# Examples

## Bouncing particle †

// y stands for the height of the ball
INIT   <=> y = 10 & y' = 0. // initial state
FALL   <=> [](y'' = -10).   // falling
BOUNCE <=> [](y- = 0 => y' = -4/5*y'-).
// if the ball reaches the ground, it bounces
INIT, FALL << BOUNCE.     // FALL is weaker than BOUNCE

## Bouncing particle with a parametric initial height †

INIT   <=> 5 < y < 15 & y' = 0. // the initial height is uncertain
FALL   <=> [](y'' = -10).
BOUNCE <=> [](y- = 0 => y' = -4/5 * y'-).

INIT, FALL << BOUNCE.

## Bouncing particle in a parabolic vase †

/**
*  bouncing particle on a curve of f(x) = (1/2) * x^2
*/

INIT <=> x = 1/2 & y = 10 & x' = 0 & y' = 0 & [](k = 1).
A    <=> [](x'' = 0 & y'' = -98/10).
/**
*     sin = f'(x) / (1+f'(x))^(1/2)
*     cos =     1 / (1+f'(x))^(1/2)
*  new x' = (-k * sin^2 + cos^2) * x' + (k+1) * sin * cos      * y'
*  new y' = (k+1) * sin * cos    * x' + (sin^2 + (-k) * cos^2) * y'
*/

SC <=> [](s = (x-)/(1+(x-)^2)^(1/2)
& c = 1   /(1+(x-)^2)^(1/2)).

BOUNCE <=> [](y- = (1/2) * (x-)^2 =>
x' = ( (-k) * s^2 + c^2 ) * x'- + ( (k+1) * s * c ) * y'-
& y' = ( (k+1) * s * c ) * x'- + ( s^2 + (-k) * c^2 ) * y'- ).

INIT, SC, A << BOUNCE.

## Bouncing particle in a circle †

INIT   <=> x = 0 /\ 0.5 < y < 0.6 /\ x' = 2 /\ y' = 1.
// the initial position is uncertain
RUN    <=> [](x'' = 0 /\ y'' = 0).
BOUNCE <=> []((x-)^2 + (y-)^2 = 1 =>
x' = x'- - (x- * x'- + y- * y'-) * 2 * (x-)
/\ y' = y'- - (x- * x'- + y- * y'-) * 2 * (y-)
).
INIT, RUN<<BOUNCE.

## Hot-Air Balloon with multiple parameters †

/* A program for a hot-air balloon that repeats rising and falling */
// The initial condition of the balloon and the timer
// h: height of the balloon
// timer: timer variable to repeat rising and falling
INIT <=> h = 10 /\ h' = 0 /\ timer = 0.

// parameters
// duration: duration of falling
// riseT: duration of rising
PARAM<=> 1 < fallT < 4 /\ 1 < riseT < 3
/\ [](riseT' = 0 /\ fallT' = 0).

// increasing of timer
TIMER <=> [](timer' = 1).

// rising of the balloon
RISE <=> [](timer- < riseT =>h'' = 1).

// falling of the balloon
FALL <=> [](timer- >= riseT => h'' = -2).

// reset the timer to repeat rising and falling
RESET <=> [](timer- >= riseT + fallT => timer=0).

// assertion for bounded model checking
ASSERT(h > 0).

// constraint hierarchies
INIT, PARAM, FALL, RISE, TIMER<<RESET.

## Bouncing particle with magnetic force †

INIT <=> y=10 & y'=0 & mag=0 & timer=0.
FALL <=> [](y''=-10+mag).
BOUNCE <=> [](y-=0=>y'=-y'-).
TRUE <=> [](1=1).
TIMER <=> [](mag'=0&timer'=1).
SWITCHON  <=> [](timer-=1=>mag=12&timer=0).
// The magnetic force may be switched on at every one second
SWITCHOFF <=> [](timer-=1=>mag=0&timer=0).
// The magnetic force may be switched off at every one second

INIT,TIMER<<(SWITCHOFF,SWITCHON)<<TRUE,FALL<<BOUNCE.


## Bouncing particle thrown toward a ceiling †

INIT  <=> 9 < y < 11 & y' = 10.
FALL  <=> [](y'' = -10).
BOUNCE <=> [](y- = 15 => y' = -(4/5)*y'-).

INIT, FALL << BOUNCE.
• In this program, the trajectories change qualitatively dependent on the initial height $y_0$.
• If $9 < y_0 < 10$, the ball doesn't reach the ceiling.
• If $y_0 = 10$, the ball touches the ceiling, but the velocity remains continuous.
• If $10 < y_0 < 11$, the ball bounces on the ceiling.
• HyLaGI performs such a case analysis automatically.

Last-modified: 2017-03-17 (Fri) 08:42:45 (1213d)