4.76/2.64	WORST_CASE(Omega(n^1), O(n^1))
4.76/2.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.76/2.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.76/2.65	
4.76/2.65	
4.76/2.65	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 2 * Arg_0) + nat(Arg_0) + nat(2 * Arg_1)).
4.76/2.65	
4.76/2.65	(0) CpxIntTrs
4.76/2.65	(1) Koat2 Proof [FINISHED, 243 ms]
4.76/2.65	(2) BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_0) + nat(2 * Arg_1))
4.76/2.65	(3) Loat Proof [FINISHED, 1038 ms]
4.76/2.65	(4) BOUNDS(n^1, INF)
4.76/2.65	
4.76/2.65	
4.76/2.65	----------------------------------------
4.76/2.65	
4.76/2.65	(0)
4.76/2.65	Obligation:
4.76/2.65	Complexity Int TRS consisting of the following rules:
4.76/2.65	eval(A, B) -> Com_1(eval(A - 1, B)) :|: A + B >= 1 && A >= B + 1
4.76/2.65	eval(A, B) -> Com_1(eval(A - 1, B)) :|: 2 * A >= 1 && B >= A && B <= A
4.76/2.65	eval(A, B) -> Com_1(eval(A, B - 1)) :|: A + B >= 1 && B >= A && B >= A + 1
4.76/2.65	eval(A, B) -> Com_1(eval(A, B - 1)) :|: A + B >= 1 && B >= A && A >= B + 1
4.76/2.65	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.76/2.65	
4.76/2.65	The start-symbols are:[start_2]
4.76/2.65	
4.76/2.65	
4.76/2.65	----------------------------------------
4.76/2.65	
4.76/2.65	(1) Koat2 Proof (FINISHED)
4.76/2.65	YES( ?, 1+max([0, 2*Arg_0])+max([0, Arg_0])+max([0, 2*Arg_1]) {O(n)})
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Initial Complexity Problem:
4.76/2.65	
4.76/2.65	  Start:  start  
4.76/2.65	
4.76/2.65	  Program_Vars:  Arg_0, Arg_1  
4.76/2.65	
4.76/2.65	  Temp_Vars:    
4.76/2.65	
4.76/2.65	  Locations:  eval, start  
4.76/2.65	
4.76/2.65	  Transitions:
4.76/2.65	
4.76/2.65	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:1 <= Arg_0+Arg_1 && Arg_1+1 <= Arg_0
4.76/2.65	
4.76/2.65	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:1 <= (2)*Arg_0 && Arg_1 <= Arg_0 && Arg_0 <= Arg_1
4.76/2.65	
4.76/2.65	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-1):|:1 <= Arg_0+Arg_1 && Arg_0 <= Arg_1 && Arg_0+1 <= Arg_1
4.76/2.65	
4.76/2.65	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Timebounds: 
4.76/2.65	
4.76/2.65	  Overall timebound: 1+max([0, 2*Arg_0])+max([0, Arg_0])+max([0, 2*Arg_1]) {O(n)}
4.76/2.65	
4.76/2.65	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.76/2.65	
4.76/2.65	  1: eval->eval: max([0, 2*Arg_0]) {O(n)}
4.76/2.65	
4.76/2.65	  2: eval->eval: max([0, 2*Arg_1]) {O(n)}
4.76/2.65	
4.76/2.65	  4: start->eval: 1 {O(1)}
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Costbounds:
4.76/2.65	
4.76/2.65	  Overall costbound: 1+max([0, 2*Arg_0])+max([0, Arg_0])+max([0, 2*Arg_1]) {O(n)}
4.76/2.65	
4.76/2.65	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.76/2.65	
4.76/2.65	  1: eval->eval: max([0, 2*Arg_0]) {O(n)}
4.76/2.65	
4.76/2.65	  2: eval->eval: max([0, 2*Arg_1]) {O(n)}
4.76/2.65	
4.76/2.65	  4: start->eval: 1 {O(1)}
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Sizebounds:
4.76/2.65	
4.76/2.65	`Lower:
4.76/2.65	
4.76/2.65	  0: eval->eval, Arg_0: 0 {O(1)}
4.76/2.65	
4.76/2.65	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	  1: eval->eval, Arg_0: 0 {O(1)}
4.76/2.65	
4.76/2.65	  1: eval->eval, Arg_1: 1 {O(1)}
4.76/2.65	
4.76/2.65	  2: eval->eval, Arg_0: min([0, Arg_0]) {O(n)}
4.76/2.65	
4.76/2.65	  2: eval->eval, Arg_1: 0 {O(1)}
4.76/2.65	
4.76/2.65	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.76/2.65	
4.76/2.65	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	`Upper:
4.76/2.65	
4.76/2.65	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.76/2.65	
4.76/2.65	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.76/2.65	
4.76/2.65	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	  2: eval->eval, Arg_0: Arg_0 {O(n)}
4.76/2.65	
4.76/2.65	  2: eval->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.76/2.65	
4.76/2.65	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.76/2.65	
4.76/2.65	
4.76/2.65	----------------------------------------
4.76/2.65	
4.76/2.65	(2)
4.76/2.65	BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_0) + nat(2 * Arg_1))
4.76/2.65	
4.76/2.65	----------------------------------------
4.76/2.65	
4.76/2.65	(3) Loat Proof (FINISHED)
4.76/2.65	
4.76/2.65	
4.76/2.65	### Pre-processing the ITS problem ###
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Initial linear ITS problem
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1+B ], cost: 1
4.76/2.65	
4.76/2.65	      1: eval -> eval : A'=-1+A, [ 2*A>=1 && B==A ], cost: 1
4.76/2.65	
4.76/2.65	      2: eval -> eval : B'=-1+B, [ A+B>=1 && B>=A && B>=1+A ], cost: 1
4.76/2.65	
4.76/2.65	      3: eval -> eval : B'=-1+B, [ A+B>=1 && B>=A && A>=1+B ], cost: 1
4.76/2.65	
4.76/2.65	      4: start -> eval : [], cost: 1
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Removed rules with unsatisfiable guard:
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1+B ], cost: 1
4.76/2.65	
4.76/2.65	      1: eval -> eval : A'=-1+A, [ 2*A>=1 && B==A ], cost: 1
4.76/2.65	
4.76/2.65	      2: eval -> eval : B'=-1+B, [ A+B>=1 && B>=A && B>=1+A ], cost: 1
4.76/2.65	
4.76/2.65	      4: start -> eval : [], cost: 1
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Simplified all rules, resulting in:
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1+B ], cost: 1
4.76/2.65	
4.76/2.65	      1: eval -> eval : A'=-1+A, [ 2*A>=1 && B==A ], cost: 1
4.76/2.65	
4.76/2.65	      2: eval -> eval : B'=-1+B, [ A+B>=1 && B>=1+A ], cost: 1
4.76/2.65	
4.76/2.65	      4: start -> eval : [], cost: 1
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	### Simplification by acceleration and chaining ###
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Accelerating simple loops of location 0.
4.76/2.65	
4.76/2.65	   Accelerating the following rules:
4.76/2.65	
4.76/2.65	      0: eval -> eval : A'=-1+A, [ A+B>=1 && A>=1+B ], cost: 1
4.76/2.65	
4.76/2.65	      1: eval -> eval : A'=-1+A, [ 2*A>=1 && B==A ], cost: 1
4.76/2.65	
4.76/2.65	      2: eval -> eval : B'=-1+B, [ A+B>=1 && B>=1+A ], cost: 1
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	   Accelerated rule 0 with backward acceleration, yielding the new rule 5.
4.76/2.65	
4.76/2.65	   Accelerated rule 0 with backward acceleration, yielding the new rule 6.
4.76/2.65	
4.76/2.65	   Accelerated rule 1 with metering function A-B, yielding the new rule 7.
4.76/2.65	
4.76/2.65	   Accelerated rule 2 with backward acceleration, yielding the new rule 8.
4.76/2.65	
4.76/2.65	   Accelerated rule 2 with backward acceleration, yielding the new rule 9.
4.76/2.65	
4.76/2.65	   Removing the simple loops: 0 1 2.
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Accelerated all simple loops using metering functions (where possible):
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	      5: eval -> eval : A'=-B, [ A+B>=1 && A>=1+B && 1-B>=1+B ], cost: A+B
4.76/2.65	
4.76/2.65	      6: eval -> eval : A'=B, [ A+B>=1 && A>=1+B && 1+2*B>=1 ], cost: A-B
4.76/2.65	
4.76/2.65	      7: eval -> eval : A'=B, [ 2*A>=1 && B==A && A-B>=1 ], cost: A-B
4.76/2.65	
4.76/2.65	      8: eval -> eval : B'=-A, [ A+B>=1 && B>=1+A && 1-A>=1+A ], cost: A+B
4.76/2.65	
4.76/2.65	      9: eval -> eval : B'=A, [ A+B>=1 && B>=1+A && 1+2*A>=1 ], cost: -A+B
4.76/2.65	
4.76/2.65	      4: start -> eval : [], cost: 1
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Chained accelerated rules (with incoming rules):
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	      4: start -> eval : [], cost: 1
4.76/2.65	
4.76/2.65	     10: start -> eval : A'=-B, [ A+B>=1 && A>=1+B && 1-B>=1+B ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     11: start -> eval : A'=B, [ A+B>=1 && A>=1+B && 1+2*B>=1 ], cost: 1+A-B
4.76/2.65	
4.76/2.65	     12: start -> eval : B'=-A, [ A+B>=1 && B>=1+A && 1-A>=1+A ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     13: start -> eval : B'=A, [ A+B>=1 && B>=1+A && 1+2*A>=1 ], cost: 1-A+B
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Removed unreachable locations (and leaf rules with constant cost):
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	     10: start -> eval : A'=-B, [ A+B>=1 && A>=1+B && 1-B>=1+B ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     11: start -> eval : A'=B, [ A+B>=1 && A>=1+B && 1+2*B>=1 ], cost: 1+A-B
4.76/2.65	
4.76/2.65	     12: start -> eval : B'=-A, [ A+B>=1 && B>=1+A && 1-A>=1+A ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     13: start -> eval : B'=A, [ A+B>=1 && B>=1+A && 1+2*A>=1 ], cost: 1-A+B
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	### Computing asymptotic complexity ###
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Fully simplified ITS problem
4.76/2.65	
4.76/2.65	   Start location: start
4.76/2.65	
4.76/2.65	     10: start -> eval : A'=-B, [ A+B>=1 && A>=1+B && 1-B>=1+B ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     11: start -> eval : A'=B, [ A+B>=1 && A>=1+B && 1+2*B>=1 ], cost: 1+A-B
4.76/2.65	
4.76/2.65	     12: start -> eval : B'=-A, [ A+B>=1 && B>=1+A && 1-A>=1+A ], cost: 1+A+B
4.76/2.65	
4.76/2.65	     13: start -> eval : B'=A, [ A+B>=1 && B>=1+A && 1+2*A>=1 ], cost: 1-A+B
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Computing asymptotic complexity for rule 10
4.76/2.65	
4.76/2.65	   Solved the limit problem by the following transformations:
4.76/2.65	
4.76/2.65	      Created initial limit problem:
4.76/2.65	
4.76/2.65	      1+A+B (+), A+B (+/+!), 1-2*B (+/+!), A-B (+/+!) [not solved]
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	      removing all constraints (solved by SMT)
4.76/2.65	
4.76/2.65	      resulting limit problem: [solved]
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	      applying transformation rule (C) using substitution {A==2*n,B==-n}
4.76/2.65	
4.76/2.65	      resulting limit problem:
4.76/2.65	
4.76/2.65	      [solved]
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	   Solution:
4.76/2.65	
4.76/2.65	      A / 2*n
4.76/2.65	
4.76/2.65	      B / -n
4.76/2.65	
4.76/2.65	   Resulting cost 1+n has complexity: Poly(n^1)
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Found new complexity Poly(n^1).
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	Obtained the following overall complexity (w.r.t. the length of the input n):
4.76/2.65	
4.76/2.65	   Complexity:  Poly(n^1)
4.76/2.65	
4.76/2.65	   Cpx degree:  1
4.76/2.65	
4.76/2.65	   Solved cost: 1+n
4.76/2.65	
4.76/2.65	   Rule cost:   1+A+B
4.76/2.65	
4.76/2.65	   Rule guard:  [ A+B>=1 && A>=1+B && 1-B>=1+B ]
4.76/2.65	
4.76/2.65	
4.76/2.65	
4.76/2.65	WORST_CASE(Omega(n^1),?)
4.76/2.65	
4.76/2.65	
4.76/2.65	----------------------------------------
4.76/2.65	
4.76/2.65	(4)
4.76/2.65	BOUNDS(n^1, INF)
4.76/2.69	EOF