3.83/2.33	WORST_CASE(?, O(n^1))
3.83/2.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.83/2.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.83/2.33	
3.83/2.33	
3.83/2.33	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(2, 2 + 5 * Arg_3)).
3.83/2.33	
3.83/2.33	(0) CpxIntTrs
3.83/2.33	(1) Koat2 Proof [FINISHED, 349 ms]
3.83/2.33	(2) BOUNDS(1, max(2, 2 + 5 * Arg_3))
3.83/2.33	
3.83/2.33	
3.83/2.33	----------------------------------------
3.83/2.33	
3.83/2.33	(0)
3.83/2.33	Obligation:
3.83/2.33	Complexity Int TRS consisting of the following rules:
3.83/2.33	sqrt(A, B, C, D) -> Com_1(f(0, 1, 1, D)) :|: TRUE
3.83/2.33	f(A, B, C, D) -> Com_1(f(A + 1, B + 2, C + B + 2, D)) :|: D >= C && B >= 0
3.83/2.33	f(A, B, C, D) -> Com_1(end(A, B, C, D)) :|: C >= D + 1
3.83/2.33	
3.83/2.33	The start-symbols are:[sqrt_4]
3.83/2.33	
3.83/2.33	
3.83/2.33	----------------------------------------
3.83/2.33	
3.83/2.33	(1) Koat2 Proof (FINISHED)
3.83/2.33	YES( ?, max([2, 2+5*Arg_3]) {O(n)})
3.83/2.33	
3.83/2.33	
3.83/2.33	
3.83/2.33	Initial Complexity Problem:
3.83/2.33	
3.83/2.33	  Start:  sqrt  
3.83/2.33	
3.83/2.33	  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3  
3.83/2.33	
3.83/2.33	  Temp_Vars:    
3.83/2.33	
3.83/2.33	  Locations:  end, f, sqrt  
3.83/2.33	
3.83/2.33	  Transitions:
3.83/2.33	
3.83/2.33	  f(Arg_0,Arg_1,Arg_2,Arg_3) -> end(Arg_0,Arg_1,Arg_2,Arg_3):|:1 <= Arg_2 && 2 <= Arg_1+Arg_2 && Arg_1 <= Arg_2 && 1 <= Arg_0+Arg_2 && 1+Arg_0 <= Arg_2 && 1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1+Arg_0 <= Arg_1 && 0 <= Arg_0 && Arg_3+1 <= Arg_2
3.83/2.33	
3.83/2.33	  f(Arg_0,Arg_1,Arg_2,Arg_3) -> f(Arg_0+1,Arg_1+2,Arg_2+Arg_1+2,Arg_3):|:1 <= Arg_2 && 2 <= Arg_1+Arg_2 && Arg_1 <= Arg_2 && 1 <= Arg_0+Arg_2 && 1+Arg_0 <= Arg_2 && 1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1+Arg_0 <= Arg_1 && 0 <= Arg_0 && Arg_2 <= Arg_3 && 0 <= Arg_1
3.83/2.33	
3.83/2.33	  sqrt(Arg_0,Arg_1,Arg_2,Arg_3) -> f(0,1,1,Arg_3):|:  
3.83/2.33	
3.83/2.33	
3.83/2.33	
3.83/2.33	Timebounds: 
3.83/2.33	
3.83/2.33	  Overall timebound: max([2, 2+5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  1: f->f: max([0, 5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  2: f->end: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f: 1 {O(1)}
3.83/2.33	
3.83/2.33	
3.83/2.33	
3.83/2.33	Costbounds:
3.83/2.33	
3.83/2.33	  Overall costbound: max([2, 2+5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  1: f->f: max([0, 5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  2: f->end: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f: 1 {O(1)}
3.83/2.33	
3.83/2.33	
3.83/2.33	
3.83/2.33	Sizebounds:
3.83/2.33	
3.83/2.33	`Lower:
3.83/2.33	
3.83/2.33	  1: f->f, Arg_0: 1 {O(1)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_1: 3 {O(1)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_2: 4 {O(1)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_3: 1 {O(1)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_0: 0 {O(1)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_1: 1 {O(1)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_2: 1 {O(1)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_3: min([1, Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_0: 0 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_1: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_2: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_3: Arg_3 {O(n)}
3.83/2.33	
3.83/2.33	`Upper:
3.83/2.33	
3.83/2.33	  1: f->f, Arg_0: max([0, 5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_1: max([1, 1+10*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_2: max([1, 2^(5*Arg_3)])*max([1, 1+5*Arg_3]) {O(EXP)}
3.83/2.33	
3.83/2.33	  1: f->f, Arg_3: Arg_3 {O(n)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_0: max([0, 5*Arg_3]) {O(n)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_1: max([1, max([1, 1+10*Arg_3])]) {O(n)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_2: max([1, max([1, 2^(5*Arg_3)])*max([1, 1+5*Arg_3])]) {O(EXP)}
3.83/2.33	
3.83/2.33	  2: f->end, Arg_3: Arg_3 {O(n)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_0: 0 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_1: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_2: 1 {O(1)}
3.83/2.33	
3.83/2.33	  0: sqrt->f, Arg_3: Arg_3 {O(n)}
3.83/2.33	
3.83/2.33	
3.83/2.33	----------------------------------------
3.83/2.33	
3.83/2.33	(2)
3.83/2.33	BOUNDS(1, max(2, 2 + 5 * Arg_3))
3.83/2.35	EOF