4.63/2.36	WORST_CASE(Omega(n^1), O(n^1))
4.63/2.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.63/2.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.63/2.37	
4.63/2.37	
4.63/2.37	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 4 * Arg_1)).
4.63/2.37	
4.63/2.37	(0) CpxIntTrs
4.63/2.37	(1) Koat2 Proof [FINISHED, 436 ms]
4.63/2.37	(2) BOUNDS(1, max(1, 1 + 4 * Arg_1))
4.63/2.37	(3) Loat Proof [FINISHED, 820 ms]
4.63/2.37	(4) BOUNDS(n^1, INF)
4.63/2.37	
4.63/2.37	
4.63/2.37	----------------------------------------
4.63/2.37	
4.63/2.37	(0)
4.63/2.37	Obligation:
4.63/2.37	Complexity Int TRS consisting of the following rules:
4.63/2.37	eval(A, B) -> Com_1(eval(A, A)) :|: 0 >= A && B >= 1 && B <= 1
4.63/2.37	eval(A, B) -> Com_1(eval(A, A)) :|: B >= 1 && 1 + B >= 0 && B >= A + 1
4.63/2.37	eval(A, B) -> Com_1(eval(A, 0)) :|: A >= 1 && B >= 1 && B <= 1
4.63/2.37	eval(A, B) -> Com_1(eval(A, B - 1)) :|: B >= 1 && 1 + B >= 0 && A >= B
4.63/2.37	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.63/2.37	
4.63/2.37	The start-symbols are:[start_2]
4.63/2.37	
4.63/2.37	
4.63/2.37	----------------------------------------
4.63/2.37	
4.63/2.37	(1) Koat2 Proof (FINISHED)
4.63/2.37	YES( ?, 1+2*2*max([0, Arg_1]) {O(n)})
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Initial Complexity Problem:
4.63/2.37	
4.63/2.37	  Start:  start  
4.63/2.37	
4.63/2.37	  Program_Vars:  Arg_0, Arg_1  
4.63/2.37	
4.63/2.37	  Temp_Vars:    
4.63/2.37	
4.63/2.37	  Locations:  eval, start  
4.63/2.37	
4.63/2.37	  Transitions:
4.63/2.37	
4.63/2.37	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_0):|:Arg_0 <= 0 && Arg_1 <= 1 && 1 <= Arg_1
4.63/2.37	
4.63/2.37	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_0):|:1 <= Arg_1 && 0 <= 1+Arg_1 && Arg_0+1 <= Arg_1
4.63/2.37	
4.63/2.37	  eval(Arg_0,Arg_1) -> eval(Arg_0,0):|:1 <= Arg_0 && Arg_1 <= 1 && 1 <= Arg_1
4.63/2.37	
4.63/2.37	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-1):|:1 <= Arg_1 && 0 <= 1+Arg_1 && Arg_1 <= Arg_0
4.63/2.37	
4.63/2.37	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Timebounds: 
4.63/2.37	
4.63/2.37	  Overall timebound: 1+2*2*max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  0: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  2: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  4: start->eval: 1 {O(1)}
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Costbounds:
4.63/2.37	
4.63/2.37	  Overall costbound: 1+2*2*max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  0: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  2: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  4: start->eval: 1 {O(1)}
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Sizebounds:
4.63/2.37	
4.63/2.37	`Lower:
4.63/2.37	
4.63/2.37	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  0: eval->eval, Arg_1: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  1: eval->eval, Arg_1: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  2: eval->eval, Arg_0: 1 {O(1)}
4.63/2.37	
4.63/2.37	  2: eval->eval, Arg_1: 0 {O(1)}
4.63/2.37	
4.63/2.37	  3: eval->eval, Arg_0: 1 {O(1)}
4.63/2.37	
4.63/2.37	  3: eval->eval, Arg_1: 0 {O(1)}
4.63/2.37	
4.63/2.37	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.63/2.37	
4.63/2.37	`Upper:
4.63/2.37	
4.63/2.37	  0: eval->eval, Arg_0: 0 {O(1)}
4.63/2.37	
4.63/2.37	  0: eval->eval, Arg_1: 0 {O(1)}
4.63/2.37	
4.63/2.37	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  1: eval->eval, Arg_1: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  2: eval->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  2: eval->eval, Arg_1: 0 {O(1)}
4.63/2.37	
4.63/2.37	  3: eval->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  3: eval->eval, Arg_1: max([Arg_0, Arg_1]) {O(n)}
4.63/2.37	
4.63/2.37	  4: start->eval, Arg_0: Arg_0 {O(n)}
4.63/2.37	
4.63/2.37	  4: start->eval, Arg_1: Arg_1 {O(n)}
4.63/2.37	
4.63/2.37	
4.63/2.37	----------------------------------------
4.63/2.37	
4.63/2.37	(2)
4.63/2.37	BOUNDS(1, max(1, 1 + 4 * Arg_1))
4.63/2.37	
4.63/2.37	----------------------------------------
4.63/2.37	
4.63/2.37	(3) Loat Proof (FINISHED)
4.63/2.37	
4.63/2.37	
4.63/2.37	### Pre-processing the ITS problem ###
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Initial linear ITS problem
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	      0: eval -> eval : B'=A, [ 0>=A && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      1: eval -> eval : B'=A, [ B>=1 && 1+B>=0 && B>=1+A ], cost: 1
4.63/2.37	
4.63/2.37	      2: eval -> eval : B'=0, [ A>=1 && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      3: eval -> eval : B'=-1+B, [ B>=1 && 1+B>=0 && A>=B ], cost: 1
4.63/2.37	
4.63/2.37	      4: start -> eval : [], cost: 1
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Simplified all rules, resulting in:
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	      0: eval -> eval : B'=A, [ 0>=A && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      1: eval -> eval : B'=A, [ B>=1 && B>=1+A ], cost: 1
4.63/2.37	
4.63/2.37	      2: eval -> eval : B'=0, [ A>=1 && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      3: eval -> eval : B'=-1+B, [ B>=1 && A>=B ], cost: 1
4.63/2.37	
4.63/2.37	      4: start -> eval : [], cost: 1
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	### Simplification by acceleration and chaining ###
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Accelerating simple loops of location 0.
4.63/2.37	
4.63/2.37	   Accelerating the following rules:
4.63/2.37	
4.63/2.37	      0: eval -> eval : B'=A, [ 0>=A && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      1: eval -> eval : B'=A, [ B>=1 && B>=1+A ], cost: 1
4.63/2.37	
4.63/2.37	      2: eval -> eval : B'=0, [ A>=1 && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      3: eval -> eval : B'=-1+B, [ B>=1 && A>=B ], cost: 1
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	   Accelerated rule 0 with NONTERM (after strengthening guard), yielding the new rule 5.
4.63/2.37	
4.63/2.37	   Found no metering function for rule 1.
4.63/2.37	
4.63/2.37	   Accelerated rule 2 with metering function B, yielding the new rule 6.
4.63/2.37	
4.63/2.37	   Accelerated rule 3 with metering function B, yielding the new rule 7.
4.63/2.37	
4.63/2.37	   Nested simple loops 1 (outer loop) and 6 (inner loop) with metering function 1-A, resulting in the new rules: 8.
4.63/2.37	
4.63/2.37	   Removing the simple loops: 1 2 3.
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Accelerated all simple loops using metering functions (where possible):
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	      0: eval -> eval : B'=A, [ 0>=A && B==1 ], cost: 1
4.63/2.37	
4.63/2.37	      5: eval -> [2] : [ 0>=A && B==1 && A==1 ], cost: INF
4.63/2.37	
4.63/2.37	      6: eval -> eval : B'=0, [ A>=1 && B==1 ], cost: B
4.63/2.37	
4.63/2.37	      7: eval -> eval : B'=0, [ B>=1 && A>=B ], cost: B
4.63/2.37	
4.63/2.37	      8: eval -> eval : B'=0, [ B>=1 && B>=1+A && A==1 && 1-A>=1 ], cost: 1-(-1+A)*A-A
4.63/2.37	
4.63/2.37	      4: start -> eval : [], cost: 1
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Chained accelerated rules (with incoming rules):
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	      4: start -> eval : [], cost: 1
4.63/2.37	
4.63/2.37	      9: start -> eval : B'=A, [ 0>=A && B==1 ], cost: 2
4.63/2.37	
4.63/2.37	     10: start -> eval : B'=0, [ A>=1 && B==1 ], cost: 1+B
4.63/2.37	
4.63/2.37	     11: start -> eval : B'=0, [ B>=1 && A>=B ], cost: 1+B
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Removed unreachable locations (and leaf rules with constant cost):
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	     10: start -> eval : B'=0, [ A>=1 && B==1 ], cost: 1+B
4.63/2.37	
4.63/2.37	     11: start -> eval : B'=0, [ B>=1 && A>=B ], cost: 1+B
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	### Computing asymptotic complexity ###
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Fully simplified ITS problem
4.63/2.37	
4.63/2.37	   Start location: start
4.63/2.37	
4.63/2.37	     10: start -> eval : B'=0, [ A>=1 && B==1 ], cost: 1+B
4.63/2.37	
4.63/2.37	     11: start -> eval : B'=0, [ B>=1 && A>=B ], cost: 1+B
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Computing asymptotic complexity for rule 10
4.63/2.37	
4.63/2.37	   Could not solve the limit problem.
4.63/2.37	
4.63/2.37	   Resulting cost 0 has complexity: Unknown
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Computing asymptotic complexity for rule 11
4.63/2.37	
4.63/2.37	   Solved the limit problem by the following transformations:
4.63/2.37	
4.63/2.37	      Created initial limit problem:
4.63/2.37	
4.63/2.37	      1+B (+), 1+A-B (+/+!), B (+/+!) [not solved]
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	      removing all constraints (solved by SMT)
4.63/2.37	
4.63/2.37	      resulting limit problem: [solved]
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	      applying transformation rule (C) using substitution {A==n,B==n}
4.63/2.37	
4.63/2.37	      resulting limit problem:
4.63/2.37	
4.63/2.37	      [solved]
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	   Solution:
4.63/2.37	
4.63/2.37	      A / n
4.63/2.37	
4.63/2.37	      B / n
4.63/2.37	
4.63/2.37	   Resulting cost 1+n has complexity: Poly(n^1)
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Found new complexity Poly(n^1).
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	Obtained the following overall complexity (w.r.t. the length of the input n):
4.63/2.37	
4.63/2.37	   Complexity:  Poly(n^1)
4.63/2.37	
4.63/2.37	   Cpx degree:  1
4.63/2.37	
4.63/2.37	   Solved cost: 1+n
4.63/2.37	
4.63/2.37	   Rule cost:   1+B
4.63/2.37	
4.63/2.37	   Rule guard:  [ B>=1 && A>=B ]
4.63/2.37	
4.63/2.37	
4.63/2.37	
4.63/2.37	WORST_CASE(Omega(n^1),?)
4.63/2.37	
4.63/2.37	
4.63/2.37	----------------------------------------
4.63/2.37	
4.63/2.37	(4)
4.63/2.37	BOUNDS(n^1, INF)
4.63/2.39	EOF