I'm trying to prove (a+b) / c == (a/c) + (b/c) in dafny.
I tried using a real for c, basically 1/c. dafny had trouble with real numbers though.
lemma s(a:nat, b:nat, d:nat)
requires d>0
ensures (a+b) / d == (a/d) + (b/d)
{
//Nothing in here works I tried using a calc == block, but I'm not really sure where to go with it because it really seems basic.
}
I expected Dafny to automatically get this because it's quite basic but it doesn't seem to understand.
The lemma is not true. Indeed, assuming it was true, Dafny would be able to prove false.
lemma no()
ensures false
{
s(1,1,2);
}
Perhaps you want to work with real numbers instead of natural numbers?
What are the benefits and drawbacks of the ?: operator as opposed to the standard if-else statement. The obvious ones being:
Conditional ?: Operator
Shorter and more concise when dealing with direct value comparisons and assignments
Doesn't seem to be as flexible as the if/else construct
Standard If/Else
Can be applied to more situations (such as function calls)
Often are unnecessarily long
Readability seems to vary for each depending on the statement. For a little while after first being exposed to the ?: operator, it took me some time to digest exactly how it worked. Would you recommend using it wherever possible, or sticking to if/else given that I work with many non-programmers?
I would basically recommend using it only when the resulting statement is extremely short and represents a significant increase in conciseness over the if/else equivalent without sacrificing readability.
Good example:
int result = Check() ? 1 : 0;
Bad example:
int result = FirstCheck() ? 1 : SecondCheck() ? 1 : ThirdCheck() ? 1 : 0;
This is pretty much covered by the other answers, but "it's an expression" doesn't really explain why that is so useful...
In languages like C++ and C#, you can define local readonly fields (within a method body) using them. This is not possible with a conventional if/then statement because the value of a readonly field has to be assigned within that single statement:
readonly int speed = (shiftKeyDown) ? 10 : 1;
is not the same as:
readonly int speed;
if (shifKeyDown)
speed = 10; // error - can't assign to a readonly
else
speed = 1; // error
In a similar way you can embed a tertiary expression in other code. As well as making the source code more compact (and in some cases more readable as a result) it can also make the generated machine code more compact and efficient:
MoveCar((shiftKeyDown) ? 10 : 1);
...may generate less code than having to call the same method twice:
if (shiftKeyDown)
MoveCar(10);
else
MoveCar(1);
Of course, it's also a more convenient and concise form (less typing, less repetition, and can reduce the chance of errors if you have to duplicate chunks of code in an if/else). In clean "common pattern" cases like this:
object thing = (reference == null) ? null : reference.Thing;
... it is simply faster to read/parse/understand (once you're used to it) than the long-winded if/else equivalent, so it can help you to 'grok' code faster.
Of course, just because it is useful does not mean it is the best thing to use in every case. I'd advise only using it for short bits of code where the meaning is clear (or made more clear) by using ?: - if you use it in more complex code, or nest ternary operators within each other it can make code horribly difficult to read.
I usually choose a ternary operator when I'd have a lot of duplicate code otherwise.
if (a > 0)
answer = compute(a, b, c, d, e);
else
answer = compute(-a, b, c, d, e);
With a ternary operator, this could be accomplished with the following.
answer = compute(a > 0 ? a : -a, b, c, d, e);
I find it particularly helpful when doing web development if I want to set a variable to a value sent in the request if it is defined or to some default value if it is not.
A really cool usage is:
x = foo ? 1 :
bar ? 2 :
baz ? 3 :
4;
Sometimes it can make the assignment of a bool value easier to read at first glance:
// With
button.IsEnabled = someControl.HasError ? false : true;
// Without
button.IsEnabled = !someControl.HasError;
I'd recommend limiting the use of the ternary(?:) operator to simple single line assignment if/else logic. Something resembling this pattern:
if(<boolCondition>) {
<variable> = <value>;
}
else {
<variable> = <anotherValue>;
}
Could be easily converted to:
<variable> = <boolCondition> ? <value> : <anotherValue>;
I would avoid using the ternary operator in situations that require if/else if/else, nested if/else, or if/else branch logic that results in the evaluation of multiple lines. Applying the ternary operator in these situations would likely result in unreadable, confusing, and unmanageable code. Hope this helps.
The conditional operator is great for short conditions, like this:
varA = boolB ? valC : valD;
I use it occasionally because it takes less time to write something that way... unfortunately, this branching can sometimes be missed by another developer browsing over your code. Plus, code isn't usually that short, so I usually help readability by putting the ? and : on separate lines, like this:
doSomeStuffToSomething(shouldSomethingBeDone()
? getTheThingThatNeedsStuffDone()
: getTheOtherThingThatNeedsStuffDone());
However, the big advantage to using if/else blocks (and why I prefer them) is that it's easier to come in later and add some additional logic to the branch,
if (shouldSomethingBeDone()) {
doSomeStuffToSomething(getTheThingThatNeedsStuffDone());
doSomeAdditionalStuff();
} else {
doSomeStuffToSomething(getTheOtherThingThatNeedsStuffDone());
}
or add another condition:
if (shouldSomethingBeDone()) {
doSomeStuffToSomething(getTheThingThatNeedsStuffDone());
doSomeAdditionalStuff();
} else if (shouldThisOtherThingBeDone()){
doSomeStuffToSomething(getTheOtherThingThatNeedsStuffDone());
}
So, in the end, it's about convenience for you now (shorter to use :?) vs. convenience for you (and others) later. It's a judgment call... but like all other code-formatting issues, the only real rule is to be consistent, and be visually courteous to those who have to maintain (or grade!) your code.
(all code eye-compiled)
One thing to recognize when using the ternary operator that it is an expression not a statement.
In functional languages like scheme the distinction doesn't exists:
(if (> a b) a b)
Conditional ?: Operator
"Doesn't seem to be as flexible as the if/else construct"
In functional languages it is.
When programming in imperative languages I apply the ternary operator in situations where I typically would use expressions (assignment, conditional statements, etc).
While the above answers are valid, and I agree with readability being important, there are 2 further points to consider:
In C#6, you can have expression-bodied methods.
This makes it particularly concise to use the ternary:
string GetDrink(DayOfWeek day)
=> day == DayOfWeek.Friday
? "Beer" : "Tea";
Behaviour differs when it comes to implicit type conversion.
If you have types T1 and T2 that can both be implicitly converted to T, then the below does not work:
T GetT() => true ? new T1() : new T2();
(because the compiler tries to determine the type of the ternary expression, and there is no conversion between T1 and T2.)
On the other hand, the if/else version below does work:
T GetT()
{
if (true) return new T1();
return new T2();
}
because T1 is converted to T and so is T2
If I'm setting a value and I know it will always be one line of code to do so, I typically use the ternary (conditional) operator. If there's a chance my code and logic will change in the future, I use an if/else as it's more clear to other programmers.
Of further interest to you may be the ?? operator.
The advantage of the conditional operator is that it is an operator. In other words, it returns a value. Since if is a statement, it cannot return a value.
There is some performance benefit of using the the ? operator in eg. MS Visual C++, but this is a really a compiler specific thing. The compiler can actually optimize out the conditional branch in some cases.
The scenario I most find myself using it is for defaulting values and especially in returns
return someIndex < maxIndex ? someIndex : maxIndex;
Those are really the only places I find it nice, but for them I do.
Though if you're looking for a boolean this might sometimes look like an appropriate thing to do:
bool hey = whatever < whatever_else ? true : false;
Because it's so easy to read and understand, but that idea should always be tossed for the more obvious:
bool hey = (whatever < whatever_else);
If you need multiple branches on the same condition, use an if:
if (A == 6)
f(1, 2, 3);
else
f(4, 5, 6);
If you need multiple branches with different conditions, then if statement count would snowball, you'll want to use the ternary:
f( (A == 6)? 1: 4, (B == 6)? 2: 5, (C == 6)? 3: 6 );
Also, you can use the ternary operator in initialization.
const int i = (A == 6)? 1 : 4;
Doing that with if is very messy:
int i_temp;
if (A == 6)
i_temp = 1;
else
i_temp = 4;
const int i = i_temp;
You can't put the initialization inside the if/else, because it changes the scope. But references and const variables can only be bound at initialization.
The ternary operator can be included within an rvalue, whereas an if-then-else cannot; on the other hand, an if-then-else can execute loops and other statements, whereas the ternary operator can only execute (possibly void) rvalues.
On a related note, the && and || operators allow some execution patterns which are harder to implement with if-then-else. For example, if one has several functions to call and wishes to execute a piece of code if any of them fail, it can be done nicely using the && operator. Doing it without that operator will either require redundant code, a goto, or an extra flag variable.
With C# 7, you can use the new ref locals feature to simplify the conditional assignment of ref-compatible variables. So now, not only can you do:
int i = 0;
T b = default(T), c = default(T);
// initialization of C#7 'ref-local' variable using a conditional r-value⁽¹⁾
ref T a = ref (i == 0 ? ref b : ref c);
...but also the extremely wonderful:
// assignment of l-value⁽²⁾ conditioned by C#7 'ref-locals'
(i == 0 ? ref b : ref c) = a;
That line of code assigns the value of a to either b or c, depending on the value of i.
Notes
1. r-value is the right-hand side of an assignment, the value that gets assigned.
2. l-value is the left-hand side of an assignment, the variable that receives the assigned value.
I am trying to verify my code in Dafny and I encountered a problem:
I have a method that is iterating over a sequence and changes it. The method changes the sequence according to the elements in the sequence. I would like to add a post condition like this: "if the elements in the sequence are X then something should happen". The problem is that the method changes the set (adds element etc.) and I want to check the condition of the original sequence. Is there an elegant way of doing that in Dafny? (The only way I could think of right now is keeping global var of the original condition of the sequence, but I am looking for the right way of doing that).
Code example:
method changeSeq(p: class1, s: seq<class1>)
ensures |s| == 10 ==> p in s
{
if (|s| == 10){
s := s + [p];
}
}
In the code, I want the post condition to check original s stat, and not its stat after we changed it.
you can use old for old value of a variable like s == old(s).
Here is one example: http://rise4fun.com/Dafny/fhQgD
From Dafny Documentation 22.18. Old Expressions
OldExpression_ = "old" "(" Expression(allowLemma: true, allowLambda: true) ")"
An old expression is used in postconditions. old(e) evaluates to the value expression e had on entry to the current method. Note that old only affects heap dereferences, like o.f and a[i]. In particular, old has no effect on the value returned for local variables or out-parameters.
I have a record declared as
T3DVector = record
X,Y,Z: Integer;
end;
One variable V of type T3DVector holds:
V.X= -25052
V.Y= 34165
V.Z= 37730
I then try to the following line. D is declared as Double.
D:= (V.X*V.X) + (V.Y*V.Y) + (V.Z*V.Z);
The return value is: -1076564467 (0xFFFFFFFFBFD4EE0D)
The following code should be equivalent:
D:= (V.X*V.X);
D:= D + (V.Y*V.Y);
D:= D + (V.Z*V.Z);
But this,however, returns 3218402829 (0x00000000BFD4EE0D), which actually is the correct value.
By looking at the high bits, I thought this was an overflow problem. When I turned on overflow checking, the first line halted with the exception "Integer overflow". This is even more confusing to me because D is Double, and I am only storing values into D
Can anyone clarify please ?
The target of an assignment statement has no bearing on how the right side is evaluated. On the right side, all you have are of type Integer, so the expression is evaluated as that type.
If you want the right side evaluated as some other type, then at least one operand must have that type. You witnessed this when you broke the statement into multiple steps, incorporating D into the right side of the expression. The value of V.Y * V.Y is still evaluated as type Integer, but the result is promoted to have type Double so that it matches the type of the other operand in the addition term (D).
The fact that D is a double doesn't affect the type of X, Y and Z. Those are Integers, and are apparently not large enough to store the squares of such large numbers, and their multiplication is what overflows. Why don't you declare them as doubles, too?
Why delphi perform mathematical 'not' of integer rather than force to cast to boolean value in while do looping ?
Example
var myint:integer;
...
myint:=1;
while not myint=5 do
begin
myint:=myint+1;
showmessage('myint now is : '+inttostr(myint));
end;
Your expression uses two operators: not and =. In order to understand how it is parsed you need to consult the table of operator precedence.
Operators Precedence
----------------------------
# first (highest)
not
----------------------------
* second
/
div
mod
and
shl
shr
as
----------------------------
+ third
-
or
xor
----------------------------
= fourth (lowest)
<>
<
>
<=
>=
in
is
----------------------------
This shows that not has the highest precedence of all operators, and is of higher precedence than =. Which means that your expression is parsed as if it were:
(not myint) = 5
In this expression, because it is bound to an integral variable, not is biwise negation.
In order to get the result that you desire you must use parentheses to indicate that you wish to perform the equality test before the not:
not (myint = 5)
Now, in this expression, (myint = 5) is a logical expression and so the not operator is now logical negation.
The answer to a question of this nature is always found in the table of operator precedence and it will pay dividends to become familiar with this table. This table will tell you how any expression that omits parentheses is parsed.
Finally as Rudy points out, your specific expression is most cleanly written using the <> operator:
myint <> 5
The reason why delphi is doing this is that the not operator has a higher order than the equal comparison operator. It is an unary operator which is calculated first. In fact, is has to be calculated first before the = operator comes into action (which -of course- has the lowest order).
To force delphi calculating what you want, use parenthes:
if not (myint = 5) then....
// vs
if (not myint) = 5 then
// vs
if not myint = 5 then
As you know by now the latter two are equivalent.
BTW: Something similar (order precedence) happens with calculations like 3+4*5. Here, the multiplication is carried out before the addition because the * operator has a higher order compared to the + operator.