4.45/2.30	WORST_CASE(Omega(n^1), O(n^1))
4.53/2.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.53/2.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.53/2.31	
4.53/2.31	
4.53/2.31	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 2 * Arg_0) + nat(Arg_1)).
4.53/2.31	
4.53/2.31	(0) CpxIntTrs
4.53/2.31	(1) Koat2 Proof [FINISHED, 242 ms]
4.53/2.31	(2) BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_1))
4.53/2.31	(3) Loat Proof [FINISHED, 622 ms]
4.53/2.31	(4) BOUNDS(n^1, INF)
4.53/2.31	
4.53/2.31	
4.53/2.31	----------------------------------------
4.53/2.31	
4.53/2.31	(0)
4.53/2.31	Obligation:
4.53/2.31	Complexity Int TRS consisting of the following rules:
4.53/2.31	eval(A, B) -> Com_1(eval(A - 1, B)) :|: A >= 1
4.53/2.31	eval(A, B) -> Com_1(eval(A - 1, B)) :|: B >= 1 && A >= 1
4.53/2.31	eval(A, B) -> Com_1(eval(A, B - 1)) :|: A >= 1 && 0 >= A && B >= 1
4.53/2.31	eval(A, B) -> Com_1(eval(A, B - 1)) :|: B >= 1 && 0 >= A
4.53/2.31	eval(A, B) -> Com_1(eval(A, B)) :|: A >= 1 && 0 >= A && 0 >= B
4.53/2.31	eval(A, B) -> Com_1(eval(A, B)) :|: B >= 1 && 0 >= A && 0 >= B
4.53/2.31	start(A, B) -> Com_1(eval(A, B)) :|: TRUE
4.53/2.31	
4.53/2.31	The start-symbols are:[start_2]
4.53/2.31	
4.53/2.31	
4.53/2.31	----------------------------------------
4.53/2.31	
4.53/2.31	(1) Koat2 Proof (FINISHED)
4.53/2.31	YES( ?, 1+2*max([0, Arg_0])+max([0, Arg_1]) {O(n)})
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Initial Complexity Problem:
4.53/2.31	
4.53/2.31	  Start:  start  
4.53/2.31	
4.53/2.31	  Program_Vars:  Arg_0, Arg_1  
4.53/2.31	
4.53/2.31	  Temp_Vars:    
4.53/2.31	
4.53/2.31	  Locations:  eval, start  
4.53/2.31	
4.53/2.31	  Transitions:
4.53/2.31	
4.53/2.31	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:1 <= Arg_0
4.53/2.31	
4.53/2.31	  eval(Arg_0,Arg_1) -> eval(Arg_0-1,Arg_1):|:1 <= Arg_1 && 1 <= Arg_0
4.53/2.31	
4.53/2.31	  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-1):|:1 <= Arg_1 && Arg_0 <= 0
4.53/2.31	
4.53/2.31	  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Timebounds: 
4.53/2.31	
4.53/2.31	  Overall timebound: 1+2*max([0, Arg_0])+max([0, Arg_1]) {O(n)}
4.53/2.31	
4.53/2.31	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.53/2.31	
4.53/2.31	  1: eval->eval: max([0, Arg_0]) {O(n)}
4.53/2.31	
4.53/2.31	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.53/2.31	
4.53/2.31	  6: start->eval: 1 {O(1)}
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Costbounds:
4.53/2.31	
4.53/2.31	  Overall costbound: 1+2*max([0, Arg_0])+max([0, Arg_1]) {O(n)}
4.53/2.31	
4.53/2.31	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.53/2.31	
4.53/2.31	  1: eval->eval: max([0, Arg_0]) {O(n)}
4.53/2.31	
4.53/2.31	  3: eval->eval: max([0, Arg_1]) {O(n)}
4.53/2.31	
4.53/2.31	  6: start->eval: 1 {O(1)}
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Sizebounds:
4.53/2.31	
4.53/2.31	`Lower:
4.53/2.31	
4.53/2.31	  0: eval->eval, Arg_0: 0 {O(1)}
4.53/2.31	
4.53/2.31	  0: eval->eval, Arg_1: min([1, Arg_1]) {O(n)}
4.53/2.31	
4.53/2.31	  1: eval->eval, Arg_0: 0 {O(1)}
4.53/2.31	
4.53/2.31	  1: eval->eval, Arg_1: 1 {O(1)}
4.53/2.31	
4.53/2.31	  3: eval->eval, Arg_0: min([0, Arg_0]) {O(n)}
4.53/2.31	
4.53/2.31	  3: eval->eval, Arg_1: 0 {O(1)}
4.53/2.31	
4.53/2.31	  6: start->eval, Arg_0: Arg_0 {O(n)}
4.53/2.31	
4.53/2.31	  6: start->eval, Arg_1: Arg_1 {O(n)}
4.53/2.31	
4.53/2.31	`Upper:
4.53/2.31	
4.53/2.31	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.53/2.31	
4.53/2.31	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.53/2.31	
4.53/2.31	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.53/2.31	
4.53/2.31	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.53/2.31	
4.53/2.31	  3: eval->eval, Arg_0: 0 {O(1)}
4.53/2.31	
4.53/2.31	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.53/2.31	
4.53/2.31	  6: start->eval, Arg_0: Arg_0 {O(n)}
4.53/2.31	
4.53/2.31	  6: start->eval, Arg_1: Arg_1 {O(n)}
4.53/2.31	
4.53/2.31	
4.53/2.31	----------------------------------------
4.53/2.31	
4.53/2.31	(2)
4.53/2.31	BOUNDS(1, max(1, 1 + 2 * Arg_0) + nat(Arg_1))
4.53/2.31	
4.53/2.31	----------------------------------------
4.53/2.31	
4.53/2.31	(3) Loat Proof (FINISHED)
4.53/2.31	
4.53/2.31	
4.53/2.31	### Pre-processing the ITS problem ###
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Initial linear ITS problem
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	      0: eval -> eval : A'=-1+A, [ A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      1: eval -> eval : A'=-1+A, [ B>=1 && A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      2: eval -> eval : B'=-1+B, [ A>=1 && 0>=A && B>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      3: eval -> eval : B'=-1+B, [ B>=1 && 0>=A ], cost: 1
4.53/2.31	
4.53/2.31	      4: eval -> eval : [ A>=1 && 0>=A && 0>=B ], cost: 1
4.53/2.31	
4.53/2.31	      5: eval -> eval : [ B>=1 && 0>=A && 0>=B ], cost: 1
4.53/2.31	
4.53/2.31	      6: start -> eval : [], cost: 1
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Removed rules with unsatisfiable guard:
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	      0: eval -> eval : A'=-1+A, [ A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      1: eval -> eval : A'=-1+A, [ B>=1 && A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      3: eval -> eval : B'=-1+B, [ B>=1 && 0>=A ], cost: 1
4.53/2.31	
4.53/2.31	      6: start -> eval : [], cost: 1
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	### Simplification by acceleration and chaining ###
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Accelerating simple loops of location 0.
4.53/2.31	
4.53/2.31	   Accelerating the following rules:
4.53/2.31	
4.53/2.31	      0: eval -> eval : A'=-1+A, [ A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      1: eval -> eval : A'=-1+A, [ B>=1 && A>=1 ], cost: 1
4.53/2.31	
4.53/2.31	      3: eval -> eval : B'=-1+B, [ B>=1 && 0>=A ], cost: 1
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	   Accelerated rule 0 with metering function A, yielding the new rule 7.
4.53/2.31	
4.53/2.31	   Accelerated rule 1 with metering function A, yielding the new rule 8.
4.53/2.31	
4.53/2.31	   Accelerated rule 3 with metering function B, yielding the new rule 9.
4.53/2.31	
4.53/2.31	   Removing the simple loops: 0 1 3.
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Accelerated all simple loops using metering functions (where possible):
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	      7: eval -> eval : A'=0, [ A>=1 ], cost: A
4.53/2.31	
4.53/2.31	      8: eval -> eval : A'=0, [ B>=1 && A>=1 ], cost: A
4.53/2.31	
4.53/2.31	      9: eval -> eval : B'=0, [ B>=1 && 0>=A ], cost: B
4.53/2.31	
4.53/2.31	      6: start -> eval : [], cost: 1
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Chained accelerated rules (with incoming rules):
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	      6: start -> eval : [], cost: 1
4.53/2.31	
4.53/2.31	     10: start -> eval : A'=0, [ A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     11: start -> eval : A'=0, [ B>=1 && A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     12: start -> eval : B'=0, [ B>=1 && 0>=A ], cost: 1+B
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Removed unreachable locations (and leaf rules with constant cost):
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	     10: start -> eval : A'=0, [ A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     11: start -> eval : A'=0, [ B>=1 && A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     12: start -> eval : B'=0, [ B>=1 && 0>=A ], cost: 1+B
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	### Computing asymptotic complexity ###
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Fully simplified ITS problem
4.53/2.31	
4.53/2.31	   Start location: start
4.53/2.31	
4.53/2.31	     10: start -> eval : A'=0, [ A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     11: start -> eval : A'=0, [ B>=1 && A>=1 ], cost: 1+A
4.53/2.31	
4.53/2.31	     12: start -> eval : B'=0, [ B>=1 && 0>=A ], cost: 1+B
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Computing asymptotic complexity for rule 10
4.53/2.31	
4.53/2.31	   Solved the limit problem by the following transformations:
4.53/2.31	
4.53/2.31	      Created initial limit problem:
4.53/2.31	
4.53/2.31	      A (+/+!), 1+A (+) [not solved]
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	      removing all constraints (solved by SMT)
4.53/2.31	
4.53/2.31	      resulting limit problem: [solved]
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	      applying transformation rule (C) using substitution {A==n}
4.53/2.31	
4.53/2.31	      resulting limit problem:
4.53/2.31	
4.53/2.31	      [solved]
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	   Solution:
4.53/2.31	
4.53/2.31	      A / n
4.53/2.31	
4.53/2.31	   Resulting cost 1+n has complexity: Poly(n^1)
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Found new complexity Poly(n^1).
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	Obtained the following overall complexity (w.r.t. the length of the input n):
4.53/2.31	
4.53/2.31	   Complexity:  Poly(n^1)
4.53/2.31	
4.53/2.31	   Cpx degree:  1
4.53/2.31	
4.53/2.31	   Solved cost: 1+n
4.53/2.31	
4.53/2.31	   Rule cost:   1+A
4.53/2.31	
4.53/2.31	   Rule guard:  [ A>=1 ]
4.53/2.31	
4.53/2.31	
4.53/2.31	
4.53/2.31	WORST_CASE(Omega(n^1),?)
4.53/2.31	
4.53/2.31	
4.53/2.31	----------------------------------------
4.53/2.31	
4.53/2.31	(4)
4.53/2.31	BOUNDS(n^1, INF)
4.53/2.34	EOF