4.30/2.25	WORST_CASE(Omega(n^1), O(n^1))
4.40/2.26	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.40/2.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.40/2.26	
4.40/2.26	
4.40/2.26	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1) + nat(Arg_0)).
4.40/2.26	
4.40/2.26	(0) CpxIntTrs
4.40/2.26	(1) Koat2 Proof [FINISHED, 128 ms]
4.40/2.26	(2) BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.40/2.26	(3) Loat Proof [FINISHED, 609 ms]
4.40/2.26	(4) BOUNDS(n^1, INF)
4.40/2.26	
4.40/2.26	
4.40/2.26	----------------------------------------
4.40/2.26	
4.40/2.26	(0)
4.40/2.26	Obligation:
4.40/2.26	Complexity Int TRS consisting of the following rules:
4.40/2.26	eval(A, B) -> Com_1(eval(A - 1, B)) :|: A + B >= 1 && A >= 1
4.40/2.26	eval(A, B) -> Com_1(eval(A, B - 1)) :|: A + B >= 1 && 0 >= A && B >= 1
4.40/2.26	eval(A, B) -> Com_1(eval(A, B)) :|: A + B >= 1 && 0 >= A && 0 >= B
4.40/2.26	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.40/2.26	
4.40/2.26	The start-symbols are:[start_2]
4.40/2.26	
4.40/2.26	
4.40/2.26	----------------------------------------
4.40/2.26	
4.40/2.26	(1) Koat2 Proof (FINISHED)
4.40/2.26	YES( ?, 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)})
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Initial Complexity Problem:
4.40/2.26	
4.40/2.26	  Start:  start  
4.40/2.26	
4.40/2.26	  Program_Vars:  Arg_0, Arg_1  
4.40/2.26	
4.40/2.26	  Temp_Vars:    
4.40/2.26	
4.40/2.26	  Locations:  eval, start  
4.40/2.26	
4.40/2.26	  Transitions:
4.40/2.26	
4.40/2.26	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:1 <= Arg_0+Arg_1 && 1 <= Arg_0
4.40/2.26	
4.40/2.26	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-1):|:1 <= Arg_0+Arg_1 && Arg_0 <= 0 && 1 <= Arg_1
4.40/2.26	
4.40/2.26	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Timebounds: 
4.40/2.26	
4.40/2.26	  Overall timebound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.40/2.26	
4.40/2.26	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.40/2.26	
4.40/2.26	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.40/2.26	
4.40/2.26	  3: start->eval: 1 {O(1)}
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Costbounds:
4.40/2.26	
4.40/2.26	  Overall costbound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.40/2.26	
4.40/2.26	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.40/2.26	
4.40/2.26	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.40/2.26	
4.40/2.26	  3: start->eval: 1 {O(1)}
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Sizebounds:
4.40/2.26	
4.40/2.26	`Lower:
4.40/2.26	
4.40/2.26	  0: eval->eval, Arg_0: 0 {O(1)}
4.40/2.26	
4.40/2.26	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.40/2.26	
4.40/2.26	  1: eval->eval, Arg_0: min([0, Arg_0]) {O(n)}
4.40/2.26	
4.40/2.26	  1: eval->eval, Arg_1: 0 {O(1)}
4.40/2.26	
4.40/2.26	  3: start->eval, Arg_0: Arg_0 {O(n)}
4.40/2.26	
4.40/2.26	  3: start->eval, Arg_1: Arg_1 {O(n)}
4.40/2.26	
4.40/2.26	`Upper:
4.40/2.26	
4.40/2.26	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.40/2.26	
4.40/2.26	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.40/2.26	
4.40/2.26	  1: eval->eval, Arg_0: 0 {O(1)}
4.40/2.26	
4.40/2.26	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.40/2.26	
4.40/2.26	  3: start->eval, Arg_0: Arg_0 {O(n)}
4.40/2.26	
4.40/2.26	  3: start->eval, Arg_1: Arg_1 {O(n)}
4.40/2.26	
4.40/2.26	
4.40/2.26	----------------------------------------
4.40/2.26	
4.40/2.26	(2)
4.40/2.26	BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.40/2.26	
4.40/2.26	----------------------------------------
4.40/2.26	
4.40/2.26	(3) Loat Proof (FINISHED)
4.40/2.26	
4.40/2.26	
4.40/2.26	### Pre-processing the ITS problem ###
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Initial linear ITS problem
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1 ], cost: 1
4.40/2.26	
4.40/2.26	      1: eval -> eval : B'=-1+B, [ A+B>=1 && 0>=A && B>=1 ], cost: 1
4.40/2.26	
4.40/2.26	      2: eval -> eval : [ A+B>=1 && 0>=A && 0>=B ], cost: 1
4.40/2.26	
4.40/2.26	      3: start -> eval : [], cost: 1
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Removed rules with unsatisfiable guard:
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1 ], cost: 1
4.40/2.26	
4.40/2.26	      1: eval -> eval : B'=-1+B, [ A+B>=1 && 0>=A && B>=1 ], cost: 1
4.40/2.26	
4.40/2.26	      3: start -> eval : [], cost: 1
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	### Simplification by acceleration and chaining ###
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Accelerating simple loops of location 0.
4.40/2.26	
4.40/2.26	   Accelerating the following rules:
4.40/2.26	
4.40/2.26	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1 ], cost: 1
4.40/2.26	
4.40/2.26	      1: eval -> eval : B'=-1+B, [ A+B>=1 && 0>=A ], cost: 1
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	   Accelerated rule 0 with backward acceleration, yielding the new rule 4.
4.40/2.26	
4.40/2.26	   Accelerated rule 0 with backward acceleration, yielding the new rule 5.
4.40/2.26	
4.40/2.26	   Accelerated rule 1 with metering function A+B, yielding the new rule 6.
4.40/2.26	
4.40/2.26	   Removing the simple loops: 0 1.
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Accelerated all simple loops using metering functions (where possible):
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      4: eval -> eval : A'=-B, [ A+B>=1 && A>=1 && 1-B>=1 ], cost: A+B
4.40/2.26	
4.40/2.26	      5: eval -> eval : A'=0, [ A+B>=1 && A>=1 && 1+B>=1 ], cost: A
4.40/2.26	
4.40/2.26	      6: eval -> eval : B'=-A, [ A+B>=1 && 0>=A ], cost: A+B
4.40/2.26	
4.40/2.26	      3: start -> eval : [], cost: 1
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Chained accelerated rules (with incoming rules):
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      3: start -> eval : [], cost: 1
4.40/2.26	
4.40/2.26	      7: start -> eval : A'=-B, [ A+B>=1 && A>=1 && 1-B>=1 ], cost: 1+A+B
4.40/2.26	
4.40/2.26	      8: start -> eval : A'=0, [ A+B>=1 && A>=1 && 1+B>=1 ], cost: 1+A
4.40/2.26	
4.40/2.26	      9: start -> eval : B'=-A, [ A+B>=1 && 0>=A ], cost: 1+A+B
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Removed unreachable locations (and leaf rules with constant cost):
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      7: start -> eval : A'=-B, [ A+B>=1 && A>=1 && 1-B>=1 ], cost: 1+A+B
4.40/2.26	
4.40/2.26	      8: start -> eval : A'=0, [ A+B>=1 && A>=1 && 1+B>=1 ], cost: 1+A
4.40/2.26	
4.40/2.26	      9: start -> eval : B'=-A, [ A+B>=1 && 0>=A ], cost: 1+A+B
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	### Computing asymptotic complexity ###
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Fully simplified ITS problem
4.40/2.26	
4.40/2.26	   Start location: start
4.40/2.26	
4.40/2.26	      7: start -> eval : A'=-B, [ A+B>=1 && A>=1 && 1-B>=1 ], cost: 1+A+B
4.40/2.26	
4.40/2.26	      8: start -> eval : A'=0, [ A+B>=1 && A>=1 && 1+B>=1 ], cost: 1+A
4.40/2.26	
4.40/2.26	      9: start -> eval : B'=-A, [ A+B>=1 && 0>=A ], cost: 1+A+B
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Computing asymptotic complexity for rule 7
4.40/2.26	
4.40/2.26	   Solved the limit problem by the following transformations:
4.40/2.26	
4.40/2.26	      Created initial limit problem:
4.40/2.26	
4.40/2.26	      1+A+B (+), A+B (+/+!), 1-B (+/+!), A (+/+!) [not solved]
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	      removing all constraints (solved by SMT)
4.40/2.26	
4.40/2.26	      resulting limit problem: [solved]
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	      applying transformation rule (C) using substitution {A==2*n,B==-n}
4.40/2.26	
4.40/2.26	      resulting limit problem:
4.40/2.26	
4.40/2.26	      [solved]
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	   Solution:
4.40/2.26	
4.40/2.26	      A / 2*n
4.40/2.26	
4.40/2.26	      B / -n
4.40/2.26	
4.40/2.26	   Resulting cost 1+n has complexity: Poly(n^1)
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Found new complexity Poly(n^1).
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	Obtained the following overall complexity (w.r.t. the length of the input n):
4.40/2.26	
4.40/2.26	   Complexity:  Poly(n^1)
4.40/2.26	
4.40/2.26	   Cpx degree:  1
4.40/2.26	
4.40/2.26	   Solved cost: 1+n
4.40/2.26	
4.40/2.26	   Rule cost:   1+A+B
4.40/2.26	
4.40/2.26	   Rule guard:  [ A+B>=1 && A>=1 && 1-B>=1 ]
4.40/2.26	
4.40/2.26	
4.40/2.26	
4.40/2.26	WORST_CASE(Omega(n^1),?)
4.40/2.26	
4.40/2.26	
4.40/2.26	----------------------------------------
4.40/2.26	
4.40/2.26	(4)
4.40/2.26	BOUNDS(n^1, INF)
4.40/2.28	EOF