 && B >= 1
5.21/2.60	eval(A, B) -> Com_1(eval(A, B - 1)) :|: A >= 1 && B >= 1
5.21/2.60	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
5.21/2.60	
5.21/2.60	The start-symbols are:[start_2]
5.21/2.60	
5.21/2.60	
5.21/2.60	----------------------------------------
5.21/2.60	
5.21/2.60	(1) Loat Proof (FINISHED)
5.21/2.60	
5.21/2.60	
5.21/2.60	### Pre-processing the ITS problem ###
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Initial linear ITS problem
5.21/2.60	
5.21/2.60	   Start location: start
5.21/2.60	
5.21/2.60	      0: eval -> eval : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.21/2.60	
5.21/2.60	      1: eval -> eval : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
5.21/2.60	
5.21/2.60	      2: start -> eval : [], cost: 1
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	### Simplification by acceleration and chaining ###
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Accelerating simple loops of location 0.
5.21/2.60	
5.21/2.60	   Accelerating the following rules:
5.21/2.60	
5.21/2.60	      0: eval -> eval : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.21/2.60	
5.21/2.60	      1: eval -> eval : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	   Accelerated rule 0 with metering function A (after strengthening guard), yielding the new rule 3.
5.21/2.60	
5.21/2.60	   Accelerated rule 1 with metering function B, yielding the new rule 4.
5.21/2.60	
5.21/2.60	   Removing the simple loops: 1.
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Accelerated all simple loops using metering functions (where possible):
5.21/2.60	
5.21/2.60	   Start location: start
5.21/2.60	
5.21/2.60	      0: eval -> eval : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.21/2.60	
5.21/2.60	      3: eval -> eval : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: A
5.21/2.60	
5.21/2.60	      4: eval -> eval : B'=0, [ A>=1 && B>=1 ], cost: B
5.21/2.60	
5.21/2.60	      2: start -> eval : [], cost: 1
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Chained accelerated rules (with incoming rules):
5.21/2.60	
5.21/2.60	   Start location: start
5.21/2.60	
5.21/2.60	      2: start -> eval : [], cost: 1
5.21/2.60	
5.21/2.60	      5: start -> eval : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 2
5.21/2.60	
5.21/2.60	      6: start -> eval : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.21/2.60	
5.21/2.60	      7: start -> eval : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Removed unreachable locations (and leaf rules with constant cost):
5.21/2.60	
5.21/2.60	   Start location: start
5.21/2.60	
5.21/2.60	      6: start -> eval : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.21/2.60	
5.21/2.60	      7: start -> eval : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	### Computing asymptotic complexity ###
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Fully simplified ITS problem
5.21/2.60	
5.21/2.60	   Start location: start
5.21/2.60	
5.21/2.60	      6: start -> eval : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.21/2.60	
5.21/2.60	      7: start -> eval : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Computing asymptotic complexity for rule 6
5.21/2.60	
5.21/2.60	   Solved the limit problem by the following transformations:
5.21/2.60	
5.21/2.60	      Created initial limit problem:
5.21/2.60	
5.21/2.60	      free (+/+!), A (+/+!), B (+/+!), 1+A (+) [not solved]
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	      removing all constraints (solved by SMT)
5.21/2.60	
5.21/2.60	      resulting limit problem: [solved]
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	      applying transformation rule (C) using substitution {free==n,A==n,B==n}
5.21/2.60	
5.21/2.60	      resulting limit problem:
5.21/2.60	
5.21/2.60	      [solved]
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	   Solution:
5.21/2.60	
5.21/2.60	      free / n
5.21/2.60	
5.21/2.60	      A / n
5.21/2.60	
5.21/2.60	      B / n
5.21/2.60	
5.21/2.60	   Resulting cost 1+n has complexity: Poly(n^1)
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Found new complexity Poly(n^1).
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	Obtained the following overall complexity (w.r.t. the length of the input n):
5.21/2.60	
5.21/2.60	   Complexity:  Poly(n^1)
5.21/2.60	
5.21/2.60	   Cpx degree:  1
5.21/2.60	
5.21/2.60	   Solved cost: 1+n
5.21/2.60	
5.21/2.60	   Rule cost:   1+A
5.21/2.60	
5.21/2.60	   Rule guard:  [ A>=1 && B>=1 && free>=1 ]
5.21/2.60	
5.21/2.60	
5.21/2.60	
5.21/2.60	WORST_CASE(Omega(n^1),?)
5.21/2.60	
5.21/2.60	
5.21/2.60	----------------------------------------
5.21/2.60	
5.21/2.60	(2)
5.21/2.60	BOUNDS(n^1, INF)
5.24/2.63	EOF