3.56/2.06	WORST_CASE(Omega(n^1), O(n^1))
3.56/2.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.56/2.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.56/2.06	
3.56/2.06	
3.56/2.06	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 3 + Arg_0 + -1 * Arg_1) + nat(Arg_0 + -1 * Arg_1)).
3.56/2.06	
3.56/2.06	(0) CpxIntTrs
3.56/2.06	(1) Koat2 Proof [FINISHED, 328 ms]
3.56/2.06	(2) BOUNDS(1, max(2, 3 + Arg_0 + -1 * Arg_1) + nat(Arg_0 + -1 * Arg_1))
3.56/2.06	(3) Loat Proof [FINISHED, 359 ms]
3.56/2.06	(4) BOUNDS(n^1, INF)
3.56/2.06	
3.56/2.06	
3.56/2.06	----------------------------------------
3.56/2.06	
3.56/2.06	(0)
3.56/2.06	Obligation:
3.56/2.06	Complexity Int TRS consisting of the following rules:
3.56/2.06	f2(A, B, C) -> Com_1(f1(A, B, C)) :|: TRUE
3.56/2.06	f1(A, B, C) -> Com_1(f1(1 + A, B, C)) :|: B >= 1 + A
3.56/2.06	f1(A, B, C) -> Com_1(f1(1 + A, A, C)) :|: B >= D && A >= B && A <= B
3.56/2.06	f1(A, B, C) -> Com_1(f300(A, B, D)) :|: A >= B && A >= B + 1
3.56/2.06	f1(A, B, C) -> Com_1(f300(A, B, D)) :|: A >= B && B >= A + 1
3.56/2.06	
3.56/2.06	The start-symbols are:[f2_3]
3.56/2.06	
3.56/2.06	
3.56/2.06	----------------------------------------
3.56/2.06	
3.56/2.06	(1) Koat2 Proof (FINISHED)
3.56/2.06	YES( ?, 2+max([0, 1+Arg_0-Arg_1])+max([0, Arg_0-Arg_1]) {O(n)})
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Initial Complexity Problem:
3.56/2.07	
3.56/2.07	  Start:  f2  
3.56/2.07	
3.56/2.07	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.56/2.07	
3.56/2.07	  Temp_Vars:  D  
3.56/2.07	
3.56/2.07	  Locations:  f1, f2, f300  
3.56/2.07	
3.56/2.07	  Transitions:
3.56/2.07	
3.56/2.07	  f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,1+Arg_1,Arg_2):|:1+Arg_1 <= Arg_0
3.56/2.07	
3.56/2.07	  f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_1,1+Arg_1,Arg_2):|:D <= Arg_0 && Arg_1 <= Arg_0 && Arg_0 <= Arg_1
3.56/2.07	
3.56/2.07	  f1(Arg_0,Arg_1,Arg_2) -> f300(Arg_0,Arg_1,D):|:Arg_0 <= Arg_1 && Arg_0+1 <= Arg_1
3.56/2.07	
3.56/2.07	  f2(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2):|:  
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Timebounds: 
3.56/2.07	
3.56/2.07	  Overall timebound: 2+max([0, 1+Arg_0-Arg_1])+max([0, Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1: max([0, Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1: max([0, 1+Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300: 1 {O(1)}
3.56/2.07	
3.56/2.07	  0: f2->f1: 1 {O(1)}
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Costbounds:
3.56/2.07	
3.56/2.07	  Overall costbound: 2+max([0, 1+Arg_0-Arg_1])+max([0, Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1: max([0, Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1: max([0, 1+Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300: 1 {O(1)}
3.56/2.07	
3.56/2.07	  0: f2->f1: 1 {O(1)}
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Sizebounds:
3.56/2.07	
3.56/2.07	`Lower:
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_0: Arg_0 {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_1: Arg_1 {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_0: Arg_1 {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_1: 1+Arg_1 {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300, Arg_0: min([Arg_0, Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300, Arg_1: min([Arg_1, -(-1-Arg_1)]) {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_0: Arg_0 {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_1: Arg_1 {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	`Upper:
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_0: Arg_0 {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
3.56/2.07	
3.56/2.07	  1: f1->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_0: max([Arg_1, Arg_1+max([0, Arg_0-Arg_1])]) {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_1: max([1+Arg_1, 1+Arg_1+max([0, Arg_0-Arg_1])]) {O(n)}
3.56/2.07	
3.56/2.07	  2: f1->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300, Arg_0: max([Arg_0, max([Arg_1, Arg_1+max([0, Arg_0-Arg_1])])]) {O(n)}
3.56/2.07	
3.56/2.07	  3: f1->f300, Arg_1: max([Arg_1, max([1+Arg_1, 1+Arg_1+max([0, Arg_0-Arg_1])])]) {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_0: Arg_0 {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_1: Arg_1 {O(n)}
3.56/2.07	
3.56/2.07	  0: f2->f1, Arg_2: Arg_2 {O(n)}
3.56/2.07	
3.56/2.07	
3.56/2.07	----------------------------------------
3.56/2.07	
3.56/2.07	(2)
3.56/2.07	BOUNDS(1, max(2, 3 + Arg_0 + -1 * Arg_1) + nat(Arg_0 + -1 * Arg_1))
3.56/2.07	
3.56/2.07	----------------------------------------
3.56/2.07	
3.56/2.07	(3) Loat Proof (FINISHED)
3.56/2.07	
3.56/2.07	
3.56/2.07	### Pre-processing the ITS problem ###
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Initial linear ITS problem
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      0: f2 -> f1 : [], cost: 1
3.56/2.07	
3.56/2.07	      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
3.56/2.07	
3.56/2.07	      2: f1 -> f1 : A'=B, B'=1+B, [ A>=free && B==A ], cost: 1
3.56/2.07	
3.56/2.07	      3: f1 -> f300 : C'=free_1, [ B>=A && B>=1+A ], cost: 1
3.56/2.07	
3.56/2.07	      4: f1 -> f300 : C'=free_2, [ B>=A && A>=1+B ], cost: 1
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Removed unreachable and leaf rules:
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      0: f2 -> f1 : [], cost: 1
3.56/2.07	
3.56/2.07	      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
3.56/2.07	
3.56/2.07	      2: f1 -> f1 : A'=B, B'=1+B, [ A>=free && B==A ], cost: 1
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Simplified all rules, resulting in:
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      0: f2 -> f1 : [], cost: 1
3.56/2.07	
3.56/2.07	      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
3.56/2.07	
3.56/2.07	      2: f1 -> f1 : A'=B, B'=1+B, [ B==A ], cost: 1
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	### Simplification by acceleration and chaining ###
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Accelerating simple loops of location 1.
3.56/2.07	
3.56/2.07	   Accelerating the following rules:
3.56/2.07	
3.56/2.07	      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1
3.56/2.07	
3.56/2.07	      2: f1 -> f1 : A'=B, B'=1+B, [ B==A ], cost: 1
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	   Accelerated rule 1 with metering function A-B, yielding the new rule 5.
3.56/2.07	
3.56/2.07	   Accelerated rule 2 with metering function 1+A-B, yielding the new rule 6.
3.56/2.07	
3.56/2.07	   Removing the simple loops: 1 2.
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Accelerated all simple loops using metering functions (where possible):
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      0: f2 -> f1 : [], cost: 1
3.56/2.07	
3.56/2.07	      5: f1 -> f1 : B'=A, [ A>=1+B ], cost: A-B
3.56/2.07	
3.56/2.07	      6: f1 -> f1 : A'=A, B'=1+A, [ B==A ], cost: 1+A-B
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Chained accelerated rules (with incoming rules):
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      0: f2 -> f1 : [], cost: 1
3.56/2.07	
3.56/2.07	      7: f2 -> f1 : B'=A, [ A>=1+B ], cost: 1+A-B
3.56/2.07	
3.56/2.07	      8: f2 -> f1 : B'=1+A, [ B==A ], cost: 2+A-B
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Removed unreachable locations (and leaf rules with constant cost):
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      7: f2 -> f1 : B'=A, [ A>=1+B ], cost: 1+A-B
3.56/2.07	
3.56/2.07	      8: f2 -> f1 : B'=1+A, [ B==A ], cost: 2+A-B
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	### Computing asymptotic complexity ###
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Fully simplified ITS problem
3.56/2.07	
3.56/2.07	   Start location: f2
3.56/2.07	
3.56/2.07	      7: f2 -> f1 : B'=A, [ A>=1+B ], cost: 1+A-B
3.56/2.07	
3.56/2.07	      8: f2 -> f1 : B'=1+A, [ B==A ], cost: 2+A-B
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Computing asymptotic complexity for rule 7
3.56/2.07	
3.56/2.07	   Solved the limit problem by the following transformations:
3.56/2.07	
3.56/2.07	      Created initial limit problem:
3.56/2.07	
3.56/2.07	      1+A-B (+), A-B (+/+!) [not solved]
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	      removing all constraints (solved by SMT)
3.56/2.07	
3.56/2.07	      resulting limit problem: [solved]
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	      applying transformation rule (C) using substitution {A==0,B==-n}
3.56/2.07	
3.56/2.07	      resulting limit problem:
3.56/2.07	
3.56/2.07	      [solved]
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	   Solution:
3.56/2.07	
3.56/2.07	      A / 0
3.56/2.07	
3.56/2.07	      B / -n
3.56/2.07	
3.56/2.07	   Resulting cost 1+n has complexity: Poly(n^1)
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Found new complexity Poly(n^1).
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	Obtained the following overall complexity (w.r.t. the length of the input n):
3.56/2.07	
3.56/2.07	   Complexity:  Poly(n^1)
3.56/2.07	
3.56/2.07	   Cpx degree:  1
3.56/2.07	
3.56/2.07	   Solved cost: 1+n
3.56/2.07	
3.56/2.07	   Rule cost:   1+A-B
3.56/2.07	
3.56/2.07	   Rule guard:  [ A>=1+B ]
3.56/2.07	
3.56/2.07	
3.56/2.07	
3.56/2.07	WORST_CASE(Omega(n^1),?)
3.56/2.07	
3.56/2.07	
3.56/2.07	----------------------------------------
3.56/2.07	
3.56/2.07	(4)
3.56/2.07	BOUNDS(n^1, INF)
3.56/2.08	EOF