3.71/1.96	WORST_CASE(Omega(n^1), O(n^1))
3.71/1.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.71/1.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.96	
3.71/1.96	
3.71/1.96	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(3, 3 + 2 * Arg_1)).
3.71/1.96	
3.71/1.96	(0) CpxIntTrs
3.71/1.96	(1) Koat2 Proof [FINISHED, 173 ms]
3.71/1.96	(2) BOUNDS(1, max(3, 3 + 2 * Arg_1))
3.71/1.96	(3) Loat Proof [FINISHED, 166 ms]
3.71/1.96	(4) BOUNDS(n^1, INF)
3.71/1.96	
3.71/1.96	
3.71/1.96	----------------------------------------
3.71/1.96	
3.71/1.96	(0)
3.71/1.96	Obligation:
3.71/1.96	Complexity Int TRS consisting of the following rules:
3.71/1.96	div(A, B) -> Com_1(end(A, B)) :|: 0 >= A
3.71/1.96	div(A, B) -> Com_1(end(A, B)) :|: A >= B
3.71/1.96	div(A, B) -> Com_1(div(A, B - A)) :|: B >= A + 1 && A >= 1
3.71/1.96	start(A, B) -> Com_1(div(A, B)) :|: TRUE
3.71/1.96	
3.71/1.96	The start-symbols are:[start_2]
3.71/1.96	
3.71/1.96	
3.71/1.96	----------------------------------------
3.71/1.96	
3.71/1.96	(1) Koat2 Proof (FINISHED)
3.71/1.96	YES( ?, max([3, 3+2*Arg_1]) {O(n)})
3.71/1.96	
3.71/1.96	
3.71/1.96	
3.71/1.96	Initial Complexity Problem:
3.71/1.96	
3.71/1.96	  Start:  start  
3.71/1.96	
3.71/1.96	  Program_Vars:  Arg_0, Arg_1  
3.71/1.96	
3.71/1.96	  Temp_Vars:    
3.71/1.96	
3.71/1.96	  Locations:  div, end, start  
3.71/1.96	
3.71/1.96	  Transitions:
3.71/1.96	
3.71/1.96	  div(Arg_0,Arg_1) -> div(Arg_0,Arg_1-Arg_0):|:Arg_0+1 <= Arg_1 && 1 <= Arg_0
3.71/1.97	
3.71/1.97	  div(Arg_0,Arg_1) -> end(Arg_0,Arg_1):|:Arg_0 <= 0
3.71/1.97	
3.71/1.97	  div(Arg_0,Arg_1) -> end(Arg_0,Arg_1):|:Arg_1 <= Arg_0
3.71/1.97	
3.71/1.97	  start(Arg_0,Arg_1) -> div(Arg_0,Arg_1):|:  
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Timebounds: 
3.71/1.97	
3.71/1.97	  Overall timebound: max([3, 3+2*Arg_1]) {O(n)}
3.71/1.97	
3.71/1.97	  0: div->end: 1 {O(1)}
3.71/1.97	
3.71/1.97	  1: div->end: 1 {O(1)}
3.71/1.97	
3.71/1.97	  2: div->div: max([0, 2*Arg_1]) {O(n)}
3.71/1.97	
3.71/1.97	  3: start->div: 1 {O(1)}
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Costbounds:
3.71/1.97	
3.71/1.97	  Overall costbound: max([3, 3+2*Arg_1]) {O(n)}
3.71/1.97	
3.71/1.97	  0: div->end: 1 {O(1)}
3.71/1.97	
3.71/1.97	  1: div->end: 1 {O(1)}
3.71/1.97	
3.71/1.97	  2: div->div: max([0, 2*Arg_1]) {O(n)}
3.71/1.97	
3.71/1.97	  3: start->div: 1 {O(1)}
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Sizebounds:
3.71/1.97	
3.71/1.97	`Lower:
3.71/1.97	
3.71/1.97	  0: div->end, Arg_0: Arg_0 {O(n)}
3.71/1.97	
3.71/1.97	  0: div->end, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	  1: div->end, Arg_0: min([1, Arg_0]) {O(n)}
3.71/1.97	
3.71/1.97	  1: div->end, Arg_1: min([1, Arg_1]) {O(n)}
3.71/1.97	
3.71/1.97	  2: div->div, Arg_0: 1 {O(1)}
3.71/1.97	
3.71/1.97	  2: div->div, Arg_1: 1 {O(1)}
3.71/1.97	
3.71/1.97	  3: start->div, Arg_0: Arg_0 {O(n)}
3.71/1.97	
3.71/1.97	  3: start->div, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	`Upper:
3.71/1.97	
3.71/1.97	  0: div->end, Arg_0: 0 {O(1)}
3.71/1.97	
3.71/1.97	  0: div->end, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	  1: div->end, Arg_0: Arg_0 {O(n)}
3.71/1.97	
3.71/1.97	  1: div->end, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	  2: div->div, Arg_0: Arg_0 {O(n)}
3.71/1.97	
3.71/1.97	  2: div->div, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	  3: start->div, Arg_0: Arg_0 {O(n)}
3.71/1.97	
3.71/1.97	  3: start->div, Arg_1: Arg_1 {O(n)}
3.71/1.97	
3.71/1.97	
3.71/1.97	----------------------------------------
3.71/1.97	
3.71/1.97	(2)
3.71/1.97	BOUNDS(1, max(3, 3 + 2 * Arg_1))
3.71/1.97	
3.71/1.97	----------------------------------------
3.71/1.97	
3.71/1.97	(3) Loat Proof (FINISHED)
3.71/1.97	
3.71/1.97	
3.71/1.97	### Pre-processing the ITS problem ###
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Initial linear ITS problem
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      0: div -> end : [ 0>=A ], cost: 1
3.71/1.97	
3.71/1.97	      1: div -> end : [ A>=B ], cost: 1
3.71/1.97	
3.71/1.97	      2: div -> div : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1
3.71/1.97	
3.71/1.97	      3: start -> div : [], cost: 1
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Removed unreachable and leaf rules:
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      2: div -> div : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1
3.71/1.97	
3.71/1.97	      3: start -> div : [], cost: 1
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	### Simplification by acceleration and chaining ###
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Accelerating simple loops of location 0.
3.71/1.97	
3.71/1.97	   Accelerating the following rules:
3.71/1.97	
3.71/1.97	      2: div -> div : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	   Accelerated rule 2 with backward acceleration, yielding the new rule 4.
3.71/1.97	
3.71/1.97	   Removing the simple loops: 2.
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Accelerated all simple loops using metering functions (where possible):
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      4: div -> div : B'=-k*A+B, [ B>=1+A && A>=1 && k>0 && -(-1+k)*A+B>=1+A ], cost: k
3.71/1.97	
3.71/1.97	      3: start -> div : [], cost: 1
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Chained accelerated rules (with incoming rules):
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      3: start -> div : [], cost: 1
3.71/1.97	
3.71/1.97	      5: start -> div : B'=-k*A+B, [ B>=1+A && A>=1 && k>0 && -(-1+k)*A+B>=1+A ], cost: 1+k
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Removed unreachable locations (and leaf rules with constant cost):
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      5: start -> div : B'=-k*A+B, [ B>=1+A && A>=1 && k>0 && -(-1+k)*A+B>=1+A ], cost: 1+k
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	### Computing asymptotic complexity ###
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Fully simplified ITS problem
3.71/1.97	
3.71/1.97	   Start location: start
3.71/1.97	
3.71/1.97	      5: start -> div : B'=-k*A+B, [ B>=1+A && A>=1 && k>0 && -(-1+k)*A+B>=1+A ], cost: 1+k
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Computing asymptotic complexity for rule 5
3.71/1.97	
3.71/1.97	   Simplified the guard:
3.71/1.97	
3.71/1.97	      5: start -> div : B'=-k*A+B, [ A>=1 && k>0 && -(-1+k)*A+B>=1+A ], cost: 1+k
3.71/1.97	
3.71/1.97	   Solved the limit problem by the following transformations:
3.71/1.97	
3.71/1.97	      Created initial limit problem:
3.71/1.97	
3.71/1.97	      -(-1+k)*A-A+B (+/+!), k (+/+!), 1+k (+), A (+/+!) [not solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      removing all constraints (solved by SMT)
3.71/1.97	
3.71/1.97	      resulting limit problem: [solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      applying transformation rule (C) using substitution {k==n,A==1,B==2*n}
3.71/1.97	
3.71/1.97	      resulting limit problem:
3.71/1.97	
3.71/1.97	      [solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	   Solved the limit problem by the following transformations:
3.71/1.97	
3.71/1.97	      Created initial limit problem:
3.71/1.97	
3.71/1.97	      -(-1+k)*A-A+B (+/+!), k (+/+!), 1+k (+), A (+/+!) [not solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      applying transformation rule (C) using substitution {A==1}
3.71/1.97	
3.71/1.97	      resulting limit problem:
3.71/1.97	
3.71/1.97	      1 (+/+!), -k+B (+/+!), k (+/+!), 1+k (+) [not solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      applying transformation rule (B), deleting 1 (+/+!)
3.71/1.97	
3.71/1.97	      resulting limit problem:
3.71/1.97	
3.71/1.97	      -k+B (+/+!), k (+/+!), 1+k (+) [not solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      removing all constraints (solved by SMT)
3.71/1.97	
3.71/1.97	      resulting limit problem: [solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	      applying transformation rule (C) using substitution {k==n,B==1+n}
3.71/1.97	
3.71/1.97	      resulting limit problem:
3.71/1.97	
3.71/1.97	      [solved]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	   Solution:
3.71/1.97	
3.71/1.97	      k / n
3.71/1.97	
3.71/1.97	      A / 1
3.71/1.97	
3.71/1.97	      B / 2*n
3.71/1.97	
3.71/1.97	   Resulting cost 1+n has complexity: Poly(n^1)
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Found new complexity Poly(n^1).
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	Obtained the following overall complexity (w.r.t. the length of the input n):
3.71/1.97	
3.71/1.97	   Complexity:  Poly(n^1)
3.71/1.97	
3.71/1.97	   Cpx degree:  1
3.71/1.97	
3.71/1.97	   Solved cost: 1+n
3.71/1.97	
3.71/1.97	   Rule cost:   1+k
3.71/1.97	
3.71/1.97	   Rule guard:  [ A>=1 && k>0 && -(-1+k)*A+B>=1+A ]
3.71/1.97	
3.71/1.97	
3.71/1.97	
3.71/1.97	WORST_CASE(Omega(n^1),?)
3.71/1.97	
3.71/1.97	
3.71/1.97	----------------------------------------
3.71/1.97	
3.71/1.97	(4)
3.71/1.97	BOUNDS(n^1, INF)
4.01/1.98	EOF