3.69/1.83	WORST_CASE(Omega(n^1), O(n^1))
3.69/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.69/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.69/1.84	
3.69/1.84	
3.69/1.84	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(3, 3 + 3 * Arg_0) + max(1, 2 + 2 * Arg_0)).
3.69/1.84	
3.69/1.84	(0) CpxIntTrs
3.69/1.84	(1) Koat2 Proof [FINISHED, 142 ms]
3.69/1.84	(2) BOUNDS(1, max(3, 3 + 3 * Arg_0) + max(1, 2 + 2 * Arg_0))
3.69/1.84	(3) Loat Proof [FINISHED, 215 ms]
3.69/1.84	(4) BOUNDS(n^1, INF)
3.69/1.84	
3.69/1.84	
3.69/1.84	----------------------------------------
3.69/1.84	
3.69/1.84	(0)
3.69/1.84	Obligation:
3.69/1.84	Complexity Int TRS consisting of the following rules:
3.69/1.84	evaleasy2start(A) -> Com_1(evaleasy2entryin(A)) :|: TRUE
3.69/1.84	evaleasy2entryin(A) -> Com_1(evaleasy2bb1in(A)) :|: TRUE
3.69/1.84	evaleasy2bb1in(A) -> Com_1(evaleasy2bbin(A)) :|: A >= 1
3.69/1.84	evaleasy2bb1in(A) -> Com_1(evaleasy2returnin(A)) :|: 0 >= A
3.69/1.84	evaleasy2bbin(A) -> Com_1(evaleasy2bb1in(A - 1)) :|: TRUE
3.69/1.84	evaleasy2returnin(A) -> Com_1(evaleasy2stop(A)) :|: TRUE
3.69/1.84	
3.69/1.84	The start-symbols are:[evaleasy2start_1]
3.69/1.84	
3.69/1.84	
3.69/1.84	----------------------------------------
3.69/1.84	
3.69/1.84	(1) Koat2 Proof (FINISHED)
3.69/1.84	YES( ?, 3+max([0, 3*Arg_0])+max([1, 2+2*Arg_0]) {O(n)})
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Initial Complexity Problem:
3.69/1.84	
3.69/1.84	  Start:  evaleasy2start  
3.69/1.84	
3.69/1.84	  Program_Vars:  Arg_0  
3.69/1.84	
3.69/1.84	  Temp_Vars:    
3.69/1.84	
3.69/1.84	  Locations:  evaleasy2bb1in, evaleasy2bbin, evaleasy2entryin, evaleasy2returnin, evaleasy2start, evaleasy2stop  
3.69/1.84	
3.69/1.84	  Transitions:
3.69/1.84	
3.69/1.84	  evaleasy2bb1in(Arg_0) -> evaleasy2bbin(Arg_0):|:1 <= Arg_0
3.69/1.84	
3.69/1.84	  evaleasy2bb1in(Arg_0) -> evaleasy2returnin(Arg_0):|:Arg_0 <= 0
3.69/1.84	
3.69/1.84	  evaleasy2bbin(Arg_0) -> evaleasy2bb1in(Arg_0-1):|:1 <= Arg_0
3.69/1.84	
3.69/1.84	  evaleasy2entryin(Arg_0) -> evaleasy2bb1in(Arg_0):|:
3.69/1.84	
3.69/1.84	  evaleasy2returnin(Arg_0) -> evaleasy2stop(Arg_0):|:Arg_0 <= 0
3.69/1.84	
3.69/1.84	  evaleasy2start(Arg_0) -> evaleasy2entryin(Arg_0):|:  
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Timebounds: 
3.69/1.84	
3.69/1.84	  Overall timebound: 3+max([0, 3*Arg_0])+max([1, 2+2*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  2: evaleasy2bb1in->evaleasy2bbin: max([0, 1+2*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  3: evaleasy2bb1in->evaleasy2returnin: 1 {O(1)}
3.69/1.84	
3.69/1.84	  4: evaleasy2bbin->evaleasy2bb1in: max([0, 3*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  1: evaleasy2entryin->evaleasy2bb1in: 1 {O(1)}
3.69/1.84	
3.69/1.84	  5: evaleasy2returnin->evaleasy2stop: 1 {O(1)}
3.69/1.84	
3.69/1.84	  0: evaleasy2start->evaleasy2entryin: 1 {O(1)}
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Costbounds:
3.69/1.84	
3.69/1.84	  Overall costbound: 3+max([0, 3*Arg_0])+max([1, 2+2*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  2: evaleasy2bb1in->evaleasy2bbin: max([0, 1+2*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  3: evaleasy2bb1in->evaleasy2returnin: 1 {O(1)}
3.69/1.84	
3.69/1.84	  4: evaleasy2bbin->evaleasy2bb1in: max([0, 3*Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  1: evaleasy2entryin->evaleasy2bb1in: 1 {O(1)}
3.69/1.84	
3.69/1.84	  5: evaleasy2returnin->evaleasy2stop: 1 {O(1)}
3.69/1.84	
3.69/1.84	  0: evaleasy2start->evaleasy2entryin: 1 {O(1)}
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Sizebounds:
3.69/1.84	
3.69/1.84	`Lower:
3.69/1.84	
3.69/1.84	  2: evaleasy2bb1in->evaleasy2bbin, Arg_0: 1 {O(1)}
3.69/1.84	
3.69/1.84	  3: evaleasy2bb1in->evaleasy2returnin, Arg_0: min([0, Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  4: evaleasy2bbin->evaleasy2bb1in, Arg_0: 0 {O(1)}
3.69/1.84	
3.69/1.84	  1: evaleasy2entryin->evaleasy2bb1in, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	  5: evaleasy2returnin->evaleasy2stop, Arg_0: min([0, Arg_0]) {O(n)}
3.69/1.84	
3.69/1.84	  0: evaleasy2start->evaleasy2entryin, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	`Upper:
3.69/1.84	
3.69/1.84	  2: evaleasy2bb1in->evaleasy2bbin, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	  3: evaleasy2bb1in->evaleasy2returnin, Arg_0: 0 {O(1)}
3.69/1.84	
3.69/1.84	  4: evaleasy2bbin->evaleasy2bb1in, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	  1: evaleasy2entryin->evaleasy2bb1in, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	  5: evaleasy2returnin->evaleasy2stop, Arg_0: 0 {O(1)}
3.69/1.84	
3.69/1.84	  0: evaleasy2start->evaleasy2entryin, Arg_0: Arg_0 {O(n)}
3.69/1.84	
3.69/1.84	
3.69/1.84	----------------------------------------
3.69/1.84	
3.69/1.84	(2)
3.69/1.84	BOUNDS(1, max(3, 3 + 3 * Arg_0) + max(1, 2 + 2 * Arg_0))
3.69/1.84	
3.69/1.84	----------------------------------------
3.69/1.84	
3.69/1.84	(3) Loat Proof (FINISHED)
3.69/1.84	
3.69/1.84	
3.69/1.84	### Pre-processing the ITS problem ###
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Initial linear ITS problem
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      0: evaleasy2start -> evaleasy2entryin : [], cost: 1
3.69/1.84	
3.69/1.84	      1: evaleasy2entryin -> evaleasy2bb1in : [], cost: 1
3.69/1.84	
3.69/1.84	      2: evaleasy2bb1in -> evaleasy2bbin : [ A>=1 ], cost: 1
3.69/1.84	
3.69/1.84	      3: evaleasy2bb1in -> evaleasy2returnin : [ 0>=A ], cost: 1
3.69/1.84	
3.69/1.84	      4: evaleasy2bbin -> evaleasy2bb1in : A'=-1+A, [], cost: 1
3.69/1.84	
3.69/1.84	      5: evaleasy2returnin -> evaleasy2stop : [], cost: 1
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Removed unreachable and leaf rules:
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      0: evaleasy2start -> evaleasy2entryin : [], cost: 1
3.69/1.84	
3.69/1.84	      1: evaleasy2entryin -> evaleasy2bb1in : [], cost: 1
3.69/1.84	
3.69/1.84	      2: evaleasy2bb1in -> evaleasy2bbin : [ A>=1 ], cost: 1
3.69/1.84	
3.69/1.84	      4: evaleasy2bbin -> evaleasy2bb1in : A'=-1+A, [], cost: 1
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	### Simplification by acceleration and chaining ###
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Eliminated locations (on linear paths):
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      6: evaleasy2start -> evaleasy2bb1in : [], cost: 2
3.69/1.84	
3.69/1.84	      7: evaleasy2bb1in -> evaleasy2bb1in : A'=-1+A, [ A>=1 ], cost: 2
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Accelerating simple loops of location 2.
3.69/1.84	
3.69/1.84	   Accelerating the following rules:
3.69/1.84	
3.69/1.84	      7: evaleasy2bb1in -> evaleasy2bb1in : A'=-1+A, [ A>=1 ], cost: 2
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	   Accelerated rule 7 with metering function A, yielding the new rule 8.
3.69/1.84	
3.69/1.84	   Removing the simple loops: 7.
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Accelerated all simple loops using metering functions (where possible):
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      6: evaleasy2start -> evaleasy2bb1in : [], cost: 2
3.69/1.84	
3.69/1.84	      8: evaleasy2bb1in -> evaleasy2bb1in : A'=0, [ A>=1 ], cost: 2*A
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Chained accelerated rules (with incoming rules):
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      6: evaleasy2start -> evaleasy2bb1in : [], cost: 2
3.69/1.84	
3.69/1.84	      9: evaleasy2start -> evaleasy2bb1in : A'=0, [ A>=1 ], cost: 2+2*A
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Removed unreachable locations (and leaf rules with constant cost):
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      9: evaleasy2start -> evaleasy2bb1in : A'=0, [ A>=1 ], cost: 2+2*A
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	### Computing asymptotic complexity ###
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Fully simplified ITS problem
3.69/1.84	
3.69/1.84	   Start location: evaleasy2start
3.69/1.84	
3.69/1.84	      9: evaleasy2start -> evaleasy2bb1in : A'=0, [ A>=1 ], cost: 2+2*A
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Computing asymptotic complexity for rule 9
3.69/1.84	
3.69/1.84	   Solved the limit problem by the following transformations:
3.69/1.84	
3.69/1.84	      Created initial limit problem:
3.69/1.84	
3.69/1.84	      A (+/+!), 2+2*A (+) [not solved]
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	      removing all constraints (solved by SMT)
3.69/1.84	
3.69/1.84	      resulting limit problem: [solved]
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	      applying transformation rule (C) using substitution {A==n}
3.69/1.84	
3.69/1.84	      resulting limit problem:
3.69/1.84	
3.69/1.84	      [solved]
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	   Solution:
3.69/1.84	
3.69/1.84	      A / n
3.69/1.84	
3.69/1.84	   Resulting cost 2+2*n has complexity: Poly(n^1)
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Found new complexity Poly(n^1).
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	Obtained the following overall complexity (w.r.t. the length of the input n):
3.69/1.84	
3.69/1.84	   Complexity:  Poly(n^1)
3.69/1.84	
3.69/1.84	   Cpx degree:  1
3.69/1.84	
3.69/1.84	   Solved cost: 2+2*n
3.69/1.84	
3.69/1.84	   Rule cost:   2+2*A
3.69/1.84	
3.69/1.84	   Rule guard:  [ A>=1 ]
3.69/1.84	
3.69/1.84	
3.69/1.84	
3.69/1.84	WORST_CASE(Omega(n^1),?)
3.69/1.84	
3.69/1.84	
3.69/1.84	----------------------------------------
3.69/1.84	
3.69/1.84	(4)
3.69/1.84	BOUNDS(n^1, INF)
3.69/1.86	EOF