3.91/2.15	WORST_CASE(Omega(n^2), O(n^2))
3.91/2.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.91/2.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.91/2.16	
3.91/2.16	
3.91/2.16	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, max(3, 1 + 2 * Arg_2) + max(2, 2 * Arg_2)^2).
3.91/2.16	
3.91/2.16	(0) CpxIntTrs
3.91/2.16	(1) Koat2 Proof [FINISHED, 438 ms]
3.91/2.16	(2) BOUNDS(1, max(3, 1 + 2 * Arg_2) + max(2, 2 * Arg_2)^2)
3.91/2.16	(3) Loat Proof [FINISHED, 437 ms]
3.91/2.16	(4) BOUNDS(n^2, INF)
3.91/2.16	
3.91/2.16	
3.91/2.16	----------------------------------------
3.91/2.16	
3.91/2.16	(0)
3.91/2.16	Obligation:
3.91/2.16	Complexity Int TRS consisting of the following rules:
3.91/2.16	f4(A, B, C) -> Com_1(f4(A, B + 1, C)) :|: A >= B + 1
3.91/2.16	f4(A, B, C) -> Com_1(f4(A + 1, 0, C)) :|: C >= A + 2 && B >= A
3.91/2.16	f0(A, B, C) -> Com_1(f4(0, 0, C)) :|: C >= 1
3.91/2.16	
3.91/2.16	The start-symbols are:[f0_3]
3.91/2.16	
3.91/2.16	
3.91/2.16	----------------------------------------
3.91/2.16	
3.91/2.16	(1) Koat2 Proof (FINISHED)
3.91/2.16	YES( ?, 1+max([2, 2*Arg_2])*max([2, 2*Arg_2])+max([2, 2*Arg_2]) {O(n^2)})
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Initial Complexity Problem:
3.91/2.16	
3.91/2.16	  Start:  f0  
3.91/2.16	
3.91/2.16	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.91/2.16	
3.91/2.16	  Temp_Vars:    
3.91/2.16	
3.91/2.16	  Locations:  f0, f4  
3.91/2.16	
3.91/2.16	  Transitions:
3.91/2.16	
3.91/2.16	  f0(Arg_0,Arg_1,Arg_2) -> f4(0,0,Arg_2):|:1 <= Arg_2
3.91/2.16	
3.91/2.16	  f4(Arg_0,Arg_1,Arg_2) -> f4(Arg_0,Arg_1+1,Arg_2):|:1 <= Arg_2 && 1 <= Arg_1+Arg_2 && 1+Arg_1 <= Arg_2 && 1 <= Arg_0+Arg_2 && 1+Arg_0 <= Arg_2 && Arg_1 <= Arg_0 && 0 <= Arg_1 && 0 <= Arg_0+Arg_1 && 0 <= Arg_0 && Arg_1+1 <= Arg_0
3.91/2.16	
3.91/2.16	  f4(Arg_0,Arg_1,Arg_2) -> f4(Arg_0+1,0,Arg_2):|:1 <= Arg_2 && 1 <= Arg_1+Arg_2 && 1+Arg_1 <= Arg_2 && 1 <= Arg_0+Arg_2 && 1+Arg_0 <= Arg_2 && Arg_1 <= Arg_0 && 0 <= Arg_1 && 0 <= Arg_0+Arg_1 && 0 <= Arg_0 && Arg_0+2 <= Arg_2 && Arg_0 <= Arg_1  
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Timebounds: 
3.91/2.16	
3.91/2.16	  Overall timebound: 1+max([2, 2*Arg_2])*max([2, 2*Arg_2])+max([2, 2*Arg_2]) {O(n^2)}
3.91/2.16	
3.91/2.16	  2: f0->f4: 1 {O(1)}
3.91/2.16	
3.91/2.16	  0: f4->f4: max([2, 2*Arg_2])*max([2, 2*Arg_2]) {O(n^2)}
3.91/2.16	
3.91/2.16	  1: f4->f4: max([2, 2*Arg_2]) {O(n)}
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Costbounds:
3.91/2.16	
3.91/2.16	  Overall costbound: 1+max([2, 2*Arg_2])*max([2, 2*Arg_2])+max([2, 2*Arg_2]) {O(n^2)}
3.91/2.16	
3.91/2.16	  2: f0->f4: 1 {O(1)}
3.91/2.16	
3.91/2.16	  0: f4->f4: max([2, 2*Arg_2])*max([2, 2*Arg_2]) {O(n^2)}
3.91/2.16	
3.91/2.16	  1: f4->f4: max([2, 2*Arg_2]) {O(n)}
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Sizebounds:
3.91/2.16	
3.91/2.16	`Lower:
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_0: 0 {O(1)}
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_1: 0 {O(1)}
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_2: 1 {O(1)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_0: 1 {O(1)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_1: 1 {O(1)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_2: 2 {O(1)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_0: 1 {O(1)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_1: 0 {O(1)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_2: 2 {O(1)}
3.91/2.16	
3.91/2.16	`Upper:
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_0: 0 {O(1)}
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_1: 0 {O(1)}
3.91/2.16	
3.91/2.16	  2: f0->f4, Arg_2: Arg_2 {O(n)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_0: max([2, 2*Arg_2]) {O(n)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_1: max([2, 2*Arg_2])*max([2, 2*Arg_2]) {O(n^2)}
3.91/2.16	
3.91/2.16	  0: f4->f4, Arg_2: Arg_2 {O(n)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_0: max([2, 2*Arg_2]) {O(n)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_1: 0 {O(1)}
3.91/2.16	
3.91/2.16	  1: f4->f4, Arg_2: Arg_2 {O(n)}
3.91/2.16	
3.91/2.16	
3.91/2.16	----------------------------------------
3.91/2.16	
3.91/2.16	(2)
3.91/2.16	BOUNDS(1, max(3, 1 + 2 * Arg_2) + max(2, 2 * Arg_2)^2)
3.91/2.16	
3.91/2.16	----------------------------------------
3.91/2.16	
3.91/2.16	(3) Loat Proof (FINISHED)
3.91/2.16	
3.91/2.16	
3.91/2.16	### Pre-processing the ITS problem ###
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Initial linear ITS problem
3.91/2.16	
3.91/2.16	   Start location: f0
3.91/2.16	
3.91/2.16	      0: f4 -> f4 : B'=1+B, [ A>=1+B ], cost: 1
3.91/2.16	
3.91/2.16	      1: f4 -> f4 : A'=1+A, B'=0, [ C>=2+A && B>=A ], cost: 1
3.91/2.16	
3.91/2.16	      2: f0 -> f4 : A'=0, B'=0, [ C>=1 ], cost: 1
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	### Simplification by acceleration and chaining ###
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Accelerating simple loops of location 0.
3.91/2.16	
3.91/2.16	   Accelerating the following rules:
3.91/2.16	
3.91/2.16	      0: f4 -> f4 : B'=1+B, [ A>=1+B ], cost: 1
3.91/2.16	
3.91/2.16	      1: f4 -> f4 : A'=1+A, B'=0, [ C>=2+A && B>=A ], cost: 1
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	   Accelerated rule 0 with metering function A-B, yielding the new rule 3.
3.91/2.16	
3.91/2.16	   Found no metering function for rule 1.
3.91/2.16	
3.91/2.16	   Nested simple loops 1 (outer loop) and 3 (inner loop) with metering function -1+C-A, resulting in the new rules: 4, 5.
3.91/2.16	
3.91/2.16	   Removing the simple loops: 0 1.
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Accelerated all simple loops using metering functions (where possible):
3.91/2.16	
3.91/2.16	   Start location: f0
3.91/2.16	
3.91/2.16	      3: f4 -> f4 : B'=A, [ A>=1+B ], cost: A-B
3.91/2.16	
3.91/2.16	      4: f4 -> f4 : A'=-1+C, B'=-1+C, [ C>=2+A && B>=A && 1+A>=1 ], cost: -3/2+3/2*C+1/2*(-1+C-A)^2-3/2*A+(-1+C-A)*A
3.91/2.16	
3.91/2.16	      5: f4 -> f4 : A'=-1+C, B'=-1+C, [ A>=1+B && C>=2+A && 1+A>=1 ], cost: -3/2+3/2*C+1/2*(-1+C-A)^2-1/2*A+(-1+C-A)*A-B
3.91/2.16	
3.91/2.16	      2: f0 -> f4 : A'=0, B'=0, [ C>=1 ], cost: 1
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Chained accelerated rules (with incoming rules):
3.91/2.16	
3.91/2.16	   Start location: f0
3.91/2.16	
3.91/2.16	      2: f0 -> f4 : A'=0, B'=0, [ C>=1 ], cost: 1
3.91/2.16	
3.91/2.16	      6: f0 -> f4 : A'=-1+C, B'=-1+C, [ C>=2 ], cost: -1/2+3/2*C+1/2*(-1+C)^2
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Removed unreachable locations (and leaf rules with constant cost):
3.91/2.16	
3.91/2.16	   Start location: f0
3.91/2.16	
3.91/2.16	      6: f0 -> f4 : A'=-1+C, B'=-1+C, [ C>=2 ], cost: -1/2+3/2*C+1/2*(-1+C)^2
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	### Computing asymptotic complexity ###
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Fully simplified ITS problem
3.91/2.16	
3.91/2.16	   Start location: f0
3.91/2.16	
3.91/2.16	      6: f0 -> f4 : A'=-1+C, B'=-1+C, [ C>=2 ], cost: -1/2+3/2*C+1/2*(-1+C)^2
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Computing asymptotic complexity for rule 6
3.91/2.16	
3.91/2.16	   Solved the limit problem by the following transformations:
3.91/2.16	
3.91/2.16	      Created initial limit problem:
3.91/2.16	
3.91/2.16	      1/2*C^2+1/2*C (+), -1+C (+/+!) [not solved]
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	      removing all constraints (solved by SMT)
3.91/2.16	
3.91/2.16	      resulting limit problem: [solved]
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	      applying transformation rule (C) using substitution {C==n}
3.91/2.16	
3.91/2.16	      resulting limit problem:
3.91/2.16	
3.91/2.16	      [solved]
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	   Solution:
3.91/2.16	
3.91/2.16	      C / n
3.91/2.16	
3.91/2.16	   Resulting cost 1/2*n+1/2*n^2 has complexity: Poly(n^2)
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Found new complexity Poly(n^2).
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	Obtained the following overall complexity (w.r.t. the length of the input n):
3.91/2.16	
3.91/2.16	   Complexity:  Poly(n^2)
3.91/2.16	
3.91/2.16	   Cpx degree:  2
3.91/2.16	
3.91/2.16	   Solved cost: 1/2*n+1/2*n^2
3.91/2.16	
3.91/2.16	   Rule cost:   -1/2+3/2*C+1/2*(-1+C)^2
3.91/2.16	
3.91/2.16	   Rule guard:  [ C>=2 ]
3.91/2.16	
3.91/2.16	
3.91/2.16	
3.91/2.16	WORST_CASE(Omega(n^2),?)
3.91/2.16	
3.91/2.16	
3.91/2.16	----------------------------------------
3.91/2.16	
3.91/2.16	(4)
3.91/2.16	BOUNDS(n^2, INF)
4.39/2.18	EOF