5.19/2.62	WORST_CASE(Omega(n^1), ?)
5.23/2.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.23/2.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.23/2.63	
5.23/2.63	
5.23/2.63	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).
5.23/2.63	
5.23/2.63	(0) CpxIntTrs
5.23/2.63	(1) Loat Proof [FINISHED, 522 ms]
5.23/2.63	(2) BOUNDS(n^1, INF)
5.23/2.63	
5.23/2.63	
5.23/2.63	----------------------------------------
5.23/2.63	
5.23/2.63	(0)
5.23/2.63	Obligation:
5.23/2.63	Complexity Int TRS consisting of the following rules:
5.23/2.63	ack(A, B) -> Com_1(ack(A - 1, C)) :|: A >= 1 && B >= 1
5.23/2.63	ack(A, B) -> Com_1(ack(A, B - 1)) :|: A >= 1 && B >= 1
5.23/2.63	start(A, B) -> Com_1(ack(A, B)) :|: TRUE
5.23/2.63	
5.23/2.63	The start-symbols are:[start_2]
5.23/2.63	
5.23/2.63	
5.23/2.63	----------------------------------------
5.23/2.63	
5.23/2.63	(1) Loat Proof (FINISHED)
5.23/2.63	
5.23/2.63	
5.23/2.63	### Pre-processing the ITS problem ###
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Initial linear ITS problem
5.23/2.63	
5.23/2.63	   Start location: start
5.23/2.63	
5.23/2.63	      0: ack -> ack : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.23/2.63	
5.23/2.63	      1: ack -> ack : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
5.23/2.63	
5.23/2.63	      2: start -> ack : [], cost: 1
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	### Simplification by acceleration and chaining ###
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Accelerating simple loops of location 0.
5.23/2.63	
5.23/2.63	   Accelerating the following rules:
5.23/2.63	
5.23/2.63	      0: ack -> ack : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.23/2.63	
5.23/2.63	      1: ack -> ack : B'=-1+B, [ A>=1 && B>=1 ], cost: 1
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	   Accelerated rule 0 with metering function A (after strengthening guard), yielding the new rule 3.
5.23/2.63	
5.23/2.63	   Accelerated rule 1 with metering function B, yielding the new rule 4.
5.23/2.63	
5.23/2.63	   Removing the simple loops: 1.
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Accelerated all simple loops using metering functions (where possible):
5.23/2.63	
5.23/2.63	   Start location: start
5.23/2.63	
5.23/2.63	      0: ack -> ack : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 1
5.23/2.63	
5.23/2.63	      3: ack -> ack : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: A
5.23/2.63	
5.23/2.63	      4: ack -> ack : B'=0, [ A>=1 && B>=1 ], cost: B
5.23/2.63	
5.23/2.63	      2: start -> ack : [], cost: 1
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Chained accelerated rules (with incoming rules):
5.23/2.63	
5.23/2.63	   Start location: start
5.23/2.63	
5.23/2.63	      2: start -> ack : [], cost: 1
5.23/2.63	
5.23/2.63	      5: start -> ack : A'=-1+A, B'=free, [ A>=1 && B>=1 ], cost: 2
5.23/2.63	
5.23/2.63	      6: start -> ack : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.23/2.63	
5.23/2.63	      7: start -> ack : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Removed unreachable locations (and leaf rules with constant cost):
5.23/2.63	
5.23/2.63	   Start location: start
5.23/2.63	
5.23/2.63	      6: start -> ack : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.23/2.63	
5.23/2.63	      7: start -> ack : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	### Computing asymptotic complexity ###
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Fully simplified ITS problem
5.23/2.63	
5.23/2.63	   Start location: start
5.23/2.63	
5.23/2.63	      6: start -> ack : A'=0, B'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1+A
5.23/2.63	
5.23/2.63	      7: start -> ack : B'=0, [ A>=1 && B>=1 ], cost: 1+B
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Computing asymptotic complexity for rule 6
5.23/2.63	
5.23/2.63	   Solved the limit problem by the following transformations:
5.23/2.63	
5.23/2.63	      Created initial limit problem:
5.23/2.63	
5.23/2.63	      free (+/+!), A (+/+!), B (+/+!), 1+A (+) [not solved]
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	      removing all constraints (solved by SMT)
5.23/2.63	
5.23/2.63	      resulting limit problem: [solved]
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	      applying transformation rule (C) using substitution {free==n,A==n,B==n}
5.23/2.63	
5.23/2.63	      resulting limit problem:
5.23/2.63	
5.23/2.63	      [solved]
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	   Solution:
5.23/2.63	
5.23/2.63	      free / n
5.23/2.63	
5.23/2.63	      A / n
5.23/2.63	
5.23/2.63	      B / n
5.23/2.63	
5.23/2.63	   Resulting cost 1+n has complexity: Poly(n^1)
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Found new complexity Poly(n^1).
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	Obtained the following overall complexity (w.r.t. the length of the input n):
5.23/2.63	
5.23/2.63	   Complexity:  Poly(n^1)
5.23/2.63	
5.23/2.63	   Cpx degree:  1
5.23/2.63	
5.23/2.63	   Solved cost: 1+n
5.23/2.63	
5.23/2.63	   Rule cost:   1+A
5.23/2.63	
5.23/2.63	   Rule guard:  [ A>=1 && B>=1 && free>=1 ]
5.23/2.63	
5.23/2.63	
5.23/2.63	
5.23/2.63	WORST_CASE(Omega(n^1),?)
5.23/2.63	
5.23/2.63	
5.23/2.63	----------------------------------------
5.23/2.63	
5.23/2.63	(2)
5.23/2.63	BOUNDS(n^1, INF)
5.23/2.66	EOF