/* Commandline execution:
 *		civl verify -inputnum_elements=5 derivative.cvl
 * */
#include<civlc.cvh>
#include<stdio.h>

$input double h;
$input int num_elements = 5;
$input double initial[num_elements];
double working[num_elements];
$abstract $contin(3) $real rho($real x);
$assume(h > 0);

void differentiate(double h, int n, double y[], double result[]){
	int i;
	
	$assume($forall {m=0 .. n-1} y[m] == rho(m*h));
	for(i = 1; i < n-1; i++)
	{
		result[i] = (y[i+1]-y[i-1])/(2*h);
	}
	result[0] = (y[1]-y[0])/h; 
	result[n-1] = (y[n-1] - y[n-2])/h;
	$assert(($uniform {k=1 .. n-2} result[k]-$D[rho,{x,1}](k*h) == $O(h*h)));
}

void main() {
	differentiate(h, num_elements, initial, working);
}