4.74/2.33	WORST_CASE(Omega(n^1), O(n^1))
4.74/2.34	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.74/2.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.74/2.34	
4.74/2.34	
4.74/2.34	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1) + nat(-2 * Arg_1 + 2 * Arg_2)).
4.74/2.34	
4.74/2.34	(0) CpxIntTrs
4.74/2.34	(1) Koat2 Proof [FINISHED, 394 ms]
4.74/2.34	(2) BOUNDS(1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1) + nat(-2 * Arg_1 + 2 * Arg_2))
4.74/2.34	(3) Loat Proof [FINISHED, 595 ms]
4.74/2.34	(4) BOUNDS(n^1, INF)
4.74/2.34	
4.74/2.34	
4.74/2.34	----------------------------------------
4.74/2.34	
4.74/2.34	(0)
4.74/2.34	Obligation:
4.74/2.34	Complexity Int TRS consisting of the following rules:
4.74/2.34	eval(A, B, C) -> Com_1(eval(A - 1, B, C)) :|: A >= B + 1
4.74/2.34	eval(A, B, C) -> Com_1(eval(A - 1, B, C)) :|: C >= B + 1 && A >= B + 1
4.74/2.34	eval(A, B, C) -> Com_1(eval(A, B, C - 1)) :|: A >= B + 1 && B >= A && C >= B + 1
4.74/2.34	eval(A, B, C) -> Com_1(eval(A, B, C - 1)) :|: C >= B + 1 && B >= A
4.74/2.34	eval(A, B, C) -> Com_1(eval(A, B, C)) :|: A >= B + 1 && B >= A && B >= C
4.74/2.34	eval(A, B, C) -> Com_1(eval(A, B, C)) :|: C >= B + 1 && B >= A && B >= C
4.74/2.34	start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE
4.74/2.34	
4.74/2.34	The start-symbols are:[start_3]
4.74/2.34	
4.74/2.34	
4.74/2.34	----------------------------------------
4.74/2.34	
4.74/2.34	(1) Koat2 Proof (FINISHED)
4.74/2.34	YES( ?, 1+2*max([0, Arg_0-Arg_1])+max([0, -2*Arg_1+2*Arg_2]) {O(n)})
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Initial Complexity Problem:
4.74/2.34	
4.74/2.34	  Start:  start  
4.74/2.34	
4.74/2.34	  Program_Vars:  Arg_0, Arg_1, Arg_2  
4.74/2.34	
4.74/2.34	  Temp_Vars:    
4.74/2.34	
4.74/2.34	  Locations:  eval, start  
4.74/2.34	
4.74/2.34	  Transitions:
4.74/2.34	
4.74/2.34	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0-1,Arg_1,Arg_2):|:Arg_1+1 <= Arg_0
4.74/2.34	
4.74/2.34	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0-1,Arg_1,Arg_2):|:Arg_1+1 <= Arg_2 && Arg_1+1 <= Arg_0
4.74/2.34	
4.74/2.34	  eval(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2-1):|:Arg_1+1 <= Arg_2 && Arg_0 <= Arg_1
4.74/2.34	
4.74/2.34	  start(Arg_0,Arg_1,Arg_2) -> eval(Arg_0,Arg_1,Arg_2):|:  
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Timebounds: 
4.74/2.34	
4.74/2.34	  Overall timebound: 1+2*max([0, Arg_0-Arg_1])+max([0, -2*Arg_1+2*Arg_2]) {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval: max([0, -2*Arg_1+2*Arg_2]) {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval: 1 {O(1)}
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Costbounds:
4.74/2.34	
4.74/2.34	  Overall costbound: 1+2*max([0, Arg_0-Arg_1])+max([0, -2*Arg_1+2*Arg_2]) {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval: max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval: max([0, -2*Arg_1+2*Arg_2]) {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval: 1 {O(1)}
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Sizebounds:
4.74/2.34	
4.74/2.34	`Lower:
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_0: Arg_0+-2*max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_0: Arg_0+-2*max([0, Arg_0-Arg_1]) {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_0: min([Arg_0, -(-(Arg_0)+2*max([0, Arg_0-Arg_1]))]) {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_2: Arg_2-max([0, -2*Arg_1+2*Arg_2]) {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_0: Arg_0 {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	`Upper:
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_0: Arg_0 {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  0: eval->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_0: Arg_0 {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  1: eval->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_0: Arg_0 {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  3: eval->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_0: Arg_0 {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_1: Arg_1 {O(n)}
4.74/2.34	
4.74/2.34	  6: start->eval, Arg_2: Arg_2 {O(n)}
4.74/2.34	
4.74/2.34	
4.74/2.34	----------------------------------------
4.74/2.34	
4.74/2.34	(2)
4.74/2.34	BOUNDS(1, max(1, 1 + 2 * Arg_0 + -2 * Arg_1) + nat(-2 * Arg_1 + 2 * Arg_2))
4.74/2.34	
4.74/2.34	----------------------------------------
4.74/2.34	
4.74/2.34	(3) Loat Proof (FINISHED)
4.74/2.34	
4.74/2.34	
4.74/2.34	### Pre-processing the ITS problem ###
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Initial linear ITS problem
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	      0: eval -> eval : A'=-1+A, [ A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      1: eval -> eval : A'=-1+A, [ C>=1+B && A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      2: eval -> eval : C'=-1+C, [ A>=1+B && B>=A && C>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      3: eval -> eval : C'=-1+C, [ C>=1+B && B>=A ], cost: 1
4.74/2.34	
4.74/2.34	      4: eval -> eval : [ A>=1+B && B>=A && B>=C ], cost: 1
4.74/2.34	
4.74/2.34	      5: eval -> eval : [ C>=1+B && B>=A && B>=C ], cost: 1
4.74/2.34	
4.74/2.34	      6: start -> eval : [], cost: 1
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Removed rules with unsatisfiable guard:
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	      0: eval -> eval : A'=-1+A, [ A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      1: eval -> eval : A'=-1+A, [ C>=1+B && A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      3: eval -> eval : C'=-1+C, [ C>=1+B && B>=A ], cost: 1
4.74/2.34	
4.74/2.34	      6: start -> eval : [], cost: 1
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	### Simplification by acceleration and chaining ###
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Accelerating simple loops of location 0.
4.74/2.34	
4.74/2.34	   Accelerating the following rules:
4.74/2.34	
4.74/2.34	      0: eval -> eval : A'=-1+A, [ A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      1: eval -> eval : A'=-1+A, [ C>=1+B && A>=1+B ], cost: 1
4.74/2.34	
4.74/2.34	      3: eval -> eval : C'=-1+C, [ C>=1+B && B>=A ], cost: 1
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	   Accelerated rule 0 with metering function A-B, yielding the new rule 7.
4.74/2.34	
4.74/2.34	   Accelerated rule 1 with metering function A-B, yielding the new rule 8.
4.74/2.34	
4.74/2.34	   Accelerated rule 3 with metering function C-B, yielding the new rule 9.
4.74/2.34	
4.74/2.34	   Removing the simple loops: 0 1 3.
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Accelerated all simple loops using metering functions (where possible):
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	      7: eval -> eval : A'=B, [ A>=1+B ], cost: A-B
4.74/2.34	
4.74/2.34	      8: eval -> eval : A'=B, [ C>=1+B && A>=1+B ], cost: A-B
4.74/2.34	
4.74/2.34	      9: eval -> eval : C'=B, [ C>=1+B && B>=A ], cost: C-B
4.74/2.34	
4.74/2.34	      6: start -> eval : [], cost: 1
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Chained accelerated rules (with incoming rules):
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	      6: start -> eval : [], cost: 1
4.74/2.34	
4.74/2.34	     10: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     11: start -> eval : A'=B, [ C>=1+B && A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     12: start -> eval : C'=B, [ C>=1+B && B>=A ], cost: 1+C-B
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Removed unreachable locations (and leaf rules with constant cost):
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	     10: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     11: start -> eval : A'=B, [ C>=1+B && A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     12: start -> eval : C'=B, [ C>=1+B && B>=A ], cost: 1+C-B
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	### Computing asymptotic complexity ###
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Fully simplified ITS problem
4.74/2.34	
4.74/2.34	   Start location: start
4.74/2.34	
4.74/2.34	     10: start -> eval : A'=B, [ A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     11: start -> eval : A'=B, [ C>=1+B && A>=1+B ], cost: 1+A-B
4.74/2.34	
4.74/2.34	     12: start -> eval : C'=B, [ C>=1+B && B>=A ], cost: 1+C-B
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Computing asymptotic complexity for rule 10
4.74/2.34	
4.74/2.34	   Solved the limit problem by the following transformations:
4.74/2.34	
4.74/2.34	      Created initial limit problem:
4.74/2.34	
4.74/2.34	      1+A-B (+), A-B (+/+!) [not solved]
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	      removing all constraints (solved by SMT)
4.74/2.34	
4.74/2.34	      resulting limit problem: [solved]
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	      applying transformation rule (C) using substitution {A==0,B==-n}
4.74/2.34	
4.74/2.34	      resulting limit problem:
4.74/2.34	
4.74/2.34	      [solved]
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	   Solution:
4.74/2.34	
4.74/2.34	      A / 0
4.74/2.34	
4.74/2.34	      B / -n
4.74/2.34	
4.74/2.34	   Resulting cost 1+n has complexity: Poly(n^1)
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Found new complexity Poly(n^1).
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	Obtained the following overall complexity (w.r.t. the length of the input n):
4.74/2.34	
4.74/2.34	   Complexity:  Poly(n^1)
4.74/2.34	
4.74/2.34	   Cpx degree:  1
4.74/2.34	
4.74/2.34	   Solved cost: 1+n
4.74/2.34	
4.74/2.34	   Rule cost:   1+A-B
4.74/2.34	
4.74/2.34	   Rule guard:  [ A>=1+B ]
4.74/2.34	
4.74/2.34	
4.74/2.34	
4.74/2.34	WORST_CASE(Omega(n^1),?)
4.74/2.34	
4.74/2.34	
4.74/2.34	----------------------------------------
4.74/2.34	
4.74/2.34	(4)
4.74/2.34	BOUNDS(n^1, INF)
4.82/3.29	EOF