3.61/2.13	WORST_CASE(Omega(n^1), O(n^1))
3.61/2.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.61/2.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.61/2.13	
3.61/2.13	
3.61/2.13	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(102, 102 + Arg_0)).
3.61/2.13	
3.61/2.13	(0) CpxIntTrs
3.61/2.13	(1) Koat2 Proof [FINISHED, 54 ms]
3.61/2.13	(2) BOUNDS(1, max(102, 102 + Arg_0))
3.61/2.13	(3) Loat Proof [FINISHED, 163 ms]
3.61/2.13	(4) BOUNDS(n^1, INF)
3.61/2.13	
3.61/2.13	
3.61/2.13	----------------------------------------
3.61/2.13	
3.61/2.13	(0)
3.61/2.13	Obligation:
3.61/2.13	Complexity Int TRS consisting of the following rules:
3.61/2.13	start(A) -> Com_1(a(A)) :|: A >= 1
3.61/2.13	start(A) -> Com_1(a(100)) :|: A >= 100 && A <= 100
3.61/2.13	a(A) -> Com_1(a(A - 1)) :|: A >= 1
3.61/2.13	
3.61/2.13	The start-symbols are:[start_1]
3.61/2.13	
3.61/2.13	
3.61/2.13	----------------------------------------
3.61/2.13	
3.61/2.13	(1) Koat2 Proof (FINISHED)
3.61/2.13	YES( ?, max([102, 102+Arg_0]) {O(n)})
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Initial Complexity Problem:
3.61/2.13	
3.61/2.13	  Start:  start  
3.61/2.13	
3.61/2.13	  Program_Vars:  Arg_0  
3.61/2.13	
3.61/2.13	  Temp_Vars:    
3.61/2.13	
3.61/2.13	  Locations:  a, start  
3.61/2.13	
3.61/2.13	  Transitions:
3.61/2.13	
3.61/2.13	  a(Arg_0) -> a(Arg_0-1):|:1 <= Arg_0
3.61/2.13	
3.61/2.13	  start(Arg_0) -> a(Arg_0):|:1 <= Arg_0
3.61/2.13	
3.61/2.13	  start(Arg_0) -> a(100):|:Arg_0 <= 100 && 100 <= Arg_0  
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Timebounds: 
3.61/2.13	
3.61/2.13	  Overall timebound: max([102, 102+Arg_0]) {O(n)}
3.61/2.13	
3.61/2.13	  2: a->a: max([100, 100+Arg_0]) {O(n)}
3.61/2.13	
3.61/2.13	  0: start->a: 1 {O(1)}
3.61/2.13	
3.61/2.13	  1: start->a: 1 {O(1)}
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Costbounds:
3.61/2.13	
3.61/2.13	  Overall costbound: max([102, 102+Arg_0]) {O(n)}
3.61/2.13	
3.61/2.13	  2: a->a: max([100, 100+Arg_0]) {O(n)}
3.61/2.13	
3.61/2.13	  0: start->a: 1 {O(1)}
3.61/2.13	
3.61/2.13	  1: start->a: 1 {O(1)}
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Sizebounds:
3.61/2.13	
3.61/2.13	`Lower:
3.61/2.13	
3.61/2.13	  2: a->a, Arg_0: 0 {O(1)}
3.61/2.13	
3.61/2.13	  0: start->a, Arg_0: 1 {O(1)}
3.61/2.13	
3.61/2.13	  1: start->a, Arg_0: 100 {O(1)}
3.61/2.13	
3.61/2.13	`Upper:
3.61/2.13	
3.61/2.13	  2: a->a, Arg_0: max([100, Arg_0]) {O(n)}
3.61/2.13	
3.61/2.13	  0: start->a, Arg_0: Arg_0 {O(n)}
3.61/2.13	
3.61/2.13	  1: start->a, Arg_0: 100 {O(1)}
3.61/2.13	
3.61/2.13	
3.61/2.13	----------------------------------------
3.61/2.13	
3.61/2.13	(2)
3.61/2.13	BOUNDS(1, max(102, 102 + Arg_0))
3.61/2.13	
3.61/2.13	----------------------------------------
3.61/2.13	
3.61/2.13	(3) Loat Proof (FINISHED)
3.61/2.13	
3.61/2.13	
3.61/2.13	### Pre-processing the ITS problem ###
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Initial linear ITS problem
3.61/2.13	
3.61/2.13	   Start location: start
3.61/2.13	
3.61/2.13	      0: start -> a : [ A>=1 ], cost: 1
3.61/2.13	
3.61/2.13	      1: start -> a : A'=100, [ A==100 ], cost: 1
3.61/2.13	
3.61/2.13	      2: a -> a : A'=-1+A, [ A>=1 ], cost: 1
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	### Simplification by acceleration and chaining ###
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Accelerating simple loops of location 1.
3.61/2.13	
3.61/2.13	   Accelerating the following rules:
3.61/2.13	
3.61/2.13	      2: a -> a : A'=-1+A, [ A>=1 ], cost: 1
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	   Accelerated rule 2 with metering function A, yielding the new rule 3.
3.61/2.13	
3.61/2.13	   Removing the simple loops: 2.
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Accelerated all simple loops using metering functions (where possible):
3.61/2.13	
3.61/2.13	   Start location: start
3.61/2.13	
3.61/2.13	      0: start -> a : [ A>=1 ], cost: 1
3.61/2.13	
3.61/2.13	      1: start -> a : A'=100, [ A==100 ], cost: 1
3.61/2.13	
3.61/2.13	      3: a -> a : A'=0, [ A>=1 ], cost: A
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Chained accelerated rules (with incoming rules):
3.61/2.13	
3.61/2.13	   Start location: start
3.61/2.13	
3.61/2.13	      0: start -> a : [ A>=1 ], cost: 1
3.61/2.13	
3.61/2.13	      1: start -> a : A'=100, [ A==100 ], cost: 1
3.61/2.13	
3.61/2.13	      4: start -> a : A'=0, [ A>=1 ], cost: 1+A
3.61/2.13	
3.61/2.13	      5: start -> a : A'=0, [ A==100 ], cost: 101
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Removed unreachable locations (and leaf rules with constant cost):
3.61/2.13	
3.61/2.13	   Start location: start
3.61/2.13	
3.61/2.13	      4: start -> a : A'=0, [ A>=1 ], cost: 1+A
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	### Computing asymptotic complexity ###
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Fully simplified ITS problem
3.61/2.13	
3.61/2.13	   Start location: start
3.61/2.13	
3.61/2.13	      4: start -> a : A'=0, [ A>=1 ], cost: 1+A
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Computing asymptotic complexity for rule 4
3.61/2.13	
3.61/2.13	   Solved the limit problem by the following transformations:
3.61/2.13	
3.61/2.13	      Created initial limit problem:
3.61/2.13	
3.61/2.13	      A (+/+!), 1+A (+) [not solved]
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	      removing all constraints (solved by SMT)
3.61/2.13	
3.61/2.13	      resulting limit problem: [solved]
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	      applying transformation rule (C) using substitution {A==n}
3.61/2.13	
3.61/2.13	      resulting limit problem:
3.61/2.13	
3.61/2.13	      [solved]
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	   Solution:
3.61/2.13	
3.61/2.13	      A / n
3.61/2.13	
3.61/2.13	   Resulting cost 1+n has complexity: Poly(n^1)
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Found new complexity Poly(n^1).
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	Obtained the following overall complexity (w.r.t. the length of the input n):
3.61/2.13	
3.61/2.13	   Complexity:  Poly(n^1)
3.61/2.13	
3.61/2.13	   Cpx degree:  1
3.61/2.13	
3.61/2.13	   Solved cost: 1+n
3.61/2.13	
3.61/2.13	   Rule cost:   1+A
3.61/2.13	
3.61/2.13	   Rule guard:  [ A>=1 ]
3.61/2.13	
3.61/2.13	
3.61/2.13	
3.61/2.13	WORST_CASE(Omega(n^1),?)
3.61/2.13	
3.61/2.13	
3.61/2.13	----------------------------------------
3.61/2.13	
3.61/2.13	(4)
3.61/2.13	BOUNDS(n^1, INF)
3.88/2.15	EOF