3.89/1.92	WORST_CASE(Omega(n^1), O(n^1))
3.89/1.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.89/1.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.89/1.92	
3.89/1.92	
3.89/1.92	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1 + Arg_2)).
3.89/1.92	
3.89/1.92	(0) CpxIntTrs
3.89/1.92	(1) Koat2 Proof [FINISHED, 114 ms]
3.89/1.92	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1 + Arg_2))
3.89/1.92	(3) Loat Proof [FINISHED, 311 ms]
3.89/1.92	(4) BOUNDS(n^1, INF)
3.89/1.92	
3.89/1.92	
3.89/1.92	----------------------------------------
3.89/1.92	
3.89/1.92	(0)
3.89/1.92	Obligation:
3.89/1.92	Complexity Int TRS consisting of the following rules:
3.89/1.92	eval(A, B, C) -> Com_1(eval(A, B + C, C)) :|: A >= B && C >= 1
3.89/1.92	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
3.89/1.92	
3.89/1.92	The start-symbols are:[start_3]
3.89/1.92	
3.89/1.92	
3.89/1.92	----------------------------------------
3.89/1.92	
3.89/1.92	(1) Koat2 Proof (FINISHED)
3.89/1.92	YES( ?, max([1, 1+Arg_0+Arg_2-Arg_1]) {O(n)})
3.89/1.92	
3.89/1.92	
3.89/1.92	
3.89/1.92	Initial Complexity Problem:
3.89/1.92	
3.89/1.92	  Start:  start  
3.89/1.92	
3.89/1.92	  Program_Vars:  Arg_0, Arg_1, Arg_2  
3.89/1.92	
3.89/1.92	  Temp_Vars:    
3.89/1.92	
3.89/1.92	  Locations:  eval, start  
3.89/1.92	
3.89/1.92	  Transitions:
3.89/1.92	
3.89/1.92	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1+Arg_2,Arg_2):|:Arg_1 <= Arg_0 && 1 <= Arg_2
3.89/1.92	
3.89/1.92	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
3.89/1.92	
3.89/1.92	
3.89/1.92	
3.89/1.92	Timebounds: 
3.89/1.92	
3.89/1.92	  Overall timebound: max([1, 1+Arg_0+Arg_2-Arg_1]) {O(n)}
3.89/1.92	
3.89/1.92	  0: eval->eval: max([0, Arg_0+Arg_2-Arg_1]) {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval: 1 {O(1)}
3.89/1.92	
3.89/1.92	
3.89/1.92	
3.89/1.92	Costbounds:
3.89/1.92	
3.89/1.92	  Overall costbound: max([1, 1+Arg_0+Arg_2-Arg_1]) {O(n)}
3.89/1.92	
3.89/1.92	  0: eval->eval: max([0, Arg_0+Arg_2-Arg_1]) {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval: 1 {O(1)}
3.89/1.92	
3.89/1.92	
3.89/1.92	
3.89/1.92	Sizebounds:
3.89/1.92	
3.89/1.92	`Lower:
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_1: Arg_1 {O(n)}
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_2: 1 {O(1)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.89/1.92	
3.89/1.92	`Upper:
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_0: Arg_0 {O(n)}
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_1: max([Arg_2, max([Arg_2, Arg_1])])+max([0, Arg_2*(Arg_0+Arg_2-Arg_1)]) {O(n^2)}
3.89/1.92	
3.89/1.92	  0: eval->eval, Arg_2: Arg_2 {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_0: Arg_0 {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_1: Arg_1 {O(n)}
3.89/1.92	
3.89/1.92	  1: start->eval, Arg_2: Arg_2 {O(n)}
3.89/1.92	
3.89/1.92	
3.89/1.92	----------------------------------------
3.89/1.92	
3.89/1.92	(2)
3.89/1.92	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1 + Arg_2))
3.89/1.93	
3.89/1.93	----------------------------------------
3.89/1.93	
3.89/1.93	(3) Loat Proof (FINISHED)
3.89/1.93	
3.89/1.93	
3.89/1.93	### Pre-processing the ITS problem ###
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Initial linear ITS problem
3.89/1.93	
3.89/1.93	   Start location: start
3.89/1.93	
3.89/1.93	      0: eval -> eval : B'=C+B, [ A>=B && C>=1 ], cost: 1
3.89/1.93	
3.89/1.93	      1: start -> eval : [], cost: 1
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	### Simplification by acceleration and chaining ###
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Accelerating simple loops of location 0.
3.89/1.93	
3.89/1.93	   Accelerating the following rules:
3.89/1.93	
3.89/1.93	      0: eval -> eval : B'=C+B, [ A>=B && C>=1 ], cost: 1
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	   Accelerated rule 0 with backward acceleration, yielding the new rule 2.
3.89/1.93	
3.89/1.93	   Removing the simple loops: 0.
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Accelerated all simple loops using metering functions (where possible):
3.89/1.93	
3.89/1.93	   Start location: start
3.89/1.93	
3.89/1.93	      2: eval -> eval : B'=C*k+B, [ A>=B && C>=1 && k>0 && A>=C*(-1+k)+B ], cost: k
3.89/1.93	
3.89/1.93	      1: start -> eval : [], cost: 1
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Chained accelerated rules (with incoming rules):
3.89/1.93	
3.89/1.93	   Start location: start
3.89/1.93	
3.89/1.93	      1: start -> eval : [], cost: 1
3.89/1.93	
3.89/1.93	      3: start -> eval : B'=C*k+B, [ A>=B && C>=1 && k>0 && A>=C*(-1+k)+B ], cost: 1+k
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Removed unreachable locations (and leaf rules with constant cost):
3.89/1.93	
3.89/1.93	   Start location: start
3.89/1.93	
3.89/1.93	      3: start -> eval : B'=C*k+B, [ A>=B && C>=1 && k>0 && A>=C*(-1+k)+B ], cost: 1+k
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	### Computing asymptotic complexity ###
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Fully simplified ITS problem
3.89/1.93	
3.89/1.93	   Start location: start
3.89/1.93	
3.89/1.93	      3: start -> eval : B'=C*k+B, [ A>=B && C>=1 && k>0 && A>=C*(-1+k)+B ], cost: 1+k
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Computing asymptotic complexity for rule 3
3.89/1.93	
3.89/1.93	   Simplified the guard:
3.89/1.93	
3.89/1.93	      3: start -> eval : B'=C*k+B, [ C>=1 && k>0 && A>=C*(-1+k)+B ], cost: 1+k
3.89/1.93	
3.89/1.93	   Solved the limit problem by the following transformations:
3.89/1.93	
3.89/1.93	      Created initial limit problem:
3.89/1.93	
3.89/1.93	      1+A-C*(-1+k)-B (+/+!), C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      removing all constraints (solved by SMT)
3.89/1.93	
3.89/1.93	      resulting limit problem: [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {C==1,A==0,k==n,B==-2*n}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	   Solved the limit problem by the following transformations:
3.89/1.93	
3.89/1.93	      Created initial limit problem:
3.89/1.93	
3.89/1.93	      1+A-C*(-1+k)-B (+/+!), C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {C==1}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1 (+/+!), 1+k (+), k (+/+!), 2+A-k-B (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {A==C*(-1+k)+B}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1 (+/+!), 1+k (+), 2+C*(-1+k)-k (+/+!), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (B), deleting 1 (+/+!)
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1+k (+), 2+C*(-1+k)-k (+/+!), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      removing all constraints (solved by SMT)
3.89/1.93	
3.89/1.93	      resulting limit problem: [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {C==2,k==n}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	   Solved the limit problem by the following transformations:
3.89/1.93	
3.89/1.93	      Created initial limit problem:
3.89/1.93	
3.89/1.93	      1+A-C*(-1+k)-B (+/+!), C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {A==C*(-1+k)+B}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1 (+/+!), C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (B), deleting 1 (+/+!)
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      removing all constraints (solved by SMT)
3.89/1.93	
3.89/1.93	      resulting limit problem: [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {C==1,k==n}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	   Solved the limit problem by the following transformations:
3.89/1.93	
3.89/1.93	      Created initial limit problem:
3.89/1.93	
3.89/1.93	      1+A-C*(-1+k)-B (+/+!), C (+/+!), 1+k (+), k (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {C==1}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1 (+/+!), 1+k (+), k (+/+!), 2+A-k-B (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (B), deleting 1 (+/+!)
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      1+k (+), k (+/+!), 2+A-k-B (+/+!) [not solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      removing all constraints (solved by SMT)
3.89/1.93	
3.89/1.93	      resulting limit problem: [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	      applying transformation rule (C) using substitution {A==0,k==n,B==-n}
3.89/1.93	
3.89/1.93	      resulting limit problem:
3.89/1.93	
3.89/1.93	      [solved]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	   Solution:
3.89/1.93	
3.89/1.93	      C / 1
3.89/1.93	
3.89/1.93	      A / 0
3.89/1.93	
3.89/1.93	      k / n
3.89/1.93	
3.89/1.93	      B / -2*n
3.89/1.93	
3.89/1.93	   Resulting cost 1+n has complexity: Poly(n^1)
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Found new complexity Poly(n^1).
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	Obtained the following overall complexity (w.r.t. the length of the input n):
3.89/1.93	
3.89/1.93	   Complexity:  Poly(n^1)
3.89/1.93	
3.89/1.93	   Cpx degree:  1
3.89/1.93	
3.89/1.93	   Solved cost: 1+n
3.89/1.93	
3.89/1.93	   Rule cost:   1+k
3.89/1.93	
3.89/1.93	   Rule guard:  [ C>=1 && k>0 && A>=C*(-1+k)+B ]
3.89/1.93	
3.89/1.93	
3.89/1.93	
3.89/1.93	WORST_CASE(Omega(n^1),?)
3.89/1.93	
3.89/1.93	
3.89/1.93	----------------------------------------
3.89/1.93	
3.89/1.93	(4)
3.89/1.93	BOUNDS(n^1, INF)
3.93/1.95	EOF