Operators

Most operators are infix operators: they have two operands, a left-hand side (lhs) operand and a right-hand side operand (rhs).

An infix operator can either have whitespace before and after the operator or have no whitespace neither before nor after the operator.

Infix operators have a precedence that indicate how strongly they bind to their operand and a left or right associativity.

A few operators are prefix operators: they only have a right-hand side. Prefix operators are followed immediately by their operand: they cannot be separated by whitespace.

A postfix operator (!, Factorial) has only a left-hand side and follows it immediately: like a prefix operator, it cannot be separated from its operand by whitespace.

info

The whitespace rules are necessary to support unambiguous parsing of expressions spanning multiple lines without requiring a separator between expressions

The table below is the complete set of operators, with their spelling, precedence and associativity — if a symbol is not listed there, it is not an operator.

Precedence

The operator at the root of the parse tree has the lowest precedence.

Precedence tiers are numbered in gaps of 10, loosest to tightest — a higher number binds tighter. Operators in the same tier have the same precedence (for example + and -, or * and /).

Tier	Operator	ASCII	Fancy	Kind	Associativity
10	Assign	:=		infix	right
—	Assign or Equal	=		infix	positional
15	MapsTo	=>	⇒	infix	right
18	Coalesce	??		infix	right
20	Pipe	|>		infix	left
20	Pipe	~>		infix	left
30	KeyValuePair	->	→	infix	left
40	Or	||	⋁	infix	left
50	And	&&	⋀	infix	left
60	Equal	==		infix	n-ary chain
60	Same	===		infix	n-ary chain
60	NotEqual	!=	≠	infix	n-ary chain
60	Less	<		infix	n-ary chain
60	Greater	>		infix	n-ary chain
60	LessEqual	<=	⩽	infix	n-ary chain
60	GreaterEqual	>=	⩾	infix	n-ary chain
60	Element	in	∈	infix	n-ary chain
60	Element (type test)	is		infix
60	NotElement	!in	∉	infix	n-ary chain
65	Range	..	‥	infix	left
70	Add	+		infix	left
70	Subtract	-	−	infix	left
80	Multiply	*	×	infix	left
80	Divide	/	÷	infix	left
80	Mod	%		infix	left
90	Negate	-	−	prefix
90	Not	!	¬	prefix
100	Power	^		infix	right
100	Power	**		infix	right
101	Sqrt, Root		√ ∛ ∜	prefix
110	Factorial	!		postfix
110	Power		x² xⁿ⁺¹	postfix
110	Subscript		xₖ₊₁	postfix

Postfix calls and indexing (f(x), xs[i]) bind tighter than every entry in this table — they are handled directly by the parser rather than through the operator table, since they are not spelled with an operator symbol.

The three Unicode-only rows — the radical signs, and the superscript and subscript runs — have no ASCII spelling. By default the serializer writes the ASCII forms sqrt(x), x ^ 2 and Subscript(x, k + 1). In its fancy-symbol mode (the fancySymbols option of serializeEpsil, or epsil --epsil --fancy-symbols) it writes √x, ∛x, ∜x, and an integer-literal exponent as a superscript (x², x⁻¹); a symbolic exponent keeps ^, and a subscript expression keeps Subscript(…). See Radical signs and Superscripts and subscripts.

The conditional expression a if c else b is not an operator row either, but it has a place in this order: between KeyValuePair (30) and Or (40), so it binds looser than every operator that computes and tighter than the forms that bind or pair (=, =>, |>, ->). See Control Flow.

The whitespace rule

An infix operator must have whitespace on both sides or on neither side. A prefix operator must have no whitespace before its operand. These rules let a multi-line program parse deterministically without a separator between every expression:

a + b     // infix addition
a+b       // same: whitespace on neither side

a +b

Here + has whitespace before but not after: it is not treated as infix. The expression a ends there; +b is left over on the same line with no separator before it, which is a diagnostic (unexpected-symbol) rather than a silently-inferred sequence — see Statements and Sequencing. On its own line (after a linebreak or ;), +b is a valid new statement: unary + is the identity, so a and +b are simply two statements.

a+ b

Here + has whitespace after but not before: an asymmetric case. The parser recovers as infix Add but reports an asymmetric-operator-whitespace diagnostic (with a fix-it), since this is more useful to the author than silently ending the statement.

Pipe: |> and ~>

x |> f is f(x). Chained, it lets a sequence of transformations be read in the order they happen instead of inside-out:

[3, 1, 2] |> sort |> reverse // ➔ [3, 2, 1]

⌘/Ctrl + Enter

A stage that takes more than one argument is written as a call, with _ in the slot the piped value fills:

1..10 |> filter(_, n => n % 2 == 1) |> map(n => n^2, _) |> sum // ➔ 165

⌘/Ctrl + Enter

The _ may be left out: a call stage that is missing required arguments receives the piped value in the first slot its type fits, so xs |> take(10) means xs |> take(_, 10) and xs |> map(f) means xs |> map(f, _) (the mapping function is map's first argument). This only fills a hole — a call that is already complete keeps its ordinary meaning, and an explicit _ anywhere in the call says exactly where the piped value goes.

A stage may also be a lambda, written inline without parentheses — after |> the arrow binds tighter than the pipe, and the lambda's body ends at the next |>. A pipe is a call written the other way round: xs |> f is f(xs), for a lambda as for a named function. So when the piped value is a list (or a range), a one-parameter lambda whose body uses its parameter as a scalar (x^2, x + 1) is applied to each element, at every depth of a nested list, as calling it on the list does; _^2 is shorthand for such a lambda. A set, a tuple (a point) or a string is passed whole. The following three pipelines are equivalent:

1..oo |> take(_, 10) |> map(_^2, _) |> sum // ➔ 385

⌘/Ctrl + Enter

1..oo |> take(10) |> x => x^2 |> sum
1..oo |> take(10) |> _^2 |> sum

Note the two readings of _: in a call stage it is the piped value (take(_, 10)); in an operator-written stage (_^2, _ + 1) it is the parameter of the lambda. A lambda that uses its parameter as a collection receives the whole value, as the same call would: xs |> l => length(l) and xs |> (l: list<number>) => length(l) are the length of xs. A named function stage is called the same way: xs |> sum sums the collection.

A pipe hands its stage exactly one value, so a stage that declares more than one parameter is a pipe-stage-arity error rather than a partial application — a leftover function is never what a pipeline was written to produce:

[100, 200] |> (x, y, z) => x + y + z
// ✘ A pipe passes its stage exactly 1 value; `(x, y, z) => …` declares 3

The fix is the call form above, with _ marking the piped value's slot (xs |> fold(f, 0, _)). The same applies to a named stage: xs |> add on a two-parameter add is this error, not a partially applied add.

When the piped value is a collection whose elements are tuples and you want to name their components, use a tuple pattern parameter — the extra parentheses are what make it one parameter taking a pair:

[(1, 2), (3, 4)] |> ((p, q)) => p + q // ➔ [3, 7]

⌘/Ctrl + Enter

|> and ~> are aliases for Pipe and sit at the loosest precedence tier, right below Assign — looser than arithmetic, relational, and boolean operators (Elixir-style). It is left-associative, so a |> f |> g is g(f(a)):

a + b |> f       // (a + b) |> f
a || b |> f      // (a || b) |> f
x = a |> f       // x = (a |> f)

When to reach for a pipeline — and when a nested call or a named intermediate reads better.

Absence coalescing: ??

a ?? b is Coalesce(a, b): the value of a unless a is absent (missing or NaN), in which case the value of b. It is lazy — b is not evaluated when a is present.

let timeout = config.timeout ?? 30
let first = xs[1] ?? 0

?? discharges absence. It does not rescue an Error: an error operand is an error, not a missing value, and propagates.

A function does not accept an absent value at a parameter annotated with a type, unless the type says so. With function f(p: tuple<number, number>), the call f(first(filter(xs, c => c[1] > 0))) is reported by the static check, because a filter can find nothing, and it is an incompatible-type error when the value is absent. Write f(first(filter(…)) ?? (0, 0)) to give a fallback, or annotate the parameter tuple<number, number> | missing and test isMissing(p) in the body. A parameter with no annotation receives the absent value, and a parameter annotated with a numeric type (number, integer, real) reads it as NaN, which such a parameter accepts. The first element of a list literal that has one (first([(1, 2), (3, 4)])) cannot be absent and needs no fallback.

It is right-associative, so a chain falls through left to right:

a ?? b ?? c      // Coalesce(a, Coalesce(b, c))

Its precedence (18) sits between => and |>, which fixes the two groupings that matter:

xs |> f ?? 0     // (xs |> f) ?? 0 — the default is for the pipeline's RESULT
x => x.a ?? 0   // x => (x.a ?? 0) — the default is inside the body

Like |>, it is looser than ->, so a dictionary value needs parentheses:

{a -> 1, b -> x ?? 2}

Write {a -> 1, b -> (x ?? 2)} instead. It is also looser than || and && (the C# position), so a ?? b || c is a ?? (b || c).

Type test: is

x is integer tests at runtime whether a value inhabits a type. It is the same test a match type pattern performs, and lowers to the same Element(value, type) expression:

x is integer
x is string && y is boolean

The right operand is a type name, not an expression, so a typo is a parse-time diagnostic rather than a comparison against an undeclared symbol. This first version resolves simple named types only: a compound type (!error, integer | string, list<integer>) parses but reports type-pattern-unsupported, exactly as the equivalent typed pattern does.

is is a contextual word, not a reserved one — it is recognized only between an operand and a type name, so let is = 5 and f(is) remain legal.

Since is and in express the same membership test, a program written back out from its parsed form uses in for both.

Anonymous functions: =>

The mapsto operator constructs an anonymous function:

x => x^2
(x, y) => x + y

It is right-associative, so x => y => x + y constructs a function that returns another function. It binds tighter than assignment but more loosely than the other expression operators, so f = x => x + 1 assigns the complete function to f. Typed parameters can be written in parentheses:

(x: integer) => x + 1

A parameter can instead be a tuple pattern, written with a second pair of parentheses. It is still ONE parameter — it takes one argument, a tuple, and binds a name to each component:

[(True, True), (True, False)] |> map(((p, q)) => p && q, _)
// ➔ [True, False]

The doubled parentheses are the whole difference: (p, q) => p && q is the two-parameter function it has always been, and ((p, q)) => p && q is the one-parameter function that takes a pair apart. Patterns mix with plain parameters and nest, exactly as in let (a, b) = v — bare names, _ to skip a position, nested (…) patterns, and nothing else (a literal or a per-element type annotation is a diagnostic):

(x, (p, q)) => x + p + q   // two parameters, the second destructured
((a, (b, c))) => a + b + c // one parameter, nested
((p, _)) => p              // one parameter, second component discarded

The argument must be a tuple of the pattern's shape; anything else yields an incompatible-type error value, the same one the destructuring let produces. A destructuring lambda is interpreted, never compiled: no compile target lowers the tuple match, so a compiled context falls back rather than emit code that binds the wrong names.

Callbacks and arity. An ordinary call with too few arguments partially applies the function — f(1) on a two-parameter f is a function awaiting the second argument. Inside a collection operator that never happens: the operator decides how many arguments the callback receives (map supplies one element per source collection, filter/any/all/count/takeWhile supply one, reduce/fold supply the accumulator and the element), and a lambda whose parameter count cannot match is a callback-arity error at parse/canonicalization time rather than a list of leftover closures. The message names both sides and, for the pair case, the fix:

map((p, q) => p + q, [(1, 2), (3, 4)])
// error: Map calls its callback with 1 argument (each element of the
// collection); `(p, q) => p + q` declares 2 parameters. To take a pair
// apart, use a tuple pattern parameter: ((p, q)) => …

sort (a key or a comparator) and iterate (f(previous) or f(index, previous)) accept either of their two arities; a () => … literal is a constant and fits any slot. A callback whose arity is not statically known — a value typed function or callback<…>, a generic function — is not checked here and is applied as before.

The MapsTo name in the table is internal to parsing: it names the operator, not the function value the expression produces.

The same arrow separates a match case's pattern from its body (pattern [if guard] => body) — one glyph meaning "yields", in both places. Nothing is ambiguous: a case reserves the first => at its own level for itself, so a pattern and a guard always end there, while a case BODY is an ordinary expression in which => builds a lambda:

match n {
  0 => x => x + 1   // body is the lambda `x => x + 1`
  n if n > 0 => n   // guard is `n > 0`, body is `n`
}

A lambda genuinely wanted inside a pattern or a guard is parenthesized: n if (f => f)(n) => n.

The Unicode arrows ⇒ (U+21D2) and ↦ (U+21A6) are both accepted as input aliases for =>. ⇒ is the one the serializer emits — for a lambda and for a match case alike — in its fancy-symbol mode; ↦, the traditional mathematical mapsto glyph, is accepted but never produced.

Earlier versions spelled this arrow |->. That spelling now reports the mapsto-arrow-legacy diagnostic, with a fixit rewriting it to =>; the expression is still parsed as the function it meant.

A -> whose left side is shaped like a parameter list — (x, y) -> x + y, (n: integer) -> n^2, f = x -> x + 1 — is diagnosed as a wrong-arrow typo (with a fixit) and recovered as the intended function: -> builds a key -> value pair, and none of those shapes is a valid key.

Ranges: ..

The range operator is a compact spelling of a two-argument Range:

1..5          // Range(1, 5)
1..n - 1      // Range(1, n - 1)
k in 1..5     // k in Range(1, 5)

It binds tighter than relational operators and more loosely than addition and subtraction. The Unicode two-dot leader ‥ is an input alias. Serialization uses Range(a, b), and a stepped range continues to use the three-argument call Range(a, b, step).

Spread: ...

A prefix ... splices a value's elements into the surrounding sequence. It is accepted in two positions — a call argument list and a list literal — and the three-dot token is distinct from the range operator ..; anywhere else ... is a diagnostic. (In a match pattern, ...rest is the same idea in reverse: it collects the remaining elements.)

In a call argument list, ... spreads a tuple into the call's arguments: the tuple's elements become ordinary positional arguments.

f(...t)          // t's elements become f's arguments
f(1, ...t, q)    // splices between positional arguments
g(...p, ...q)    // several spreads splice in order
max(...t)        // variadic built-ins accept spreads

In a call, only tuples spread — argument lists are tuple-shaped, so a List (or any other value) is an incompatible-type error. A literal tuple splices immediately; a symbolic argument is spliced when the call evaluates, and until then the call stays symbolic (the spread never binds positionally to a single parameter).

In a list or set literal, ... splices a collection — a list, a set, a range:

let xs = [1, 2] [...xs, 3, ...(4..6)] // ➔ [1, 2, 3, 4, 5, 6]

⌘/Ctrl + Enter

let s = {2, 3} {1, ...s, ...[3, 4]} // ➔ Set(1, 2, 3, 4)

⌘/Ctrl + Enter

The splice happens at canonicalization: literal collections splice immediately, and a symbolic or lazy segment lowers to the equivalent join expression — a lone spread [...xs] is join(xs), the list materialization of xs, and an infinite segment stays lazy ([...(1..oo), 5] |> Take(3) is [1, 2, 3]). Set literals deduplicate as usual.

Tuples do not spread here — a tuple is a unit (a point, a pair), and splicing it would quietly discard that; spreading one is a spread-tuple error. To use a tuple's elements, convert explicitly: [...ListFrom(t), 3]. A scalar or string operand is an incompatible-type error (a string is a scalar, not a character collection). Note the mirror-image rule in calls: argument lists are tuple-shaped, so there exactly tuples spread.

In a dictionary literal, ... merges the entries of a dictionary. Entries combine left to right and a later entry wins on a key collision — so a literal entry after a spread overrides it, the usual defaults idiom:

let defaults = {"verbose" -> false, "depth" -> 3} {...defaults, "verbose" -> true} // ➔ {"verbose" -> "True", "depth" -> 3}

⌘/Ctrl + Enter

(Duplicate literal keys are different: they are almost certainly typos, so they keep the literal rule — first wins, with a duplicate-dictionary-key diagnostic.)

A brace of only spreads has nothing to mark it as a dictionary, so it is read as a set-spread. To write a pure merge, lead with the bare -> marker (the same marker as the empty dictionary {->}):

{...a, ...b}        // a SET containing the elements of a and b
{->, ...d1, ...d2}  // a dictionary merge of d1 and d2

Unary prefix: - and !

- (Negate) and ! (Not) are prefix operators. They must abut their operand with no whitespace:

-x        // negation
!a        // logical not
!!a       // double negation — `!!` lexes as one token that peels into two

Negate/Not bind looser than Power, so a leading minus does not reach inside an exponent:

-x^2      // -(x^2), not (-x)^2

A unary minus applied directly to a number literal folds into the literal rather than becoming a negation:

-2        // the literal -2, not a negation of 2

Unary + is accepted the same way but is the identity: +(2 + 1) is just 2 + 1.

Power: ^ and **

Power is the tightest operator in the table and is right-associative. ** is an accepted alias for ^ (same table row, same precedence):

x^2       // exponentiation
x**2      // the same
2^3^2     // 2^(3^2) — right-associative

Because Power binds tighter than Multiply/Divide:

x^1/2     // (x^1)/2, not x^(1/2)

Radical signs: √, ∛, ∜

A radical sign is a prefix operator: √x is sqrt(x), ∛x is root(x, 3), ∜x is root(x, 4). Its operand is what a function call would take — a primary with its postfix clauses and scripts, or another prefix operator — but not an infix operator:

√3          // Sqrt(3)
√(x + 1)    // Sqrt(x + 1)
√x²         // Sqrt(x^2) — the script belongs to the operand
√f(x)       // Sqrt(f(x))
√√2         // Sqrt(Sqrt(2))
-√2         // Negate(Sqrt(2))
√x^2        // Sqrt(x)^2 — `^` does not
√x + 1      // Sqrt(x) + 1

This is how Lean reads √, and it keeps √2x the "√2 times x" every reader expects (see Invisible multiplication). Unlike - and !, a radical sign may be separated from its operand by whitespace (√ 2): it has no infix reading, so there is nothing for the whitespace to disambiguate.

Superscripts and subscripts

A run of superscript characters written against an operand is its exponent:

x²          // x^2
x¹⁰         // x^10
x⁻¹         // x^(-1)
xⁿ⁺¹        // x^(n + 1)
xʸ          // x^y
(x + 1)²    // (x + 1)^2
f(x)²       // f(x)^2
2²          // 2^2

The exponent may use the superscript digits ⁰–⁹, the signs ⁺ ⁻, the parentheses ⁽ ⁾, and the superscript Latin letters (ⁱ, ⁿ, ˣ, ʸ, …). A script binds like the postfix factorial — tighter than ^ and than the prefix minus — and composes with ! in written order:

-x²         // -(x^2)
2x²         // 2·(x^2)
2^x²        // 2^(x^2)
x²^3        // (x^2)^3
x²!         // (x^2)!
3!²         // (3!)^2

A subscript run of letters and digits directly after a name is part of the name (xₙ is the symbol x_n; see Naming). Any other subscript run — one holding a sign or a parenthesis, or one written after a non-symbol operand — is a Subscript:

xₖ₊₁        // Subscript(x, k + 1)
(a + b)ₖ    // Subscript(a + b, k)
xₙ²         // (x_n)^2

A Subscript whose index evaluates to an integer or a symbol names the same symbol the folded spelling names: with k = 3, xₖ₊₁ evaluates to x_4, and to the value of x_4 if it has one. An index that stays unknown, or is not an integer, keeps the expression symbolic.

Like the factorial, a script must abut its operand: x ² ends the expression at x, and the stray ² is a diagnostic. A run that is not an expression (x⁺) is diagnosed at the run.

Modulo: %

% is Mod, an infix operator at the multiplicative tier (the same precedence as * and /), left-associative:

a % b       // remainder
a + b % c   // a + (b % c)
a % b % c   // (a % b) % c — left-associative

Factorial: postfix !

! in postfix position is Factorial. Position disambiguates it from the prefix ! (Not): a ! that abuts the preceding operand is a factorial (x!), while a ! at the start of an operand is Not (!x).

5!          // factorial
n!          // factorial
!x          // prefix not, unchanged

Factorial binds tighter than Power (tier 110 vs. 100), so it reaches inside a Power operand, and a leading minus stays outside it:

2^3!        // 2^(3!)
3! ^ 2      // (3!)^2
-3!         // -(3!)

It also applies after a parenthesized expression, a call, or an index:

(a + b)!    // factorial of the sum
f(x)!       // factorial of the result

Like a prefix operator, a postfix ! must abut its operand: x! is a factorial, but x !y is not — the space before ! ends the x expression, leaving !y (a prefix Not) with no separator, which is a diagnostic. Because the lexer maximal-munches a run of operator characters into one token, a ! directly followed by another operator character is not seen as a lone ! (write 3! ^ 2, not 3!^2; x! + 1, not x!+1). The != (NotEqual) and !in (NotElement) operators are unaffected: the lexer keeps != whole and !in is recognized as a compound before the postfix !.

Invisible multiplication

A number literal immediately followed — with no whitespace — by a symbol or an opening parenthesis is read as an implicit Multiply:

2x        // 2 * x
3x^3      // 3·(x^3)
2i        // 2 * i, where `i` is the imaginary unit
2(2 + 1)  // 2 * (2 + 1)

A number literal under radical signs and superscript exponents is a numeric coefficient too, so it leads (and continues) an invisible multiplication exactly as a bare literal does, and a radical sign may follow a literal:

2√3       // 2 * Sqrt(3)
√2x       // Sqrt(2) * x
√2(x + 1) // Sqrt(2) * (x + 1)
2²x       // 2^2 * x
2√3x      // 2 * Sqrt(3) * x

Only the glyph spellings qualify: Sqrt(2)x and 2^2x are still diagnostics.

Note that a symbol immediately followed by ( is a function call, not an invisible multiplication: x(2+1) calls x, and (a+b)(2+1) calls the value of a+b. Only a number on the left means multiplication. See Calls and Indexing.

Whitespace between the number and the symbol suppresses invisible multiplication and is instead a statement boundary: 2 1/2 is a diagnostic (unexpected-symbol), not 2 * (1/2).

Chained relational operators

Relational operators (precedence tier 60) are chainable, matching how mathematicians write inequalities: a < b < c means what it looks like, and so does a chain that mixes operators —

a < b <= c

means a < b && b <= c. A mixed chain is rewritten into that pairwise conjunction before it is evaluated, so both kinds of chain have the usual mathematical chained-comparison semantics. Both kinds also short-circuit like &&: the operands are evaluated left to right and evaluation stops at the first adjacent pair that is false, so in a < b < c the operand c is not evaluated when a < b is false.

Logic operators

• && (And), || (Or), ! (Not), with the fancy Unicode forms ⋀, ⋁, ¬.
• && binds tighter than ||, matching the tiers above.
• Both short-circuit: the operands are evaluated left to right and evaluation stops at the first operand that decides the result — the first false for &&, the first true for ||. The remaining operands do not run, so k <= n && xs[k] > 0 never reads xs[k] when k is out of range, and false && f() never calls f(). Because the written order is meaningful, &&/|| operands are never reordered by canonicalization. The exception is an element-wise application — an operand that is a list of booleans ([true, false] && xs) makes the result a list, cell by cell, and every operand is then evaluated once.

The word forms and, or, and not, and the equivalence infix operator <=>, are reserved but not implemented. The token => is not available as logical implication: it is the mapsto arrow (see Anonymous functions), which is also what separates a match pattern from its result.

Assignment vs. equality

Three spellings, two meanings:

• := always assigns. It never compares.
• == always compares (and === is Same, structural identity). A third comparison tier asks the prover whether the two sides are equal for every value of their free variables: identicallyEqual(sin(t)^2 + cos(t)^2, 1) is True, where == leaves the equation as an inert condition. It is deliberately spelled as a call, never as an operator — the equivalence glyphs ≡, ≢, and ≣ are rejected outright, because their bar counts cross the =-run lengths (≡ has three bars, ≣ four) and a visual transliteration would silently land on the wrong tier.
• = is positional. It assigns when it is the top-level operator of a statement whose left side is a binding target — a name, or a field/index path rooted at one. Everywhere else it compares.

So a statement assigns:

x = 5
count = count + 1

…while the same = inside any larger expression is an equation, which is what a reader of mathematics expects:

solve(x^2 = 4, x)        // Equal — the equation, not an assignment
if a = true { 1 } else { 2 }
[a = 1, b = 2]

This is why = needs no parentheses to be safe in a condition: if a = true cannot silently assign, and the C footgun does not exist in Epsil.

As a comparison, = binds at the relational tier (60) like ==, so if x = 5 && y groups as (x = 5) && y. As an assignment it binds loosest (10), taking the whole right-hand side.

Three consequences worth knowing:

A function head defines the function. As a statement, f(x) = body defines the function f, and f(x) := body is the same definition. This also applies to typed parameters, a return type and literal-pattern clauses (f(0) := 1).

A non-binding left side compares, even as a statement. x^2 = 4 on its own line is the equation, because x^2 is not a name. A bare name always assigns, so write == when you mean the equation:

y == 2 * x + 1           // the equation
y = 2 * x + 1            // assigns to y

A chain is diagnosed. a = b = 5 would assign a the boolean b == 5, which is never what a chained assignment means:

a = b = 5

Write a := b := 5 to chain the assignment, or a = (b = 5) if the comparison really was intended.

A tuple pattern with a bare = is diagnosed. A parenthesized left side is not a binding target, so (a, b) = (b, a) is a comparison of two tuples whose result is discarded — the swap it looks like silently does nothing:

(a, b) = (b, a)

Write (a, b) := (b, a) to destructure, or == if the comparison really was intended. The diagnostic is narrow: it fires only when the left side is shaped exactly like a destructuring pattern (bare names, _, nested tuples), so a genuine tuple equation with computed components — (x + 1, y) = t — stays silent.

An assignment in a condition is a warning. := is unconditional, so it reaches a condition where a bare = no longer can — and Epsil has no if init; cond form, so the assigned value is the test:

if flag := true { 1 }   // warning: assign-in-condition

It is a warning rather than an error, since := is the deliberate spelling. It fires only where a value is consumed as a boolean — an if/while condition — not for f(a := 1) or [a := 1], which are unambiguous.

Serialization uses the explicit spellings. An expression written back out by the formatter or serializer always uses := for assignment and == for comparison, never a bare = — so a round-trip is exact regardless of position. = is an input convenience.