4.14/2.63	WORST_CASE(Omega(n^1), O(n^1))
4.14/2.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.14/2.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.14/2.63	
4.14/2.63	
4.14/2.63	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_1)).
4.14/2.63	
4.14/2.63	(0) CpxIntTrs
4.14/2.63	(1) Koat2 Proof [FINISHED, 85 ms]
4.14/2.63	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
4.14/2.63	(3) Loat Proof [FINISHED, 480 ms]
4.14/2.63	(4) BOUNDS(n^1, INF)
4.14/2.63	
4.14/2.63	
4.14/2.63	----------------------------------------
4.14/2.63	
4.14/2.63	(0)
4.14/2.63	Obligation:
4.14/2.63	Complexity Int TRS consisting of the following rules:
4.14/2.63	eval(A, B) -> Com_1(eval(A, A + B)) :|: A >= 1 && A >= B + 1
4.14/2.63	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.14/2.63	
4.14/2.63	The start-symbols are:[start_2]
4.14/2.63	
4.14/2.63	
4.14/2.63	----------------------------------------
4.14/2.63	
4.14/2.63	(1) Koat2 Proof (FINISHED)
4.14/2.63	YES( ?, max([1, 1+Arg_0-Arg_1]) {O(n)})
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	Initial Complexity Problem:
4.14/2.63	
4.14/2.63	  Start:  start  
4.14/2.63	
4.14/2.63	  Program_Vars:  Arg_0, Arg_1  
4.14/2.63	
4.14/2.63	  Temp_Vars:    
4.14/2.63	
4.14/2.63	  Locations:  eval, start  
4.14/2.63	
4.14/2.63	  Transitions:
4.14/2.63	
4.14/2.63	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_0+Arg_1):|:1 <= Arg_0 && Arg_1+1 <= Arg_0
4.14/2.63	
4.14/2.63	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	Timebounds: 
4.14/2.63	
4.14/2.63	  Overall timebound: max([1, 1+Arg_0-Arg_1]) {O(n)}
4.14/2.63	
4.14/2.63	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval: 1 {O(1)}
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	Costbounds:
4.14/2.63	
4.14/2.63	  Overall costbound: max([1, 1+Arg_0-Arg_1]) {O(n)}
4.14/2.63	
4.14/2.63	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval: 1 {O(1)}
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	Sizebounds:
4.14/2.63	
4.14/2.63	`Lower:
4.14/2.63	
4.14/2.63	  0: eval->eval, Arg_0: 1 {O(1)}
4.14/2.63	
4.14/2.63	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.14/2.63	
4.14/2.63	`Upper:
4.14/2.63	
4.14/2.63	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.14/2.63	
4.14/2.63	  0: eval->eval, Arg_1: 2*Arg_0 {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval, Arg_0: Arg_0 {O(n)}
4.14/2.63	
4.14/2.63	  1: start->eval, Arg_1: Arg_1 {O(n)}
4.14/2.63	
4.14/2.63	
4.14/2.63	----------------------------------------
4.14/2.63	
4.14/2.63	(2)
4.14/2.63	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_1))
4.14/2.63	
4.14/2.63	----------------------------------------
4.14/2.63	
4.14/2.63	(3) Loat Proof (FINISHED)
4.14/2.63	
4.14/2.63	
4.14/2.63	### Pre-processing the ITS problem ###
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	Initial linear ITS problem
4.14/2.63	
4.14/2.63	   Start location: start
4.14/2.63	
4.14/2.63	      0: eval -> eval : B'=A+B, [ A>=1 && A>=1+B ], cost: 1
4.14/2.63	
4.14/2.63	      1: start -> eval : [], cost: 1
4.14/2.63	
4.14/2.63	
4.14/2.63	
4.14/2.63	### Simplification by acceleration and chaining ###
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Accelerating simple loops of location 0.
4.14/2.64	
4.14/2.64	   Accelerating the following rules:
4.14/2.64	
4.14/2.64	      0: eval -> eval : B'=A+B, [ A>=1 && A>=1+B ], cost: 1
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	   Accelerated rule 0 with backward acceleration, yielding the new rule 2.
4.14/2.64	
4.14/2.64	   Removing the simple loops: 0.
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Accelerated all simple loops using metering functions (where possible):
4.14/2.64	
4.14/2.64	   Start location: start
4.14/2.64	
4.14/2.64	      2: eval -> eval : B'=k*A+B, [ A>=1 && A>=1+B && k>0 && A>=1+(-1+k)*A+B ], cost: k
4.14/2.64	
4.14/2.64	      1: start -> eval : [], cost: 1
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Chained accelerated rules (with incoming rules):
4.14/2.64	
4.14/2.64	   Start location: start
4.14/2.64	
4.14/2.64	      1: start -> eval : [], cost: 1
4.14/2.64	
4.14/2.64	      3: start -> eval : B'=k*A+B, [ A>=1 && A>=1+B && k>0 && A>=1+(-1+k)*A+B ], cost: 1+k
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Removed unreachable locations (and leaf rules with constant cost):
4.14/2.64	
4.14/2.64	   Start location: start
4.14/2.64	
4.14/2.64	      3: start -> eval : B'=k*A+B, [ A>=1 && A>=1+B && k>0 && A>=1+(-1+k)*A+B ], cost: 1+k
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	### Computing asymptotic complexity ###
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Fully simplified ITS problem
4.14/2.64	
4.14/2.64	   Start location: start
4.14/2.64	
4.14/2.64	      3: start -> eval : B'=k*A+B, [ A>=1 && A>=1+B && k>0 && A>=1+(-1+k)*A+B ], cost: 1+k
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Computing asymptotic complexity for rule 3
4.14/2.64	
4.14/2.64	   Solved the limit problem by the following transformations:
4.14/2.64	
4.14/2.64	      Created initial limit problem:
4.14/2.64	
4.14/2.64	      -(-1+k)*A+A-B (+/+!), k (+/+!), 1+k (+), A (+/+!), A-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      removing all constraints (solved by SMT)
4.14/2.64	
4.14/2.64	      resulting limit problem: [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {k==1+n,A==1,B==-n}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	   Solved the limit problem by the following transformations:
4.14/2.64	
4.14/2.64	      Created initial limit problem:
4.14/2.64	
4.14/2.64	      -(-1+k)*A+A-B (+/+!), k (+/+!), 1+k (+), A (+/+!), A-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {A==1}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      1 (+/+!), k (+/+!), 1-B (+/+!), 1+k (+), 2-k-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {A==1+B}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      1 (+/+!), k (+/+!), 1-B (+/+!), 1+k (+), 2-k-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (B), deleting 1 (+/+!)
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      k (+/+!), 1-B (+/+!), 1+k (+), 2-k-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      removing all constraints (solved by SMT)
4.14/2.64	
4.14/2.64	      resulting limit problem: [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {k==n,B==-n}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	   Solved the limit problem by the following transformations:
4.14/2.64	
4.14/2.64	      Created initial limit problem:
4.14/2.64	
4.14/2.64	      -(-1+k)*A+A-B (+/+!), k (+/+!), 1+k (+), A (+/+!), A-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {A==1}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      1 (+/+!), k (+/+!), 1-B (+/+!), 1+k (+), 2-k-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (B), deleting 1 (+/+!)
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      k (+/+!), 1-B (+/+!), 1+k (+), 2-k-B (+/+!) [not solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      removing all constraints (solved by SMT)
4.14/2.64	
4.14/2.64	      resulting limit problem: [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	      applying transformation rule (C) using substitution {k==n,B==-n}
4.14/2.64	
4.14/2.64	      resulting limit problem:
4.14/2.64	
4.14/2.64	      [solved]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	   Solution:
4.14/2.64	
4.14/2.64	      k / 1+n
4.14/2.64	
4.14/2.64	      A / 1
4.14/2.64	
4.14/2.64	      B / -n
4.14/2.64	
4.14/2.64	   Resulting cost 2+n has complexity: Poly(n^1)
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Found new complexity Poly(n^1).
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	Obtained the following overall complexity (w.r.t. the length of the input n):
4.14/2.64	
4.14/2.64	   Complexity:  Poly(n^1)
4.14/2.64	
4.14/2.64	   Cpx degree:  1
4.14/2.64	
4.14/2.64	   Solved cost: 2+n
4.14/2.64	
4.14/2.64	   Rule cost:   1+k
4.14/2.64	
4.14/2.64	   Rule guard:  [ A>=1 && A>=1+B && k>0 && A>=1+(-1+k)*A+B ]
4.14/2.64	
4.14/2.64	
4.14/2.64	
4.14/2.64	WORST_CASE(Omega(n^1),?)
4.14/2.64	
4.14/2.64	
4.14/2.64	----------------------------------------
4.14/2.64	
4.14/2.64	(4)
4.14/2.64	BOUNDS(n^1, INF)
4.14/2.66	EOF