4.31/2.29	WORST_CASE(Omega(n^1), O(n^1))
4.31/2.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.31/2.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.31/2.30	
4.31/2.30	
4.31/2.30	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1) + nat(Arg_0)).
4.31/2.30	
4.31/2.30	(0) CpxIntTrs
4.31/2.30	(1) Koat2 Proof [FINISHED, 242 ms]
4.31/2.30	(2) BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.31/2.30	(3) Loat Proof [FINISHED, 647 ms]
4.31/2.30	(4) BOUNDS(n^1, INF)
4.31/2.30	
4.31/2.30	
4.31/2.30	----------------------------------------
4.31/2.30	
4.31/2.30	(0)
4.31/2.30	Obligation:
4.31/2.30	Complexity Int TRS consisting of the following rules:
4.31/2.30	eval(A, B, C) -> Com_1(eval(A - 1, B, C)) :|: A + B >= C + 1 && C >= 0 && A >= 1
4.31/2.30	eval(A, B, C) -> Com_1(eval(A, B - 1, C)) :|: A + B >= C + 1 && C >= 0 && 0 >= A && B >= 1
4.31/2.30	eval(A, B, C) -> Com_1(eval(A, B, C)) :|: A + B >= C + 1 && C >= 0 && 0 >= A && 0 >= B
4.31/2.30	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
4.31/2.30	
4.31/2.30	The start-symbols are:[start_3]
4.31/2.30	
4.31/2.30	
4.31/2.30	----------------------------------------
4.31/2.30	
4.31/2.30	(1) Koat2 Proof (FINISHED)
4.31/2.30	YES( ?, 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)})
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Initial Complexity Problem:
4.31/2.30	
4.31/2.30	  Start:  start  
4.31/2.30	
4.31/2.30	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.31/2.30	
4.31/2.30	  Temp_Vars:    
4.31/2.30	
4.31/2.30	  Locations:  eval, start  
4.31/2.30	
4.31/2.30	  Transitions:
4.31/2.30	
4.31/2.30	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0-1,Arg_1,Arg_2):|:Arg_2+1 <= Arg_0+Arg_1 && 0 <= Arg_2 && 1 <= Arg_0
4.31/2.30	
4.31/2.30	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1-1,Arg_2):|:Arg_2+1 <= Arg_0+Arg_1 && 0 <= Arg_2 && Arg_0 <= 0 && 1 <= Arg_1
4.31/2.30	
4.31/2.30	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Timebounds: 
4.31/2.30	
4.31/2.30	  Overall timebound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.31/2.30	
4.31/2.30	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.31/2.30	
4.31/2.30	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval: 1 {O(1)}
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Costbounds:
4.31/2.30	
4.31/2.30	  Overall costbound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}
4.31/2.30	
4.31/2.30	  0: eval->eval: max([0, Arg_0]) {O(n)}
4.31/2.30	
4.31/2.30	  1: eval->eval: max([0, Arg_1]) {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval: 1 {O(1)}
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Sizebounds:
4.31/2.30	
4.31/2.30	`Lower:
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_0: 0 {O(1)}
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_2: 0 {O(1)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_0: min([0, Arg_0]) {O(n)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_1: 0 {O(1)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_2: 0 {O(1)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_0: Arg_0 {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_1: Arg_1 {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_2: Arg_2 {O(n)}
4.31/2.30	
4.31/2.30	`Upper:
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.31/2.30	
4.31/2.30	  0: eval->eval, Arg_2: Arg_2 {O(n)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_0: 0 {O(1)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.31/2.30	
4.31/2.30	  1: eval->eval, Arg_2: Arg_2 {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_0: Arg_0 {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_1: Arg_1 {O(n)}
4.31/2.30	
4.31/2.30	  3: start->eval, Arg_2: Arg_2 {O(n)}
4.31/2.30	
4.31/2.30	
4.31/2.30	----------------------------------------
4.31/2.30	
4.31/2.30	(2)
4.31/2.30	BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
4.31/2.30	
4.31/2.30	----------------------------------------
4.31/2.30	
4.31/2.30	(3) Loat Proof (FINISHED)
4.31/2.30	
4.31/2.30	
4.31/2.30	### Pre-processing the ITS problem ###
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Initial linear ITS problem
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      0: eval -> eval : A'=-1+A, [ A+B>=1+C && C>=0 && A>=1 ], cost: 1
4.31/2.30	
4.31/2.30	      1: eval -> eval : B'=-1+B, [ A+B>=1+C && C>=0 && 0>=A && B>=1 ], cost: 1
4.31/2.30	
4.31/2.30	      2: eval -> eval : [ A+B>=1+C && C>=0 && 0>=A && 0>=B ], cost: 1
4.31/2.30	
4.31/2.30	      3: start -> eval : [], cost: 1
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Removed rules with unsatisfiable guard:
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      0: eval -> eval : A'=-1+A, [ A+B>=1+C && C>=0 && A>=1 ], cost: 1
4.31/2.30	
4.31/2.30	      1: eval -> eval : B'=-1+B, [ A+B>=1+C && C>=0 && 0>=A && B>=1 ], cost: 1
4.31/2.30	
4.31/2.30	      3: start -> eval : [], cost: 1
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	### Simplification by acceleration and chaining ###
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Accelerating simple loops of location 0.
4.31/2.30	
4.31/2.30	   Accelerating the following rules:
4.31/2.30	
4.31/2.30	      0: eval -> eval : A'=-1+A, [ A+B>=1+C && C>=0 && A>=1 ], cost: 1
4.31/2.30	
4.31/2.30	      1: eval -> eval : B'=-1+B, [ A+B>=1+C && C>=0 && 0>=A ], cost: 1
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	   Accelerated rule 0 with backward acceleration, yielding the new rule 4.
4.31/2.30	
4.31/2.30	   Accelerated rule 0 with backward acceleration, yielding the new rule 5.
4.31/2.30	
4.31/2.30	   Accelerated rule 1 with metering function -C+A+B, yielding the new rule 6.
4.31/2.30	
4.31/2.30	   Removing the simple loops: 0 1.
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Accelerated all simple loops using metering functions (where possible):
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      4: eval -> eval : A'=C-B, [ A+B>=1+C && C>=0 && A>=1 && 1+C-B>=1 ], cost: -C+A+B
4.31/2.30	
4.31/2.30	      5: eval -> eval : A'=0, [ A+B>=1+C && C>=0 && A>=1 && 1+B>=1+C ], cost: A
4.31/2.30	
4.31/2.30	      6: eval -> eval : B'=C-A, [ A+B>=1+C && C>=0 && 0>=A ], cost: -C+A+B
4.31/2.30	
4.31/2.30	      3: start -> eval : [], cost: 1
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Chained accelerated rules (with incoming rules):
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      3: start -> eval : [], cost: 1
4.31/2.30	
4.31/2.30	      7: start -> eval : A'=C-B, [ A+B>=1+C && C>=0 && A>=1 && 1+C-B>=1 ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	      8: start -> eval : A'=0, [ A+B>=1+C && C>=0 && A>=1 && 1+B>=1+C ], cost: 1+A
4.31/2.30	
4.31/2.30	      9: start -> eval : B'=C-A, [ A+B>=1+C && C>=0 && 0>=A ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Removed unreachable locations (and leaf rules with constant cost):
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      7: start -> eval : A'=C-B, [ A+B>=1+C && C>=0 && A>=1 && 1+C-B>=1 ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	      8: start -> eval : A'=0, [ A+B>=1+C && C>=0 && A>=1 && 1+B>=1+C ], cost: 1+A
4.31/2.30	
4.31/2.30	      9: start -> eval : B'=C-A, [ A+B>=1+C && C>=0 && 0>=A ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	### Computing asymptotic complexity ###
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Fully simplified ITS problem
4.31/2.30	
4.31/2.30	   Start location: start
4.31/2.30	
4.31/2.30	      7: start -> eval : A'=C-B, [ A+B>=1+C && C>=0 && A>=1 && 1+C-B>=1 ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	      8: start -> eval : A'=0, [ A+B>=1+C && C>=0 && A>=1 && 1+B>=1+C ], cost: 1+A
4.31/2.30	
4.31/2.30	      9: start -> eval : B'=C-A, [ A+B>=1+C && C>=0 && 0>=A ], cost: 1-C+A+B
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Computing asymptotic complexity for rule 7
4.31/2.30	
4.31/2.30	   Solved the limit problem by the following transformations:
4.31/2.30	
4.31/2.30	      Created initial limit problem:
4.31/2.30	
4.31/2.30	      1+C (+/+!), 1+C-B (+/+!), A (+/+!), 1-C+A+B (+), -C+A+B (+/+!) [not solved]
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	      removing all constraints (solved by SMT)
4.31/2.30	
4.31/2.30	      resulting limit problem: [solved]
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	      applying transformation rule (C) using substitution {C==n,A==n,B==n}
4.31/2.30	
4.31/2.30	      resulting limit problem:
4.31/2.30	
4.31/2.30	      [solved]
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	   Solution:
4.31/2.30	
4.31/2.30	      C / n
4.31/2.30	
4.31/2.30	      A / n
4.31/2.30	
4.31/2.30	      B / n
4.31/2.30	
4.31/2.30	   Resulting cost 1+n has complexity: Poly(n^1)
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Found new complexity Poly(n^1).
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	Obtained the following overall complexity (w.r.t. the length of the input n):
4.31/2.30	
4.31/2.30	   Complexity:  Poly(n^1)
4.31/2.30	
4.31/2.30	   Cpx degree:  1
4.31/2.30	
4.31/2.30	   Solved cost: 1+n
4.31/2.30	
4.31/2.30	   Rule cost:   1-C+A+B
4.31/2.30	
4.31/2.30	   Rule guard:  [ A+B>=1+C && C>=0 && A>=1 && 1+C-B>=1 ]
4.31/2.30	
4.31/2.30	
4.31/2.30	
4.31/2.30	WORST_CASE(Omega(n^1),?)
4.31/2.30	
4.31/2.30	
4.31/2.30	----------------------------------------
4.31/2.30	
4.31/2.30	(4)
4.31/2.30	BOUNDS(n^1, INF)
4.42/2.32	EOF