Conditionals

If-Then-Else

The basic conditional expression:

if condition then value1 else value2

Examples:

define positive_check(x) = if x > 0 then "positive" else "non-positive"

define factorial(n) = if n = 0 then 1 else n * factorial(n - 1)

define abs(x) = if x ≥ 0 then x else -x

Conditionals Are Expressions

In Kleis, if-then-else is an expression that returns a value:

define doubled_abs(x) =
    let result = if x > 0 then x else -x in
    result * 2

// Both branches must have compatible types!
// if True then 42 else "hello"  // ❌ Type error!

Nested Conditionals

define sign(x) =
    if x > 0 then 1
    else if x < 0 then -1
    else 0

define grade(score) =
    if score ≥ 90 then "A"
    else if score ≥ 80 then "B"
    else if score ≥ 70 then "C"
    else if score ≥ 60 then "D"
    else "F"

Guards vs If-Then-Else

Pattern matching with guards is often cleaner:

// With if-then-else
define classify_if(n) =
    if n < 0 then "negative"
    else if n = 0 then "zero"
    else "positive"

// With pattern matching and guards
define classify_match(n) =
    match n {
        x if x < 0 => "negative"
        0 => "zero"
        _ => "positive"
    }

Piecewise Functions

Mathematicians love piecewise definitions:

// Absolute value
define abs_fn(x) =
    if x ≥ 0 then x else -x

// Heaviside step function
define heaviside(x) =
    if x < 0 then 0
    else if x = 0 then 0.5
    else 1

// Piecewise polynomial
define piecewise_f(x) =
    if x < 0 then x^2
    else if x < 1 then x
    else 2 - x

Boolean Expressions

Conditions can be complex:

define quadrant(x, y) =
    if x > 0 ∧ y > 0 then "first quadrant"
    else if x < 0 ∧ y > 0 then "second quadrant"
    else if x < 0 ∧ y < 0 then "third quadrant"
    else if x > 0 ∧ y < 0 then "fourth quadrant"
    else "on an axis"

Short-Circuit Evaluation

Kleis uses short-circuit evaluation for ∧ and ∨:

// If x = 0, division is never evaluated
define check_ratio(x, y) =
    if x ≠ 0 ∧ y/x > 1 then "big ratio" else "safe"

What’s Next?

Learn about structures for defining mathematical objects!

→ Next: Structures