4.26/2.14	WORST_CASE(Omega(n^1), O(n^1))
4.26/2.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.26/2.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.26/2.15	
4.26/2.15	
4.26/2.15	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1)).
4.26/2.15	
4.26/2.15	(0) CpxIntTrs
4.26/2.15	(1) Koat2 Proof [FINISHED, 237 ms]
4.26/2.15	(2) BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1))
4.26/2.15	(3) Loat Proof [FINISHED, 547 ms]
4.26/2.15	(4) BOUNDS(n^1, INF)
4.26/2.15	
4.26/2.15	
4.26/2.15	----------------------------------------
4.26/2.15	
4.26/2.15	(0)
4.26/2.15	Obligation:
4.26/2.15	Complexity Int TRS consisting of the following rules:
4.26/2.15	eval(A, B, C) -> Com_1(eval(A, B + 1, C)) :|: A >= B + 1 && C >= B + 1
4.26/2.15	eval(A, B, C) -> Com_1(eval(A, B, C + 1)) :|: A >= B + 1 && B >= C
4.26/2.15	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
4.26/2.15	
4.26/2.15	The start-symbols are:[start_3]
4.26/2.15	
4.26/2.15	
4.26/2.15	----------------------------------------
4.26/2.15	
4.26/2.15	(1) Koat2 Proof (FINISHED)
4.26/2.15	YES( ?, 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)})
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Initial Complexity Problem:
4.26/2.15	
4.26/2.15	  Start:  start  
4.26/2.15	
4.26/2.15	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.26/2.15	
4.26/2.15	  Temp_Vars:    
4.26/2.15	
4.26/2.15	  Locations:  eval, start  
4.26/2.15	
4.26/2.15	  Transitions:
4.26/2.15	
4.26/2.15	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1+1,Arg_2):|:Arg_1+1 <= Arg_0 && Arg_1+1 <= Arg_2
4.26/2.15	
4.26/2.15	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2+1):|:Arg_1+1 <= Arg_0 && Arg_2 <= Arg_1
4.26/2.15	
4.26/2.15	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Timebounds: 
4.26/2.15	
4.26/2.15	  Overall timebound: 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval: max([0, Arg_0-Arg_2]) {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval: 1 {O(1)}
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Costbounds:
4.26/2.15	
4.26/2.15	  Overall costbound: 1+max([0, Arg_0-Arg_2])+max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval: max([0, Arg_0-Arg_2]) {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval: 1 {O(1)}
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Sizebounds:
4.26/2.15	
4.26/2.15	`Lower:
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_2: Arg_2 {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_2: Arg_2 {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_1: Arg_1 {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_2: Arg_2 {O(n)}
4.26/2.15	
4.26/2.15	`Upper:
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  0: eval->eval, Arg_2: Arg_2+max([0, Arg_0-Arg_2]) {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_1: Arg_1+max([0, Arg_0-Arg_1]) {O(n)}
4.26/2.15	
4.26/2.15	  1: eval->eval, Arg_2: Arg_2+max([0, Arg_0-Arg_2]) {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_0: Arg_0 {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_1: Arg_1 {O(n)}
4.26/2.15	
4.26/2.15	  2: start->eval, Arg_2: Arg_2 {O(n)}
4.26/2.15	
4.26/2.15	
4.26/2.15	----------------------------------------
4.26/2.15	
4.26/2.15	(2)
4.26/2.15	BOUNDS(1, max(1, 1 + Arg_0 + -1 * Arg_2) + nat(Arg_0 + -1 * Arg_1))
4.26/2.15	
4.26/2.15	----------------------------------------
4.26/2.15	
4.26/2.15	(3) Loat Proof (FINISHED)
4.26/2.15	
4.26/2.15	
4.26/2.15	### Pre-processing the ITS problem ###
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Initial linear ITS problem
4.26/2.15	
4.26/2.15	   Start location: start
4.26/2.15	
4.26/2.15	      0: eval -> eval : B'=1+B, [ A>=1+B && C>=1+B ], cost: 1
4.26/2.15	
4.26/2.15	      1: eval -> eval : C'=1+C, [ A>=1+B && B>=C ], cost: 1
4.26/2.15	
4.26/2.15	      2: start -> eval : [], cost: 1
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	### Simplification by acceleration and chaining ###
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Accelerating simple loops of location 0.
4.26/2.15	
4.26/2.15	   Accelerating the following rules:
4.26/2.15	
4.26/2.15	      0: eval -> eval : B'=1+B, [ A>=1+B && C>=1+B ], cost: 1
4.26/2.15	
4.26/2.15	      1: eval -> eval : C'=1+C, [ A>=1+B && B>=C ], cost: 1
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	   Accelerated rule 0 with backward acceleration, yielding the new rule 3.
4.26/2.15	
4.26/2.15	   Accelerated rule 0 with backward acceleration, yielding the new rule 4.
4.26/2.15	
4.26/2.15	   Accelerated rule 1 with metering function 1-C+B, yielding the new rule 5.
4.26/2.15	
4.26/2.15	   Removing the simple loops: 0 1.
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Accelerated all simple loops using metering functions (where possible):
4.26/2.15	
4.26/2.15	   Start location: start
4.26/2.15	
4.26/2.15	      3: eval -> eval : B'=A, [ A>=1+B && C>=1+B && C>=A ], cost: A-B
4.26/2.15	
4.26/2.15	      4: eval -> eval : B'=C, [ A>=1+B && C>=1+B && A>=C ], cost: C-B
4.26/2.15	
4.26/2.15	      5: eval -> eval : C'=1+B, [ A>=1+B && B>=C ], cost: 1-C+B
4.26/2.15	
4.26/2.15	      2: start -> eval : [], cost: 1
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Chained accelerated rules (with incoming rules):
4.26/2.15	
4.26/2.15	   Start location: start
4.26/2.15	
4.26/2.15	      2: start -> eval : [], cost: 1
4.26/2.15	
4.26/2.15	      6: start -> eval : B'=A, [ A>=1+B && C>=1+B && C>=A ], cost: 1+A-B
4.26/2.15	
4.26/2.15	      7: start -> eval : B'=C, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
4.26/2.15	
4.26/2.15	      8: start -> eval : C'=1+B, [ A>=1+B && B>=C ], cost: 2-C+B
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Removed unreachable locations (and leaf rules with constant cost):
4.26/2.15	
4.26/2.15	   Start location: start
4.26/2.15	
4.26/2.15	      6: start -> eval : B'=A, [ A>=1+B && C>=1+B && C>=A ], cost: 1+A-B
4.26/2.15	
4.26/2.15	      7: start -> eval : B'=C, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
4.26/2.15	
4.26/2.15	      8: start -> eval : C'=1+B, [ A>=1+B && B>=C ], cost: 2-C+B
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	### Computing asymptotic complexity ###
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Fully simplified ITS problem
4.26/2.15	
4.26/2.15	   Start location: start
4.26/2.15	
4.26/2.15	      6: start -> eval : B'=A, [ A>=1+B && C>=1+B && C>=A ], cost: 1+A-B
4.26/2.15	
4.26/2.15	      7: start -> eval : B'=C, [ A>=1+B && C>=1+B && A>=C ], cost: 1+C-B
4.26/2.15	
4.26/2.15	      8: start -> eval : C'=1+B, [ A>=1+B && B>=C ], cost: 2-C+B
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Computing asymptotic complexity for rule 6
4.26/2.15	
4.26/2.15	   Solved the limit problem by the following transformations:
4.26/2.15	
4.26/2.15	      Created initial limit problem:
4.26/2.15	
4.26/2.15	      C-B (+/+!), 1+C-A (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	      removing all constraints (solved by SMT)
4.26/2.15	
4.26/2.15	      resulting limit problem: [solved]
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	      applying transformation rule (C) using substitution {C==0,A==0,B==-n}
4.26/2.15	
4.26/2.15	      resulting limit problem:
4.26/2.15	
4.26/2.15	      [solved]
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	   Solution:
4.26/2.15	
4.26/2.15	      C / 0
4.26/2.15	
4.26/2.15	      A / 0
4.26/2.15	
4.26/2.15	      B / -n
4.26/2.15	
4.26/2.15	   Resulting cost 1+n has complexity: Poly(n^1)
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Found new complexity Poly(n^1).
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	Obtained the following overall complexity (w.r.t. the length of the input n):
4.26/2.15	
4.26/2.15	   Complexity:  Poly(n^1)
4.26/2.15	
4.26/2.15	   Cpx degree:  1
4.26/2.15	
4.26/2.15	   Solved cost: 1+n
4.26/2.15	
4.26/2.15	   Rule cost:   1+A-B
4.26/2.15	
4.26/2.15	   Rule guard:  [ A>=1+B && C>=1+B && C>=A ]
4.26/2.15	
4.26/2.15	
4.26/2.15	
4.26/2.15	WORST_CASE(Omega(n^1),?)
4.26/2.15	
4.26/2.15	
4.26/2.15	----------------------------------------
4.26/2.15	
4.26/2.15	(4)
4.26/2.15	BOUNDS(n^1, INF)
4.36/2.17	EOF