Documentation

Init.Prelude

Init.Prelude #

This is the first file in the Lean import hierarchy. It is responsible for setting up basic definitions, most of which Lean already has "built in knowledge" about, so it is important that they be set up in exactly this way. (For example, Lean will use PUnit in the desugaring of do notation, or in the pattern match compiler.)

@[inline]

def id {α : Sort u} (a : α) :

α

The identity function. id takes an implicit argument α : Sort u (a type in any universe), and an argument a : α, and returns a.

Although this may look like a useless function, one application of the identity function is to explicitly put a type on an expression. If e has type T, and T' is definitionally equal to T, then @id T' e typechecks, and Lean knows that this expression has type T' rather than T. This can make a difference for typeclass inference, since T and T' may have different typeclass instances on them. show T' from e is sugar for an @id T' e expression.

Equations

id a = a

Instances For

@[inline]

def Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) :

α → δ

Function composition, usually written with the infix operator ∘. A new function is created from two existing functions, where one function's output is used as input to the other.

Examples:

Function.comp List.reverse (List.drop 2) [3, 2, 4, 1] = [1, 4]
(List.reverse ∘ List.drop 2) [3, 2, 4, 1] = [1, 4]

Conventions for notations in identifiers:

The recommended spelling of ∘ in identifiers is comp.

Equations

(f ∘ g) x = f (g x)

Instances For

@[inline]

def Function.const {α : Sort u} (β : Sort v) (a : α) :

β → α

The constant function that ignores its argument.

If a : α, then Function.const β a : β → α is the “constant function with value a”. For all arguments b : β, Function.const β a b = a.

Examples:

Function.const Bool 10 true = 10
Function.const Bool 10 false = 10
Function.const String 10 "any string" = 10

Equations

Function.const β a x✝ = a

@[irreducible]

def letFun {α : Sort u} {β : α → Sort v} (v : α) (f : (x : α) → β x) :

β v

The encoding of let_fun x := v; b is letFun v (fun x => b). This is equal to (fun x => b) v, so the value of x is not accessible to b. This is in contrast to let x := v; b, where the value of x is accessible to b.

There is special support for letFun. Both WHNF and simp are aware of letFun and can reduce it when zeta reduction is enabled, despite the fact it is marked irreducible. For metaprogramming, the function Lean.Expr.letFun? can be used to recognize a let_fun expression to extract its parts as if it were a let expression.

Equations

letFun v f = f v

@[reducible, inline]

abbrev inferInstance {α : Sort u} [i : α] :

α

inferInstance synthesizes a value of any target type by typeclass inference. This function has the same type signature as the identity function, but the square brackets on the [i : α] argument means that it will attempt to construct this argument by typeclass inference. (This will fail if α is not a class.) Example:

#check (inferInstance : Inhabited Nat) -- Inhabited Nat

def foo : Inhabited (Nat × Nat) :=
  inferInstance

example : foo.default = (default, default) :=
  rfl

Equations

inferInstance = i

Instances For

@[reducible, inline]

abbrev inferInstanceAs (α : Sort u) [i : α] :

α

inferInstanceAs α synthesizes a value of any target type by typeclass inference. This is just like inferInstance except that α is given explicitly instead of being inferred from the target type. It is especially useful when the target type is some α' which is definitionally equal to α, but the instance we are looking for is only registered for α (because typeclass search does not unfold most definitions, but definitional equality does.) Example:

#check inferInstanceAs (Inhabited Nat) -- Inhabited Nat

Equations

inferInstanceAs α = i

inductive PUnit :

The canonical universe-polymorphic type with just one element.

It should be used in contexts that require a type to be universe polymorphic, thus disallowing Unit.

unit : PUnit
The only element of the universe-polymorphic unit type.

Instances For

@[reducible, inline]

abbrev Unit :

The canonical type with one element. This element is written ().

Unit has a number of uses:

It can be used to model control flow that returns from a function call without providing other information.
Monadic actions that return Unit have side effects without computing values.
In polymorphic types, it can be used to indicate that no data is to be stored in a particular field.

Equations

Unit = PUnit

Instances For

@[reducible, match_pattern, inline]

abbrev Unit.unit :

The only element of the unit type.

It can be written as an empty tuple: ().

Equations

() = PUnit.unit

unsafe axiom lcErased :

Marker for information that has been erased by the code generator.

unsafe axiom lcAny :

Marker for type dependency that has been erased by the code generator.

unsafe axiom lcProof {α : Prop} :

α

Auxiliary unsafe constant used by the Compiler when erasing proofs from code.

It may look strange to have an axiom that says "every proposition is true", since this is obviously unsound, but the unsafe marker ensures that the kernel will not let this through into regular proofs. The lower levels of the code generator don't need proofs in terms, so this is used to stub the proofs out.

unsafe axiom lcCast {α : Sort u} {β : Sort v} (a : α) :

β

Auxiliary unsafe constant used by the Compiler when erasing casts.

unsafe axiom lcUnreachable {α : Sort u} :

α

Auxiliary unsafe constant used by the Compiler to mark unreachable code.

Like lcProof, this is an unsafe axiom, which means that even though it is not sound, the kernel will not let us use it for regular proofs.

Executing this expression to actually synthesize a value of type α causes immediate undefined behavior, and the compiler does take advantage of this to optimize the code assuming that it is not called. If it is not optimized out, it is likely to appear as a print message saying "unreachable code", but this behavior is not guaranteed or stable in any way.

inductive True :

True is a proposition and has only an introduction rule, True.intro : True. In other words, True is simply true, and has a canonical proof, True.intro For more information: Propositional Logic

intro : True
True is true, and True.intro (or more commonly, trivial) is the proof.

Instances For

inductive False :

False is the empty proposition. Thus, it has no introduction rules. It represents a contradiction. False elimination rule, False.rec, expresses the fact that anything follows from a contradiction. This rule is sometimes called ex falso (short for ex falso sequitur quodlibet), or the principle of explosion. For more information: Propositional Logic

Instances For

inductive Empty :

The empty type. It has no constructors.

Use Empty.elim in contexts where a value of type Empty is in scope.

Instances For

inductive PEmpty :

The universe-polymorphic empty type, with no constructors.

PEmpty can be used in any universe, but this flexibility can lead to worse error messages and more challenges with universe level unification. Prefer the type Empty or the proposition False when possible.

Instances For

def Not (a : Prop) :

Not p, or ¬p, is the negation of p. It is defined to be p → False, so if your goal is ¬p you can use intro h to turn the goal into h : p ⊢ False, and if you have hn : ¬p and h : p then hn h : False and (hn h).elim will prove anything. For more information: Propositional Logic

Conventions for notations in identifiers:

The recommended spelling of ¬ in identifiers is not.

Equations

(¬a) = (a → False)

Instances For

@[macro_inline]

def False.elim {C : Sort u} (h : False) :

C

False.elim : False → C says that from False, any desired proposition C holds. Also known as ex falso quodlibet (EFQ) or the principle of explosion.

The target type is actually C : Sort u which means it works for both propositions and types. When executed, this acts like an "unreachable" instruction: it is undefined behavior to run, but it will probably print "unreachable code". (You would need to construct a proof of false to run it anyway, which you can only do using sorry or unsound axioms.)

Equations

h.elim = False.rec (fun (x : False) => C) h

@[macro_inline]

def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : ¬a) :

b

Anything follows from two contradictory hypotheses. Example:

example (hp : p) (hnp : ¬p) : q := absurd hp hnp

For more information: Propositional Logic

Equations

absurd h₁ h₂ = False.rec (fun (x : False) => b) ⋯

inductive Eq {α : Sort u_1} :

α → α → Prop

The equality relation. It has one introduction rule, Eq.refl. We use a = b as notation for Eq a b. A fundamental property of equality is that it is an equivalence relation.

variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd

Equality is much more than an equivalence relation, however. It has the important property that every assertion respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value. That is, given h1 : a = b and h2 : p a, we can construct a proof for p b using substitution: Eq.subst h1 h2. Example:

example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2

The triangle in the second presentation is a macro built on top of Eq.subst and Eq.symm, and you can enter it by typing \t. For more information: Equality

Conventions for notations in identifiers:

The recommended spelling of = in identifiers is eq.

refl {α : Sort u_1} (a : α) : a = a
Eq.refl a : a = a is reflexivity, the unique constructor of the equality type. See also rfl, which is usually used instead.

Instances For

@[match_pattern]

def rfl {α : Sort u} {a : α} :

a = a

rfl : a = a is the unique constructor of the equality type. This is the same as Eq.refl except that it takes a implicitly instead of explicitly.

This is a more powerful theorem than it may appear at first, because although the statement of the theorem is a = a, Lean will allow anything that is definitionally equal to that type. So, for instance, 2 + 2 = 4 is proven in Lean by rfl, because both sides are the same up to definitional equality.

Equations

⋯ = ⋯

@[simp]

theorem id_eq {α : Sort u_1} (a : α) :

id a = a

id x = x, as a @[simp] lemma.

theorem Eq.subst {α : Sort u} {motive : α → Prop} {a b : α} (h₁ : a = b) (h₂ : motive a) :

motive b

The substitution principle for equality. If a = b and P a holds, then P b also holds. We conventionally use the name motive for P here, so that you can specify it explicitly using e.g. Eq.subst (motive := fun x => x < 5) if it is not otherwise inferred correctly.

This theorem is the underlying mechanism behind the rw tactic, which is essentially a fancy algorithm for finding good motive arguments to usefully apply this theorem to replace occurrences of a with b in the goal or hypotheses.

For more information: Equality

theorem Eq.symm {α : Sort u} {a b : α} (h : a = b) :

b = a

Equality is symmetric: if a = b then b = a.

Because this is in the Eq namespace, if you have a variable h : a = b, h.symm can be used as shorthand for Eq.symm h as a proof of b = a.

For more information: Equality

theorem Eq.trans {α : Sort u} {a b c : α} (h₁ : a = b) (h₂ : b = c) :

a = c

Equality is transitive: if a = b and b = c then a = c.

Because this is in the Eq namespace, if you have variables or expressions h₁ : a = b and h₂ : b = c, you can use h₁.trans h₂ : a = c as shorthand for Eq.trans h₁ h₂.

For more information: Equality

@[macro_inline]

def cast {α β : Sort u} (h : α = β) (a : α) :

β

Cast across a type equality. If h : α = β is an equality of types, and a : α, then a : β will usually not typecheck directly, but this function will allow you to work around this and embed a in type β as cast h a : β.

It is best to avoid this function if you can, because it is more complicated to reason about terms containing casts, but if the types don't match up definitionally sometimes there isn't anything better you can do.

For more information: Equality

Equations

cast h a = h ▸ a

theorem congrArg {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β) (h : a₁ = a₂) :

f a₁ = f a₂

Congruence in the function argument: if a₁ = a₂ then f a₁ = f a₂ for any (nondependent) function f. This is more powerful than it might look at first, because you can also use a lambda expression for f to prove that <something containing a₁> = <something containing a₂>. This function is used internally by tactics like congr and simp to apply equalities inside subterms.

For more information: Equality

theorem congr {α : Sort u} {β : Sort v} {f₁ f₂ : α → β} {a₁ a₂ : α} (h₁ : f₁ = f₂) (h₂ : a₁ = a₂) :

f₁ a₁ = f₂ a₂

Congruence in both function and argument. If f₁ = f₂ and a₁ = a₂ then f₁ a₁ = f₂ a₂. This only works for nondependent functions; the theorem statement is more complex in the dependent case.

For more information: Equality

theorem congrFun {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x} (h : f = g) (a : α) :

f a = g a

Congruence in the function part of an application: If f = g then f a = g a.

Initialize the Quotient Module, which effectively adds the following definitions:

opaque Quot {α : Sort u} (r : α → α → Prop) : Sort u

opaque Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) : Quot r

opaque Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :
  (∀ a b : α, r a b → Eq (f a) (f b)) → Quot r → β

opaque Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} :
  (∀ a : α, β (Quot.mk r a)) → ∀ q : Quot r, β q

opaque Quot {α : Sort u} (r : α → α → Prop) :

Low-level quotient types. Quotient types coarsen the propositional equality for a type α, so that terms related by some relation r are considered equal in Quot r.

Set-theoretically, Quot r can seen as the set of equivalence classes of α modulo r. Functions from Quot r must prove that they respect r: to define a function f : Quot r → β, it is necessary to provide f' : α → β and prove that for all x : α and y : α, r x y → f' x = f' y.

Quot is a built-in primitive:

Quot.mk places elements of the underlying type α into the quotient.
Quot.lift allows the definition of functions from the quotient to some other type.
Quot.sound asserts the equality of elements related by r.
Quot.ind is used to write proofs about quotients by assuming that all elements are constructed with Quot.mk.

The relation r is not required to be an equivalence relation; the resulting quotient type's equality extends r to an equivalence as a consequence of the rules for equality and quotients. When r is an equivalence relation, it can be more convenient to use the higher-level type Quotient.

opaque Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) :

Places an element of a type into the quotient that equates terms according to the provided relation.

Given v : α and relation r : α → α → Prop, Quot.mk r v : Quot r is like v, except all observations of v's value must respect r.

Quot.mk is a built-in primitive:

Quot is the built-in quotient type.
Quot.lift allows the definition of functions from the quotient to some other type.
Quot.sound asserts the equality of elements related by r.
Quot.ind is used to write proofs about quotients by assuming that all elements are constructed with Quot.mk.

opaque Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} (mk : ∀ (a : α), β (mk r a)) (q : Quot r) :

β q

A reasoning principle for quotients that allows proofs about quotients to assume that all values are constructed with Quot.mk.

Quot.rec is analogous to the recursor for a structure, and can be used when the resulting type is not necessarily a proposition.

Quot.ind is a built-in primitive:

Quot is the built-in quotient type.
Quot.mk places elements of the underlying type α into the quotient.
Quot.lift allows the definition of functions from the quotient to some other type.
Quot.sound asserts the equality of elements related by r.

opaque Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) (a : ∀ (a b : α), r a b → f a = f b) :

Quot r → β

Lifts a function from an underlying type to a function on a quotient, requiring that it respects the quotient's relation.

Given a relation r : α → α → Prop and a quotient Quot r, applying a function f : α → β requires a proof a that f respects r. In this case, Quot.lift f a : Quot r → β computes the same values as f.

Lean's type theory includes a definitional reduction from Quot.lift f h (Quot.mk r v) to f v.

Quot.lift is a built-in primitive:

Quot is the built-in quotient type.
Quot.mk places elements of the underlying type α into the quotient.
Quot.sound asserts the equality of elements related by r
Quot.ind is used to write proofs about quotients by assuming that all elements are constructed with Quot.mk; it is analogous to the recursor for a structure.

unsafe axiom Quot.lcInv {α : Sort u} {r : α → α → Prop} (q : Quot r) :

α

Unsafe auxiliary constant used by the compiler to erase Quot.lift.

inductive HEq {α : Sort u} :

α → {β : Sort u} → β → Prop

Heterogeneous equality. HEq a b asserts that a and b have the same type, and casting a across the equality yields b, and vice versa.

You should avoid using this type if you can. Heterogeneous equality does not have all the same properties as Eq, because the assumption that the types of a and b are equal is often too weak to prove theorems of interest. One important non-theorem is the analogue of congr: If HEq f g and HEq x y and f x and g y are well typed it does not follow that HEq (f x) (g y). (This does follow if you have f = g instead.) However if a and b have the same type then a = b and HEq a b are equivalent.

refl {α : Sort u} (a : α) : HEq a a
Reflexivity of heterogeneous equality.

@[match_pattern]

def HEq.rfl {α : Sort u} {a : α} :

HEq a a

A version of HEq.refl with an implicit argument.

Equations

⋯ = ⋯

theorem eq_of_heq {α : Sort u} {a a' : α} (h : HEq a a') :

a = a'

If two heterogeneously equal terms have the same type, then they are propositionally equal.

@[unbox]

structure Prod (α : Type u) (β : Type v) :

Type (max u v)

The product type, usually written α × β. Product types are also called pair or tuple types. Elements of this type are pairs in which the first element is an α and the second element is a β.

Products nest to the right, so (x, y, z) : α × β × γ is equivalent to (x, (y, z)) : α × (β × γ).

Conventions for notations in identifiers:

The recommended spelling of × in identifiers is Prod.

fst : α
The first element of a pair.
snd : β
The second element of a pair.

Instances For

structure PProd (α : Sort u) (β : Sort v) :

Sort (max (max 1 u) v)

A product type in which the types may be propositions, usually written α ×' β.

This type is primarily used internally and as an implementation detail of proof automation. It is rarely useful in hand-written code.

Conventions for notations in identifiers:

The recommended spelling of ×' in identifiers is PProd.

fst : α
The first element of a pair.
snd : β
The second element of a pair.

Instances For

structure MProd (α β : Type u) :

A product type in which both α and β are in the same universe.

It is called MProd is because it is the universe-monomorphic product type.

fst : α
The first element of a pair.
snd : β
The second element of a pair.

Instances For

structure And (a b : Prop) :

And a b, or a ∧ b, is the conjunction of propositions. It can be constructed and destructed like a pair: if ha : a and hb : b then ⟨ha, hb⟩ : a ∧ b, and if h : a ∧ b then h.left : a and h.right : b.

Conventions for notations in identifiers:

The recommended spelling of ∧ in identifiers is and.
The recommended spelling of /\ in identifiers is and (prefer ∧ over /\).

intro :: (

left : a
Extract the left conjunct from a conjunction. h : a ∧ b then h.left, also notated as h.1, is a proof of a.
right : b
Extract the right conjunct from a conjunction. h : a ∧ b then h.right, also notated as h.2, is a proof of b.

)

Instances For

inductive Or (a b : Prop) :

Or a b, or a ∨ b, is the disjunction of propositions. There are two constructors for Or, called Or.inl : a → a ∨ b and Or.inr : b → a ∨ b, and you can use match or cases to destruct an Or assumption into the two cases.

Conventions for notations in identifiers:

The recommended spelling of ∨ in identifiers is or.
The recommended spelling of \/ in identifiers is or (prefer ∨ over \/).

inl {a b : Prop} (h : a) : a ∨ b
Or.inl is "left injection" into an Or. If h : a then Or.inl h : a ∨ b.
inr {a b : Prop} (h : b) : a ∨ b
Or.inr is "right injection" into an Or. If h : b then Or.inr h : a ∨ b.

Instances For

theorem Or.intro_left {a : Prop} (b : Prop) (h : a) :

a ∨ b

Alias for Or.inl.

theorem Or.intro_right {b : Prop} (a : Prop) (h : b) :

a ∨ b

Alias for Or.inr.

theorem Or.elim {a b c : Prop} (h : a ∨ b) (left : a → c) (right : b → c) :

c

Proof by cases on an Or. If a ∨ b, and both a and b imply proposition c, then c is true.

theorem Or.resolve_left {a b : Prop} (h : a ∨ b) (na : ¬a) :

b

theorem Or.resolve_right {a b : Prop} (h : a ∨ b) (nb : ¬b) :

a

theorem Or.neg_resolve_left {a b : Prop} (h : ¬a ∨ b) (ha : a) :

b

theorem Or.neg_resolve_right {a b : Prop} (h : a ∨ ¬b) (nb : b) :

a

inductive Bool :

The Boolean values, true and false.

Logically speaking, this is equivalent to Prop (the type of propositions). The distinction is important for programming: both propositions and their proofs are erased in the code generator, while Bool corresponds to the Boolean type in most programming languages and carries precisely one bit of run-time information.

false : Bool
The Boolean value false, not to be confused with the proposition False.
true : Bool
The Boolean value true, not to be confused with the proposition True.

Instances For

structure Subtype {α : Sort u} (p : α → Prop) :

Sort (max 1 u)

All the elements of a type that satisfy a predicate.

Subtype p, usually written { x : α // p x } or { x // p x }, contains all elements x : α for which p x is true. Its constructor is a pair of the value and the proof that it satisfies the predicate. In run-time code, { x : α // p x } is represented identically to α.

There is a coercion from { x : α // p x } to α, so elements of a subtype may be used where the underlying type is expected.

Examples:

{ n : Nat // n % 2 = 0 } is the type of even numbers.
{ xs : Array String // xs.size = 5 } is the type of arrays with five Strings.
Given xs : List α, List { x : α // x ∈ xs } is the type of lists in which all elements are contained in xs.

Conventions for notations in identifiers:

The recommended spelling of { x // p x } in identifiers is subtype.

val : α
The value in the underlying type that satisfies the predicate.
property : p self.val
The proof that val satisfies the predicate p.

Instances For

@[reducible]

def optParam (α : Sort u) (default : α) :

Gadget for optional parameter support.

A binder like (x : α := default) in a declaration is syntax sugar for x : optParam α default, and triggers the elaborator to attempt to use default to supply the argument if it is not supplied.

Equations

optParam α default = α

@[reducible]

def outParam (α : Sort u) :

Gadget for marking output parameters in type classes.

For example, the Membership class is defined as:

class Membership (α : outParam (Type u)) (γ : Type v)

This means that whenever a typeclass goal of the form Membership ?α ?γ comes up, Lean will wait to solve it until ?γ is known, but then it will run typeclass inference, and take the first solution it finds, for any value of ?α, which thereby determines what ?α should be.

This expresses that in a term like a ∈ s, s might be a Set α or List α or some other type with a membership operation, and in each case the "member" type α is determined by looking at the container type.

Equations

outParam α = α

@[reducible]

def semiOutParam (α : Sort u) :

Gadget for marking semi output parameters in type classes.

Semi-output parameters influence the order in which arguments to type class instances are processed. Lean determines an order where all non-(semi-)output parameters to the instance argument have to be figured out before attempting to synthesize an argument (that is, they do not contain assignable metavariables created during TC synthesis). This rules out instances such as [Mul β] : Add α (because β could be anything). Marking a parameter as semi-output is a promise that instances of the type class will always fill in a value for that parameter.

For example, the Coe class is defined as:

class Coe (α : semiOutParam (Sort u)) (β : Sort v)

This means that all Coe instances should provide a concrete value for α (i.e., not an assignable metavariable). An instance like Coe Nat Int or Coe α (Option α) is fine, but Coe α Nat is not since it does not provide a value for α.

Equations

semiOutParam α = α

@[reducible]

def namedPattern {α : Sort u} (x a : α) (h : x = a) :

α

Auxiliary declaration used to implement named patterns like x@h:p.

Equations

a = a

@[never_extract, extern lean_sorry]

axiom sorryAx (α : Sort u) (synthetic : Bool) :

α

Auxiliary axiom used to implement the sorry term and tactic.

The sorry term/tactic expands to sorryAx _ (synthetic := false). It is intended for stubbing-out incomplete parts of a value or proof while still having a syntactically correct skeleton. Lean will give a warning whenever a declaration uses sorry, so you aren't likely to miss it, but you can check if a declaration depends on sorry either directly or indirectly by looking for sorryAx in the output of the #print axioms my_thm command.

The synthetic flag is false when a sorry is written explicitly by the user, but it is set to true when a tactic fails to prove a goal, or if there is a type error in the expression. A synthetic sorry acts like a regular one, except that it suppresses follow-up errors in order to prevent an error from causing a cascade of other errors because the desired term was not constructed.

theorem eq_false_of_ne_true {b : Bool} :

¬b = true → b = false

theorem eq_true_of_ne_false {b : Bool} :

¬b = false → b = true

theorem ne_false_of_eq_true {b : Bool} :

b = true → ¬b = false

theorem ne_true_of_eq_false {b : Bool} :

b = false → ¬b = true

class Inhabited (α : Sort u) :

Sort (max 1 u)

Inhabited α is a typeclass that says that α has a designated element, called (default : α). This is sometimes referred to as a "pointed type".

This class is used by functions that need to return a value of the type when called "out of domain". For example, Array.get! arr i : α returns a value of type α when arr : Array α, but if i is not in range of the array, it reports a panic message, but this does not halt the program, so it must still return a value of type α (and in fact this is required for logical consistency), so in this case it returns default.

default : α
default is a function that produces a "default" element of any Inhabited type. This element does not have any particular specified properties, but it is often an all-zeroes value.

Instances

class inductive Nonempty (α : Sort u) :

Nonempty α is a typeclass that says that α is not an empty type, that is, there exists an element in the type. It differs from Inhabited α in that Nonempty α is a Prop, which means that it does not actually carry an element of α, only a proof that there exists such an element. Given Nonempty α, you can construct an element of α nonconstructively using Classical.choice.

intro {α : Sort u} (val : α) : Nonempty α
If val : α, then α is nonempty.

Instances

axiom Classical.choice {α : Sort u} :

Nonempty α → α

The axiom of choice. Nonempty α is a proof that α has an element, but the element itself is erased. The axiom choice supplies a particular element of α given only this proof.

The textbook axiom of choice normally makes a family of choices all at once, but that is implied from this formulation, because if α : ι → Type is a family of types and h : ∀ i, Nonempty (α i) is a proof that they are all nonempty, then fun i => Classical.choice (h i) : ∀ i, α i is a family of chosen elements. This is actually a bit stronger than the ZFC choice axiom; this is sometimes called "global choice".

In Lean, we use the axiom of choice to derive the law of excluded middle (see Classical.em), so it will often show up in axiom listings where you may not expect. You can use #print axioms my_thm to find out if a given theorem depends on this or other axioms.

This axiom can be used to construct "data", but obviously there is no algorithm to compute it, so Lean will require you to mark any definition that would involve executing Classical.choice or other axioms as noncomputable, and will not produce any executable code for such definitions.

theorem Nonempty.elim {α : Sort u} {p : Prop} (h₁ : Nonempty α) (h₂ : ∀ (a : α), p) :

p

The elimination principle for Nonempty α. If Nonempty α, and we can prove p given any element x : α, then p holds. Note that it is essential that p is a Prop here; the version with p being a Sort u is equivalent to Classical.choice.

instance instNonemptyOfInhabited {α : Sort u} [Inhabited α] :

noncomputable def Classical.ofNonempty {α : Sort u} [Nonempty α] :

α

A variation on Classical.choice that uses typeclass inference to infer the proof of Nonempty α.

Equations

Classical.ofNonempty = Classical.choice ⋯

instance instNonemptyForall {α : Sort u} {β : Sort v} [Nonempty β] :

Nonempty (α → β)

instance Pi.instNonempty {α : Sort u} {β : α → Sort v} [∀ (a : α), Nonempty (β a)] :

Nonempty ((a : α) → β a)

instance instInhabitedSort :

Inhabited (Sort u)

Equations

instInhabitedSort = { default := PUnit }

instance instInhabitedForall (α : Sort u) {β : Sort v} [Inhabited β] :

Inhabited (α → β)

Equations

instInhabitedForall α = { default := fun (x : α) => default }

instance Pi.instInhabited {α : Sort u} {β : α → Sort v} [(a : α) → Inhabited (β a)] :

Inhabited ((a : α) → β a)

Equations

Pi.instInhabited = { default := fun (x : α) => default }

instance instInhabitedBool :

Equations

instInhabitedBool = { default := false }

structure PLift (α : Sort u) :

Lifts a proposition or type to a higher universe level.

PLift α wraps a proof or value of type α. The resulting type is in the next largest universe after that of α. In particular, propositions become data.

The related type ULift can be used to lift a non-proposition type by any number of levels.

Examples:

(False : Prop)
(PLift False : Type)
([.up (by trivial), .up (by simp), .up (by decide)] : List (PLift True))
(Nat : Type 0)
(PLift Nat : Type 1)

up :: (

down : α
Extracts a wrapped proof or value from a universe-lifted proposition or type.

)

Instances For

theorem PLift.up_down {α : Sort u} (b : PLift α) :

{ down := b.down } = b

Bijection between α and PLift α

theorem PLift.down_up {α : Sort u} (a : α) :

{ down := a }.down = a

Bijection between α and PLift α

def NonemptyType :

Type (u + 1)

NonemptyType.{u} is the type of nonempty types in universe u. It is mainly used in constant declarations where we wish to introduce a type and simultaneously assert that it is nonempty, but otherwise make the type opaque.

Equations

NonemptyType = { α : Type ?u.3 // Nonempty α }

Instances For

instInhabitedNonemptyType

@[reducible, inline]

abbrev NonemptyType.type (type : NonemptyType) :

The underlying type of a NonemptyType.

Equations

type.type = type.val

instance instInhabitedNonemptyType :

Inhabited NonemptyType

NonemptyType is inhabited, because PUnit is a nonempty type.

Equations

instInhabitedNonemptyType = { default := ⟨PUnit, ⋯⟩ }

structure ULift (α : Type s) :

Type (max s r)

Lifts a type to a higher universe level.

ULift α wraps a value of type α. Instead of occupying the same universe as α, which would be the minimal level, it takes a further level parameter and occupies their minimum. The resulting type may occupy any universe that's at least as large as that of α.

The resulting universe of the lifting operator is the first parameter, and may be written explicitly while allowing α's level to be inferred.

The related type PLift can be used to lift a proposition or type by one level.

Examples:

(Nat : Type 0)
(ULift Nat : Type 0)
(ULift Nat : Type 1)
(ULift Nat : Type 5)
(ULift.{7} (PUnit : Type 3) : Type 7)

up :: (

down : α
Extracts a wrapped value from a universe-lifted type.

)

Instances For

theorem ULift.up_down {α : Type u} (b : ULift α) :

{ down := b.down } = b

Bijection between α and ULift.{v} α

theorem ULift.down_up {α : Type u} (a : α) :

{ down := a }.down = a

Bijection between α and ULift.{v} α

class inductive Decidable (p : Prop) :

Either a proof that p is true or a proof that p is false. This is equivalent to a Bool paired with a proof that the Bool is true if and only if p is true.

Decidable instances are primarily used via if-expressions and the tactic decide. In conditional expressions, the Decidable instance for the proposition is used to select a branch. At run time, this case distinction code is identical to that which would be generated for a Bool-based conditional. In proofs, the tactic decide synthesizes an instance of Decidable p, attempts to reduce it to isTrue h, and then succeeds with the proof h if it can.

Because Decidable carries data, when writing @[simp] lemmas which include a Decidable instance on the LHS, it is best to use {_ : Decidable p} rather than [Decidable p] so that non-canonical instances can be found via unification rather than instance synthesis.

isFalse {p : Prop} (h : ¬p) : Decidable p
Proves that p is decidable by supplying a proof of ¬p
isTrue {p : Prop} (h : p) : Decidable p
Proves that p is decidable by supplying a proof of p

Instances

@[inline_if_reduce]

def Decidable.decide (p : Prop) [h : Decidable p] :

Converts a decidable proposition into a Bool.

If p : Prop is decidable, then decide p : Bool is the Boolean value that is true if p is true and false if p is false.

Equations

decide p = Decidable.casesOn h (fun (x : ¬p) => false) fun (x : p) => true

@[reducible, inline]

abbrev DecidablePred {α : Sort u} (r : α → Prop) :

Sort (max 1 u)

A decidable predicate.

A predicate is decidable if the corresponding proposition is Decidable for each possible argument.

Equations

DecidablePred r = ((a : α) → Decidable (r a))

@[reducible, inline]

abbrev DecidableRel {α : Sort u} {β : Sort v} (r : α → β → Prop) :

Sort (max (max 1 u) v)

A decidable relation.

A relation is decidable if the corresponding proposition is Decidable for all possible arguments.

Equations

DecidableRel r = ((a : α) → (b : β) → Decidable (r a b))

@[reducible, inline]

abbrev DecidableEq (α : Sort u) :

Sort (max 1 u)

Propositional equality is Decidable for all elements of a type.

In other words, an instance of DecidableEq α is a means of deciding the proposition a = b is for all a b : α.

Equations

DecidableEq α = ((a b : α) → Decidable (a = b))

def decEq {α : Sort u} [inst : DecidableEq α] (a b : α) :

Decidable (a = b)

Checks whether two terms of a type are equal using the type's DecidableEq instance.

Equations

decEq a b = inst a b

theorem decide_eq_true {p : Prop} [inst : Decidable p] :

p → decide p = true

theorem decide_eq_false {p : Prop} [Decidable p] :

¬p → decide p = false

theorem of_decide_eq_true {p : Prop} [inst : Decidable p] :

decide p = true → p

theorem of_decide_eq_false {p : Prop} [inst : Decidable p] :

decide p = false → ¬p

theorem of_decide_eq_self_eq_true {α : Sort u_1} [inst : DecidableEq α] (a : α) :

decide (a = a) = true

@[inline]

def Bool.decEq (a b : Bool) :

Decidable (a = b)

Decides whether two Booleans are equal.

This function should normally be called via the DecidableEq Bool instance that it exists to support.

Equations

@[inline]

instance instDecidableEqBool :

DecidableEq Bool

Equations

instDecidableEqBool = Bool.decEq

class BEq (α : Type u) :

BEq α is a typeclass for supplying a boolean-valued equality relation on α, notated as a == b. Unlike DecidableEq α (which uses a = b), this is Bool valued instead of Prop valued, and it also does not have any axioms like being reflexive or agreeing with =. It is mainly intended for programming applications. See LawfulBEq for a version that requires that == and = coincide.

Typically we prefer to put the "more variable" term on the left, and the "more constant" term on the right.

beq : α → α → Bool
Boolean equality, notated as a == b.
Conventions for notations in identifiers:
- The recommended spelling of == in identifiers is beq.

Instances

@[instance 500]

instance instBEqOfDecidableEq {α : Type u_1} [DecidableEq α] :

BEq α

Equations

instBEqOfDecidableEq = { beq := fun (a b : α) => decide (a = b) }

@[macro_inline]

def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : ¬c → α) :

α

"Dependent" if-then-else, normally written via the notation if h : c then t(h) else e(h), is sugar for dite c (fun h => t(h)) (fun h => e(h)), and it is the same as if c then t else e except that t is allowed to depend on a proof h : c, and e can depend on h : ¬c. (Both branches use the same name for the hypothesis, even though it has different types in the two cases.)

We use this to be able to communicate the if-then-else condition to the branches. For example, Array.get arr i h expects a proof h : i < arr.size in order to avoid a bounds check, so you can write if h : i < arr.size then arr.get i h else ... to avoid the bounds check inside the if branch. (Of course in this case we have only lifted the check into an explicit if, but we could also use this proof multiple times or derive i < arr.size from some other proposition that we are checking in the if.)

Equations

dite c t e = Decidable.casesOn h e t

Instances For

instDecidableDite

if-then-else #

@[macro_inline]

def ite {α : Sort u} (c : Prop) [h : Decidable c] (t e : α) :

α

if c then t else e is notation for ite c t e, "if-then-else", which decides to return t or e depending on whether c is true or false. The explicit argument c : Prop does not have any actual computational content, but there is an additional [Decidable c] argument synthesized by typeclass inference which actually determines how to evaluate c to true or false. Write if h : c then t else e instead for a "dependent if-then-else" dite, which allows t/e to use the fact that c is true/false.

Equations

(if c then t else e) = Decidable.casesOn h (fun (x : ¬c) => e) fun (x : c) => t

Instances For

@[macro_inline]

instance instDecidableAnd {p q : Prop} [dp : Decidable p] [dq : Decidable q] :

Decidable (p ∧ q)

Equations

@[macro_inline]

instance instDecidableOr {p q : Prop} [dp : Decidable p] [dq : Decidable q] :

Decidable (p ∨ q)

Equations

instance instDecidableNot {p : Prop} [dp : Decidable p] :

Equations

instDecidableNot = isFalse ⋯
instDecidableNot = isTrue hq

Boolean operators #

@[macro_inline]

def cond {α : Sort u} (c : Bool) (x y : α) :

α

The conditional function.

cond c x y is the same as if c then x else y, but optimized for a Boolean condition rather than a decidable proposition. It can also be written using the notation bif c then x else y.

Just like ite, cond is declared @[macro_inline], which causes applications of cond to be unfolded. As a result, x and y are not evaluated at runtime until one of them is selected, and only the selected branch is evaluated.

Equations

(bif true then x else y) = x
(bif false then x else y) = y

@[macro_inline]

def Bool.dcond {α : Sort u} (c : Bool) (x : c = true → α) (y : c = false → α) :

α

The dependent conditional function, in which each branch is provided with a local assumption about the condition's value. This allows the value to be used in proofs as well as for control flow.

dcond c (fun h => x) (fun h => y) is the same as if h : c then x else y, but optimized for a Boolean condition rather than a decidable proposition. Unlike the non-dependent version cond, there is no special notation for dcond.

Just like ite, dite, and cond, dcond is declared @[macro_inline], which causes applications of dcond to be unfolded. As a result, x and y are not evaluated at runtime until one of them is selected, and only the selected branch is evaluated. dcond is intended for metaprogramming use, rather than for use in verified programs, so behavioral lemmas are not provided.

Equations

true.dcond x_2 y_2 = x_2 ⋯
false.dcond x_2 y_2 = y_2 ⋯

@[macro_inline]

def Bool.or (x y : Bool) :

Boolean “or”, also known as disjunction. or x y can be written x || y.

The corresponding propositional connective is Or : Prop → Prop → Prop, written with the ∨ operator.

The Boolean or is a @[macro_inline] function in order to give it short-circuiting evaluation: if x is true then y is not evaluated at runtime.

Equations

(true || y) = true
(false || y) = y

@[macro_inline]

def Bool.and (x y : Bool) :

Boolean “and”, also known as conjunction. and x y can be written x && y.

The corresponding propositional connective is And : Prop → Prop → Prop, written with the ∧ operator.

The Boolean and is a @[macro_inline] function in order to give it short-circuiting evaluation: if x is false then y is not evaluated at runtime.

Conventions for notations in identifiers:

The recommended spelling of && in identifiers is and.
The recommended spelling of || in identifiers is or.

Equations

(false && y) = false
(true && y) = y

@[inline]

Boolean negation, also known as Boolean complement. not x can be written !x.

This is a function that maps the value true to false and the value false to true. The propositional connective is Not : Prop → Prop.

Conventions for notations in identifiers:

The recommended spelling of ! in identifiers is not.

Equations

(!true) = false
(!false) = true

inductive Nat :

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually GMP); at runtime, Nat values that are sufficiently small are unboxed.

zero : Nat
Zero, the smallest natural number.
Using Nat.zero explicitly should usually be avoided in favor of the literal 0, which is the simp normal form.
succ (n : Nat) : Nat
The successor of a natural number n.
Using Nat.succ n should usually be avoided in favor of n + 1, which is the simp normal form.

Instances For

instance instInhabitedNat :

Equations

instInhabitedNat = { default := Nat.zero }

class OfNat (α : Type u) :

Nat → Type u

The class OfNat α n powers the numeric literal parser. If you write 37 : α, Lean will attempt to synthesize OfNat α 37, and will generate the term (OfNat.ofNat 37 : α).

There is a bit of infinite regress here since the desugaring apparently still contains a literal 37 in it. The type of expressions contains a primitive constructor for "raw natural number literals", which you can directly access using the macro nat_lit 37. Raw number literals are always of type Nat. So it would be more correct to say that Lean looks for an instance of OfNat α (nat_lit 37), and it generates the term (OfNat.ofNat (nat_lit 37) : α).

ofNat : α
The OfNat.ofNat function is automatically inserted by the parser when the user writes a numeric literal like 1 : α. Implementations of this typeclass can therefore customize the behavior of n : α based on n and α.

Instances

@[defaultInstance 100]

instance instOfNatNat (n : Nat) :

Equations

instOfNatNat n = { ofNat := n }

class LE (α : Type u) :

LE α is the typeclass which supports the notation x ≤ y where x y : α.

le : α → α → Prop
The less-equal relation: x ≤ y
Conventions for notations in identifiers:
- The recommended spelling of ≤ in identifiers is le.
- The recommended spelling of <= in identifiers is le (prefer ≤ over <=).

Instances

class LT (α : Type u) :

LT α is the typeclass which supports the notation x < y where x y : α.

lt : α → α → Prop
The less-than relation: x < y
Conventions for notations in identifiers:
- The recommended spelling of < in identifiers is lt.

Instances

@[reducible]

def GE.ge {α : Type u} [LE α] (a b : α) :

a ≥ b is an abbreviation for b ≤ a.

Conventions for notations in identifiers:

The recommended spelling of ≥ in identifiers is ge.
The recommended spelling of >= in identifiers is ge (prefer ≥ over >=).

Equations

(a ≥ b) = (b ≤ a)

Instances For

@[reducible]

def GT.gt {α : Type u} [LT α] (a b : α) :

a > b is an abbreviation for b < a.

Conventions for notations in identifiers:

The recommended spelling of > in identifiers is gt.

Equations

(a > b) = (b < a)

Instances For

@[reducible, inline]

abbrev DecidableLT (α : Type u) [LT α] :

Abbreviation for DecidableRel (· < · : α → α → Prop).

Equations

DecidableLT α = DecidableRel LT.lt

@[reducible, inline]

abbrev DecidableLE (α : Type u) [LE α] :

Abbreviation for DecidableRel (· ≤ · : α → α → Prop).

Equations

DecidableLE α = DecidableRel LE.le

class Max (α : Type u) :

An overloaded operation to find the greater of two values of type α.

max : α → α → α
Returns the greater of its two arguments.

Instances

@[inline]

def maxOfLe {α : Type u_1} [LE α] [DecidableRel LE.le] :

Max α

Constructs a Max instance from a decidable ≤ operation.

Equations

maxOfLe = { max := fun (x y : α) => if x ≤ y then y else x }

class Min (α : Type u) :

An overloaded operation to find the lesser of two values of type α.

min : α → α → α
Returns the lesser of its two arguments.

Instances

@[inline]

def minOfLe {α : Type u_1} [LE α] [DecidableRel LE.le] :

Min α

Constructs a Min instance from a decidable ≤ operation.

Equations

minOfLe = { min := fun (x y : α) => if x ≤ y then x else y }

class Trans {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (r : α → β → Sort u) (s : β → γ → Sort v) (t : outParam (α → γ → Sort w)) :

Sort (max (max (max (max (max (max 1 u) u_1) u_2) u_3) v) w)

Transitive chaining of proofs, used e.g. by calc.

It takes two relations r and s as "input", and produces an "output" relation t, with the property that r a b and s b c implies t a c. The calc tactic uses this so that when it sees a chain with a ≤ b and b < c it knows that this should be a proof of a < c because there is an instance Trans (·≤·) (·<·) (·<·).

trans {a : α} {b : β} {c : γ} : r a b → s b c → t a c
Compose two proofs by transitivity, generalized over the relations involved.

Instances

instance instTransEq {α : Sort u_1} {γ : Sort u_2} (r : α → γ → Sort u) :

Equations

instTransEq r = { trans := fun {a b : α} {c : γ} (heq : a = b) (h' : r b c) => ⋯ ▸ h' }

instance instTransEq_1 {α : Sort u_1} {β : Sort u_2} (r : α → β → Sort u) :

Equations

instTransEq_1 r = { trans := fun {a : α} {b c : β} (h' : r a b) (heq : b = c) => heq ▸ h' }

class HAdd (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous addition. This enables the notation a + b : γ where a : α, b : β.

hAdd : α → β → γ
a + b computes the sum of a and b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of + in identifiers is add.

Instances

class HSub (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous subtraction. This enables the notation a - b : γ where a : α, b : β.

hSub : α → β → γ
a - b computes the difference of a and b. The meaning of this notation is type-dependent.
- For natural numbers, this operator saturates at 0: a - b = 0 when a ≤ b.
Conventions for notations in identifiers:
- The recommended spelling of - in identifiers is sub (when used as a binary operator).

Instances

class HMul (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous multiplication. This enables the notation a * b : γ where a : α, b : β.

hMul : α → β → γ
a * b computes the product of a and b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of * in identifiers is mul.

Instances

class HDiv (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous division. This enables the notation a / b : γ where a : α, b : β.

hDiv : α → β → γ
a / b computes the result of dividing a by b. The meaning of this notation is type-dependent.
- For most types like Nat, Int, Rat, Real, a / 0 is defined to be 0.
- For Nat, a / b rounds downwards.
- For Int, a / b rounds downwards if b is positive or upwards if b is negative. It is implemented as Int.ediv, the unique function satisfying a % b + b * (a / b) = a and 0 ≤ a % b < natAbs b for b ≠ 0. Other rounding conventions are available using the functions Int.fdiv (floor rounding) and Int.tdiv (truncation rounding).
- For Float, a / 0 follows the IEEE 754 semantics for division, usually resulting in inf or nan.
Conventions for notations in identifiers:
- The recommended spelling of / in identifiers is div.

Instances

class HMod (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous modulo / remainder. This enables the notation a % b : γ where a : α, b : β.

hMod : α → β → γ
a % b computes the remainder upon dividing a by b. The meaning of this notation is type-dependent.
- For Nat and Int it satisfies a % b + b * (a / b) = a, and a % 0 is defined to be a.
Conventions for notations in identifiers:
- The recommended spelling of % in identifiers is mod.

Instances

class HPow (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous exponentiation. This enables the notation a ^ b : γ where a : α, b : β.

hPow : α → β → γ
a ^ b computes a to the power of b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of ^ in identifiers is pow.

Instances

class HAppend (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The notation typeclass for heterogeneous append. This enables the notation a ++ b : γ where a : α, b : β.

hAppend : α → β → γ
a ++ b is the result of concatenation of a and b, usually read "append". The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of ++ in identifiers is append.

Instances

class HOrElse (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a <|> b : γ where a : α, b : β. Because b is "lazy" in this notation, it is passed as Unit → β to the implementation so it can decide when to evaluate it.

hOrElse : α → (Unit → β) → γ
a <|> b executes a and returns the result, unless it fails in which case it executes and returns b. Because b is not always executed, it is passed as a thunk so it can be forced only when needed. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of <|> in identifiers is orElse.

Instances

instHOrElseOfOrElse

class HAndThen (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a >> b : γ where a : α, b : β. Because b is "lazy" in this notation, it is passed as Unit → β to the implementation so it can decide when to evaluate it.

hAndThen : α → (Unit → β) → γ
a >> b executes a, ignores the result, and then executes b. If a fails then b is not executed. Because b is not always executed, it is passed as a thunk so it can be forced only when needed. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of >> in identifiers is andThen.

Instances

instHAndThenOfAndThen

class HAnd (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a &&& b : γ where a : α, b : β.

hAnd : α → β → γ
a &&& b computes the bitwise AND of a and b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of &&& in identifiers is and.

Instances

instHAndOfAndOp

class HXor (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a ^^^ b : γ where a : α, b : β.

hXor : α → β → γ
a ^^^ b computes the bitwise XOR of a and b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of ^^^ in identifiers is xor.

Instances

instHXorOfXor

class HOr (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a ||| b : γ where a : α, b : β.

hOr : α → β → γ
a ||| b computes the bitwise OR of a and b. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of ||| in identifiers is or.

Instances

instHOrOfOrOp

class HShiftLeft (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a <<< b : γ where a : α, b : β.

hShiftLeft : α → β → γ
a <<< b computes a shifted to the left by b places. The meaning of this notation is type-dependent.
- On Nat, this is equivalent to a * 2 ^ b.
- On UInt8 and other fixed width unsigned types, this is the same but truncated to the bit width.
Conventions for notations in identifiers:
- The recommended spelling of <<< in identifiers is shiftLeft.

Instances

class HShiftRight (α : Type u) (β : Type v) (γ : outParam (Type w)) :

Type (max (max u v) w)

The typeclass behind the notation a >>> b : γ where a : α, b : β.

hShiftRight : α → β → γ
a >>> b computes a shifted to the right by b places. The meaning of this notation is type-dependent.
- On Nat and fixed width unsigned types like UInt8, this is equivalent to a / 2 ^ b.
Conventions for notations in identifiers:
- The recommended spelling of >>> in identifiers is shiftRight.

Instances

class Zero (α : Type u) :

A type with a zero element.

zero : α
The zero element of the type.

Instances

class Add (α : Type u) :

The homogeneous version of HAdd: a + b : α where a b : α.

add : α → α → α
a + b computes the sum of a and b. See HAdd.

Instances

class Sub (α : Type u) :

The homogeneous version of HSub: a - b : α where a b : α.

sub : α → α → α
a - b computes the difference of a and b. See HSub.

Instances

class Mul (α : Type u) :

The homogeneous version of HMul: a * b : α where a b : α.

mul : α → α → α
a * b computes the product of a and b. See HMul.

Instances

class Neg (α : Type u) :

The notation typeclass for negation. This enables the notation -a : α where a : α.

neg : α → α
-a computes the negative or opposite of a. The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
- The recommended spelling of - in identifiers is neg (when used as a unary operator).

Instances

class Div (α : Type u) :

The homogeneous version of HDiv: a / b : α where a b : α.

div : α → α → α
a / b computes the result of dividing a by b. See HDiv.

Instances

class Mod (α : Type u) :

The homogeneous version of HMod: a % b : α where a b : α.

mod : α → α → α
a % b computes the remainder upon dividing a by b. See HMod.

Instances

class Dvd (α : Type u_1) :

Type u_1

Notation typeclass for the ∣ operation (typed as \|), which represents divisibility.

dvd : α → α → Prop
Divisibility. a ∣ b (typed as \|) means that there is some c such that b = a * c.
Conventions for notations in identifiers:
- The recommended spelling of ∣ in identifiers is dvd.

Instances

class Pow (α : Type u) (β : Type v) :

Type (max u v)

The homogeneous version of HPow: a ^ b : α where a : α, b : β. (The right argument is not the same as the left since we often want this even in the homogeneous case.)

Types can choose to subscribe to particular defaulting behavior by providing an instance to either NatPow or HomogeneousPow:

NatPow is for types whose exponents is preferentially a Nat.
HomogeneousPow is for types whose base and exponent are preferentially the same.

pow : α → β → α
a ^ b computes a to the power of b. See HPow.

Instances

class NatPow (α : Type u) :

The homogeneous version of Pow where the exponent is a Nat. The purpose of this class is that it provides a default Pow instance, which can be used to specialize the exponent to Nat during elaboration.

For example, if x ^ 2 should preferentially elaborate with 2 : Nat then x's type should provide an instance for this class.

pow : α → Nat → α
a ^ n computes a to the power of n where n : Nat. See Pow.

Instances

class HomogeneousPow (α : Type u) :

The completely homogeneous version of Pow where the exponent has the same type as the base. The purpose of this class is that it provides a default Pow instance, which can be used to specialize the exponent to have the same type as the base's type during elaboration. This is to say, a type should provide an instance for this class in case x ^ y should be elaborated with both x and y having the same type.

For example, the Float type provides an instance of this class, which causes expressions such as (2.2 ^ 2.2 : Float) to elaborate.

pow : α → α → α
a ^ b computes a to the power of b where a and b both have the same type.

Instances

class Append (α : Type u) :

The homogeneous version of HAppend: a ++ b : α where a b : α.

append : α → α → α
a ++ b is the result of concatenation of a and b. See HAppend.

Instances

class OrElse (α : Type u) :

The homogeneous version of HOrElse: a <|> b : α where a b : α. Because b is "lazy" in this notation, it is passed as Unit → α to the implementation so it can decide when to evaluate it.

orElse : α → (Unit → α) → α
The implementation of a <|> b : α. See HOrElse.

Instances

class AndThen (α : Type u) :

The homogeneous version of HAndThen: a >> b : α where a b : α. Because b is "lazy" in this notation, it is passed as Unit → α to the implementation so it can decide when to evaluate it.

andThen : α → (Unit → α) → α
The implementation of a >> b : α. See HAndThen.

Instances

class AndOp (α : Type u) :

The homogeneous version of HAnd: a &&& b : α where a b : α. (It is called AndOp because And is taken for the propositional connective.)

and : α → α → α
The implementation of a &&& b : α. See HAnd.

Instances

class Xor (α : Type u) :

The homogeneous version of HXor: a ^^^ b : α where a b : α.

xor : α → α → α
The implementation of a ^^^ b : α. See HXor.

Instances

class OrOp (α : Type u) :

The homogeneous version of HOr: a ||| b : α where a b : α. (It is called OrOp because Or is taken for the propositional connective.)

or : α → α → α
The implementation of a ||| b : α. See HOr.

Instances

class Complement (α : Type u) :

The typeclass behind the notation ~~~a : α where a : α.

complement : α → α
The implementation of ~~~a : α.
Conventions for notations in identifiers:
- The recommended spelling of ~~~ in identifiers is not.

Instances

class ShiftLeft (α : Type u) :

The homogeneous version of HShiftLeft: a <<< b : α where a b : α.

shiftLeft : α → α → α
The implementation of a <<< b : α. See HShiftLeft.

Instances

class ShiftRight (α : Type u) :

The homogeneous version of HShiftRight: a >>> b : α where a b : α.

shiftRight : α → α → α
The implementation of a >>> b : α. See HShiftRight.

Instances

@[defaultInstance 1000]

instance instHAdd {α : Type u_1} [Add α] :

HAdd α α α

Equations

instHAdd = { hAdd := fun (a b : α) => Add.add a b }

@[defaultInstance 1000]

instance instHSub {α : Type u_1} [Sub α] :

HSub α α α

Equations

instHSub = { hSub := fun (a b : α) => Sub.sub a b }

@[defaultInstance 1000]

instance instHMul {α : Type u_1} [Mul α] :

HMul α α α

Equations

instHMul = { hMul := fun (a b : α) => Mul.mul a b }

@[defaultInstance 1000]

instance instHDiv {α : Type u_1} [Div α] :

HDiv α α α

Equations

instHDiv = { hDiv := fun (a b : α) => Div.div a b }

@[defaultInstance 1000]

instance instHMod {α : Type u_1} [Mod α] :

HMod α α α

Equations

instHMod = { hMod := fun (a b : α) => Mod.mod a b }

@[defaultInstance 1000]

instance instHPow {α : Type u_1} {β : Type u_2} [Pow α β] :

HPow α β α

Equations

instHPow = { hPow := fun (a : α) (b : β) => Pow.pow a b }

@[defaultInstance 1000]

instance instPowNat {α : Type u_1} [NatPow α] :

Equations

instPowNat = { pow := fun (a : α) (n : Nat) => NatPow.pow a n }

@[defaultInstance 1000]

instance instPowOfHomogeneousPow {α : Type u_1} [HomogeneousPow α] :

Pow α α

Equations

instPowOfHomogeneousPow = { pow := fun (a b : α) => HomogeneousPow.pow a b }

@[defaultInstance 1000]

instance instHAppendOfAppend {α : Type u_1} [Append α] :

HAppend α α α

Equations

instHAppendOfAppend = { hAppend := fun (a b : α) => Append.append a b }

@[defaultInstance 1000]

instance instHOrElseOfOrElse {α : Type u_1} [OrElse α] :

HOrElse α α α

Equations

instHOrElseOfOrElse = { hOrElse := fun (a : α) (b : Unit → α) => OrElse.orElse a b }

@[defaultInstance 1000]

instance instHAndThenOfAndThen {α : Type u_1} [AndThen α] :

HAndThen α α α

Equations

instHAndThenOfAndThen = { hAndThen := fun (a : α) (b : Unit → α) => AndThen.andThen a b }

@[defaultInstance 1000]

instance instHAndOfAndOp {α : Type u_1} [AndOp α] :

HAnd α α α

Equations

instHAndOfAndOp = { hAnd := fun (a b : α) => AndOp.and a b }

@[defaultInstance 1000]

instance instHXorOfXor {α : Type u_1} [Xor α] :

HXor α α α

Equations

instHXorOfXor = { hXor := fun (a b : α) => Xor.xor a b }

@[defaultInstance 1000]

instance instHOrOfOrOp {α : Type u_1} [OrOp α] :

HOr α α α

Equations

instHOrOfOrOp = { hOr := fun (a b : α) => OrOp.or a b }

@[defaultInstance 1000]

instance instHShiftLeftOfShiftLeft {α : Type u_1} [ShiftLeft α] :

HShiftLeft α α α

Equations

instHShiftLeftOfShiftLeft = { hShiftLeft := fun (a b : α) => ShiftLeft.shiftLeft a b }

@[defaultInstance 1000]

instance instHShiftRightOfShiftRight {α : Type u_1} [ShiftRight α] :

HShiftRight α α α

Equations

instHShiftRightOfShiftRight = { hShiftRight := fun (a b : α) => ShiftRight.shiftRight a b }

class Membership (α : outParam (Type u)) (γ : Type v) :

Type (max u v)

The typeclass behind the notation a ∈ s : Prop where a : α, s : γ. Because α is an outParam, the "container type" γ determines the type of the elements of the container.

mem : γ → α → Prop
The membership relation a ∈ s : Prop where a : α, s : γ.
Conventions for notations in identifiers:
- The recommended spelling of ∈ in identifiers is mem.

Instances

@[match_pattern, extern lean_nat_add]

Nat → Nat → Nat

Addition of natural numbers, typically used via the + operator.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Equations

x✝.add Nat.zero = x✝
x✝.add b.succ = (x✝.add b).succ

instance instAddNat :

Equations

instAddNat = { add := Nat.add }

@[extern lean_nat_mul]

Nat → Nat → Nat

Multiplication of natural numbers, usually accessed via the * operator.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Equations

x✝.mul 0 = 0
x✝.mul b.succ = (x✝.mul b).add x✝

instance instMulNat :

Equations

instMulNat = { mul := Nat.mul }

@[extern lean_nat_pow]

def Nat.pow (m : Nat) :

The power operation on natural numbers, usually accessed via the ^ operator.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Equations

m.pow 0 = 1
m.pow n.succ = (m.pow n).mul m

instance instNatPowNat :

Equations

instNatPowNat = { pow := Nat.pow }

@[extern lean_nat_dec_eq]

Nat → Nat → Bool

Boolean equality of natural numbers, usually accessed via the == operator.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Equations

theorem Nat.eq_of_beq_eq_true {n m : Nat} :

n.beq m = true → n = m

theorem Nat.ne_of_beq_eq_false {n m : Nat} :

n.beq m = false → ¬n = m

@[reducible, extern lean_nat_dec_eq]

def Nat.decEq (n m : Nat) :

Decidable (n = m)

A decision procedure for equality of natural numbers, usually accessed via the DecidableEq Nat instance.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Examples:

Nat.decEq 5 5 = isTrue rfl
(if 3 = 4 then "yes" else "no") = "no"
show 12 = 12 by decide

Equations

n.decEq m = match h : n.beq m with | true => isTrue ⋯ | false => isFalse ⋯

@[inline]

instance instDecidableEqNat :

DecidableEq Nat

Equations

instDecidableEqNat = Nat.decEq

@[extern lean_nat_dec_le]

Nat → Nat → Bool

The Boolean less-than-or-equal-to comparison on natural numbers.

This function is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Examples:

Nat.ble 2 5 = true
Nat.ble 5 2 = false
Nat.ble 5 5 = true

Equations

inductive Nat.le (n : Nat) :

Non-strict, or weak, inequality of natural numbers, usually accessed via the ≤ operator.

refl {n : Nat} : n.le n
Non-strict inequality is reflexive: n ≤ n
step {n m : Nat} : n.le m → n.le m.succ
If n ≤ m, then n ≤ m + 1.

instance instLENat :

Equations

instLENat = { le := Nat.le }

def Nat.lt (n m : Nat) :

Strict inequality of natural numbers, usually accessed via the < operator.

It is defined as n < m = n + 1 ≤ m.

Equations

n.lt m = n.succ.le m

instance instLTNat :

Equations

instLTNat = { lt := Nat.lt }

theorem Nat.not_succ_le_zero (n : Nat) :

n.succ ≤ 0 → False

@[simp]

theorem Nat.not_lt_zero (n : Nat) :

¬n < 0

@[simp]

theorem Nat.zero_le (n : Nat) :

0 ≤ n

theorem Nat.succ_le_succ {n m : Nat} :

n ≤ m → n.succ ≤ m.succ

@[simp]

theorem Nat.zero_lt_succ (n : Nat) :

0 < n.succ

theorem Nat.le_step {n m : Nat} (h : n ≤ m) :

theorem Nat.le_trans {n m k : Nat} :

n ≤ m → m ≤ k → n ≤ k

theorem Nat.lt_trans {n m k : Nat} (h₁ : n < m) :

m < k → n < k

theorem Nat.le_succ (n : Nat) :

theorem Nat.le_succ_of_le {n m : Nat} (h : n ≤ m) :

@[simp]

theorem Nat.le_refl (n : Nat) :

n ≤ n

theorem Nat.succ_pos (n : Nat) :

0 < n.succ

@[extern lean_nat_pred]

The predecessor of a natural number is one less than it. The precedessor of 0 is defined to be 0.

This definition is overridden in the compiler with an efficient implementation. This definition is the logical model.

Equations

Nat.pred 0 = 0
n.succ.pred = n

theorem Nat.pred_le_pred {n m : Nat} :

n ≤ m → n.pred ≤ m.pred

theorem Nat.le_of_succ_le_succ {n m : Nat} :

n.succ ≤ m.succ → n ≤ m

theorem Nat.le_of_lt_succ {m n : Nat} :

m < n.succ → m ≤ n

def Nat.eq_or_lt_of_le {n m : Nat} :

n ≤ m → n = m ∨ n < m

Equations

⋯ = ⋯

theorem Nat.lt_or_ge (n m : Nat) :

n < m ∨ n ≥ m

theorem Nat.not_succ_le_self (n : Nat) :

@[simp]

theorem Nat.lt_irrefl (n : Nat) :

¬n < n

theorem Nat.lt_of_le_of_lt {n m k : Nat} (h₁ : n ≤ m) (h₂ : m < k) :

n < k

theorem Nat.le_antisymm {n m : Nat} (h₁ : n ≤ m) (h₂ : m ≤ n) :

n = m

theorem Nat.lt_of_le_of_ne {n m : Nat} (h₁ : n ≤ m) (h₂ : ¬n = m) :

n < m

theorem Nat.le_of_ble_eq_true {n m : Nat} (h : n.ble m = true) :

n ≤ m

theorem Nat.ble_self_eq_true (n : Nat) :

theorem Nat.ble_succ_eq_true {n m : Nat} :

n.ble m = true → n.ble m.succ = true

theorem Nat.ble_eq_true_of_le {n m : Nat} (h : n ≤ m) :

theorem Nat.not_le_of_not_ble_eq_true {n m : Nat} (h : ¬n.ble m = true) :

@[extern lean_nat_dec_le]

instance Nat.decLe (n m : Nat) :

Decidable (n ≤ m)

A decision procedure for non-strict inequality of natural numbers, usually accessed via the DecidableLE Nat instance.

Examples:

(if 3 ≤ 4 then "yes" else "no") = "yes"
(if 6 ≤ 4 then "yes" else "no") = "no"
show 12 ≤ 12 by decide
show 5 ≤ 12 by decide

Equations

n.decLe m = if h : n.ble m = true then isTrue ⋯ else isFalse ⋯

@[extern lean_nat_dec_lt]

instance Nat.decLt (n m : Nat) :

Decidable (n < m)

A decision procedure for strict inequality of natural numbers, usually accessed via the DecidableLT Nat instance.

Examples:

(if 3 < 4 then "yes" else "no") = "yes"
(if 4 < 4 then "yes" else "no") = "no"
(if 6 < 4 then "yes" else "no") = "no"
show 5 < 12 by decide

Equations

n.decLt m = n.succ.decLe m

instance instMinNat :

Equations

instMinNat = minOfLe

@[extern lean_nat_sub]

Nat → Nat → Nat

Subtraction of natural numbers, truncated at 0. Usually used via the - operator.

If a result would be less than zero, then the result is zero.

This definition is overridden in both the kernel and the compiler to efficiently evaluate using the arbitrary-precision arithmetic library. The definition provided here is the logical model.

Examples:

5 - 3 = 2
8 - 2 = 6
8 - 8 = 0
8 - 20 = 0

Equations

x✝.sub 0 = x✝
x✝.sub b.succ = (x✝.sub b).pred

instance instSubNat :

Equations

instSubNat = { sub := Nat.sub }

@[extern lean_system_platform_nbits]

opaque System.Platform.getNumBits :

Unit → { n : Nat // n = 32 ∨ n = 64 }

Gets the word size of the curent platform. The word size may be 64 or 32 bits.

This function is opaque because there is no guarantee at compile time that the target will have the same word size as the host. It also helps avoid having type checking be architecture-dependent.

Lean only works on 64 and 32 bit systems. This fact is visible in the return type.

def System.Platform.numBits :

The word size of the current platform, which may be 64 or 32 bits.

Equations

System.Platform.numBits = (System.Platform.getNumBits ()).val

Instances For

System.Platform.instNeZeroNatNumBits

theorem System.Platform.numBits_eq :

numBits = 32 ∨ numBits = 64

The word size of the current platform may be 64 or 32 bits.

structure Fin (n : Nat) :

Natural numbers less than some upper bound.

In particular, a Fin n is a natural number i with the constraint that i < n. It is the canonical type with n elements.

val : Nat
The number that is strictly less than n.
Fin.val is a coercion, so any Fin n can be used in a position where a Nat is expected.
isLt : ↑self < n
The number val is strictly less than the bound n.

Instances For

theorem Fin.eq_of_val_eq {n : Nat} {i j : Fin n} :

↑i = ↑j → i = j

theorem Fin.val_eq_of_eq {n : Nat} {i j : Fin n} (h : i = j) :

↑i = ↑j

instance instDecidableEqFin (n : Nat) :

DecidableEq (Fin n)

Equations

instDecidableEqFin n i j = match decEq ↑i ↑j with | isTrue h => isTrue ⋯ | isFalse h => isFalse ⋯

instance instLTFin {n : Nat} :

LT (Fin n)

Equations

instLTFin = { lt := fun (a b : Fin n) => ↑a < ↑b }

instance instLEFin {n : Nat} :

LE (Fin n)

Equations

instLEFin = { le := fun (a b : Fin n) => ↑a ≤ ↑b }

instance Fin.decLt {n : Nat} (a b : Fin n) :

Decidable (a < b)

Equations

a.decLt b = (↑a).decLt ↑b

instance Fin.decLe {n : Nat} (a b : Fin n) :

Decidable (a ≤ b)

Equations

a.decLe b = (↑a).decLe ↑b

structure BitVec (w : Nat) :

A bitvector of the specified width.

This is represented as the underlying Nat number in both the runtime and the kernel, inheriting all the special support for Nat.

ofFin :: (

toFin : Fin (2 ^ w)
Interpret a bitvector as a number less than 2^w. O(1), because we use Fin as the internal representation of a bitvector.

)

Instances For

def BitVec.decEq {w : Nat} (x y : BitVec w) :

Decidable (x = y)

Bitvectors have decidable equality.

This should be used via the instance DecidableEq (BitVec w).

Equations

{ toFin := n }.decEq { toFin := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqBitVec {w : Nat} :

DecidableEq (BitVec w)

Equations

instDecidableEqBitVec = BitVec.decEq

@[match_pattern]

def BitVec.ofNatLT {w : Nat} (i : Nat) (p : i < 2 ^ w) :

The BitVec with value i, given a proof that i < 2^w.

Equations

i#'p = { toFin := ⟨i, p⟩ }

def BitVec.toNat {w : Nat} (x : BitVec w) :

Return the underlying Nat that represents a bitvector.

This is O(1) because BitVec is a (zero-cost) wrapper around a Nat.

Equations

x.toNat = ↑x.toFin

instance instLTBitVec {w : Nat} :

Equations

instLTBitVec = { lt := fun (x1 x2 : BitVec w) => x1.toNat < x2.toNat }

instance instDecidableLtBitVec {w : Nat} (x y : BitVec w) :

Decidable (x < y)

Equations

instDecidableLtBitVec x y = inferInstanceAs (Decidable (x.toNat < y.toNat))

instance instLEBitVec {w : Nat} :

Equations

instLEBitVec = { le := fun (x1 x2 : BitVec w) => x1.toNat ≤ x2.toNat }

instance instDecidableLeBitVec {w : Nat} (x y : BitVec w) :

Decidable (x ≤ y)

Equations

instDecidableLeBitVec x y = inferInstanceAs (Decidable (x.toNat ≤ y.toNat))

@[reducible, inline]

abbrev UInt8.size :

The number of distinct values representable by UInt8, that is, 2^8 = 256.

Equations

UInt8.size = 256

structure UInt8 :

Unsigned 8-bit integers.

This type has special support in the compiler so it can be represented by an unboxed 8-bit value rather than wrapping a BitVec 8.

ofBitVec :: (

toBitVec : BitVec 8
Unpacks a UInt8 into a BitVec 8. This function is overridden with a native implementation.

)

Instances For

@[extern lean_uint8_of_nat]

def UInt8.ofNatLT (n : Nat) (h : n < size) :

Converts a natural number to an 8-bit unsigned integer. Requires a proof that the number is small enough to be representable without overflow; it must be smaller than 2^8.

This function is overridden at runtime with an efficient implementation.

Equations

UInt8.ofNatLT n h = { toBitVec := n#'h }

@[extern lean_uint8_dec_eq]

def UInt8.decEq (a b : UInt8) :

Decidable (a = b)

Decides whether two 8-bit unsigned integers are equal. Usually accessed via the DecidableEq UInt8 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

UInt8.decEq 123 123 = .isTrue rfl
(if (6 : UInt8) = 7 then "yes" else "no") = "no"
show (7 : UInt8) = 7 by decide

Equations

{ toBitVec := n }.decEq { toBitVec := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqUInt8 :

DecidableEq UInt8

Equations

instDecidableEqUInt8 = UInt8.decEq

instance instInhabitedUInt8 :

Inhabited UInt8

Equations

instInhabitedUInt8 = { default := UInt8.ofNatLT 0 instInhabitedUInt8._proof_11 }

@[reducible, inline]

abbrev UInt16.size :

The number of distinct values representable by UInt16, that is, 2^16 = 65536.

Equations

UInt16.size = 65536

structure UInt16 :

Unsigned 16-bit integers.

This type has special support in the compiler so it can be represented by an unboxed 16-bit value rather than wrapping a BitVec 16.

ofBitVec :: (

toBitVec : BitVec 16
Unpacks a UInt16 into a BitVec 16. This function is overridden with a native implementation.

)

Instances For

@[extern lean_uint16_of_nat]

def UInt16.ofNatLT (n : Nat) (h : n < size) :

Converts a natural number to a 16-bit unsigned integer. Requires a proof that the number is small enough to be representable without overflow; it must be smaller than 2^16.

This function is overridden at runtime with an efficient implementation.

Equations

UInt16.ofNatLT n h = { toBitVec := n#'h }

@[extern lean_uint16_dec_eq]

def UInt16.decEq (a b : UInt16) :

Decidable (a = b)

Decides whether two 16-bit unsigned integers are equal. Usually accessed via the DecidableEq UInt16 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

UInt16.decEq 123 123 = .isTrue rfl
(if (6 : UInt16) = 7 then "yes" else "no") = "no"
show (7 : UInt16) = 7 by decide

Equations

{ toBitVec := n }.decEq { toBitVec := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqUInt16 :

DecidableEq UInt16

Equations

instDecidableEqUInt16 = UInt16.decEq

instance instInhabitedUInt16 :

Inhabited UInt16

Equations

instInhabitedUInt16 = { default := UInt16.ofNatLT 0 instInhabitedUInt16._proof_14 }

@[reducible, inline]

abbrev UInt32.size :

The number of distinct values representable by UInt32, that is, 2^32 = 4294967296.

Equations

UInt32.size = 4294967296

structure UInt32 :

Unsigned 32-bit integers.

This type has special support in the compiler so it can be represented by an unboxed 32-bit value rather than wrapping a BitVec 32.

ofBitVec :: (

toBitVec : BitVec 32
Unpacks a UInt32 into a BitVec 32. This function is overridden with a native implementation.

)

Instances For

@[extern lean_uint32_of_nat]

def UInt32.ofNatLT (n : Nat) (h : n < size) :

Converts a natural number to a 32-bit unsigned integer. Requires a proof that the number is small enough to be representable without overflow; it must be smaller than 2^32.

This function is overridden at runtime with an efficient implementation.

Equations

UInt32.ofNatLT n h = { toBitVec := n#'h }

@[extern lean_uint32_to_nat]

def UInt32.toNat (n : UInt32) :

Converts a 32-bit unsigned integer to an arbitrary-precision natural number.

This function is overridden at runtime with an efficient implementation.

Equations

n.toNat = n.toBitVec.toNat

@[extern lean_uint32_dec_eq]

def UInt32.decEq (a b : UInt32) :

Decidable (a = b)

Decides whether two 32-bit unsigned integers are equal. Usually accessed via the DecidableEq UInt32 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

UInt32.decEq 123 123 = .isTrue rfl
(if (6 : UInt32) = 7 then "yes" else "no") = "no"
show (7 : UInt32) = 7 by decide

Equations

{ toBitVec := n }.decEq { toBitVec := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqUInt32 :

DecidableEq UInt32

Equations

instDecidableEqUInt32 = UInt32.decEq

instance instInhabitedUInt32 :

Inhabited UInt32

Equations

instInhabitedUInt32 = { default := UInt32.ofNatLT 0 instInhabitedUInt32._proof_17 }

instance instLTUInt32 :

Equations

instLTUInt32 = { lt := fun (a b : UInt32) => a.toBitVec < b.toBitVec }

instance instLEUInt32 :

Equations

instLEUInt32 = { le := fun (a b : UInt32) => a.toBitVec ≤ b.toBitVec }

@[extern lean_uint32_dec_lt]

def UInt32.decLt (a b : UInt32) :

Decidable (a < b)

Decides whether one 8-bit unsigned integer is strictly less than another. Usually accessed via the DecidableLT UInt32 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

(if (6 : UInt32) < 7 then "yes" else "no") = "yes"
(if (5 : UInt32) < 5 then "yes" else "no") = "no"
show ¬((7 : UInt32) < 7) by decide

Equations

a.decLt b = inferInstanceAs (Decidable (a.toBitVec < b.toBitVec))

@[extern lean_uint32_dec_le]

def UInt32.decLe (a b : UInt32) :

Decidable (a ≤ b)

Decides whether one 32-bit signed integer is less than or equal to another. Usually accessed via the DecidableLE UInt32 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

(if (15 : UInt32) ≤ 15 then "yes" else "no") = "yes"
(if (15 : UInt32) ≤ 5 then "yes" else "no") = "no"
(if (5 : UInt32) ≤ 15 then "yes" else "no") = "yes"
show (7 : UInt32) ≤ 7 by decide

Equations

a.decLe b = inferInstanceAs (Decidable (a.toBitVec ≤ b.toBitVec))

instance instDecidableLtUInt32 (a b : UInt32) :

Decidable (a < b)

Equations

instDecidableLtUInt32 a b = a.decLt b

instance instDecidableLeUInt32 (a b : UInt32) :

Decidable (a ≤ b)

Equations

instDecidableLeUInt32 a b = a.decLe b

instance instMaxUInt32 :

Equations

instMaxUInt32 = maxOfLe

instance instMinUInt32 :

Equations

instMinUInt32 = minOfLe

@[reducible, inline]

abbrev UInt64.size :

The number of distinct values representable by UInt64, that is, 2^64 = 18446744073709551616.

Equations

UInt64.size = 18446744073709551616

structure UInt64 :

Unsigned 64-bit integers.

This type has special support in the compiler so it can be represented by an unboxed 64-bit value rather than wrapping a BitVec 64.

ofBitVec :: (

toBitVec : BitVec 64
Unpacks a UInt64 into a BitVec 64. This function is overridden with a native implementation.

)

Instances For

@[extern lean_uint64_of_nat]

def UInt64.ofNatLT (n : Nat) (h : n < size) :

Converts a natural number to a 64-bit unsigned integer. Requires a proof that the number is small enough to be representable without overflow; it must be smaller than 2^64.

This function is overridden at runtime with an efficient implementation.

Equations

UInt64.ofNatLT n h = { toBitVec := n#'h }

@[extern lean_uint64_dec_eq]

def UInt64.decEq (a b : UInt64) :

Decidable (a = b)

Decides whether two 64-bit unsigned integers are equal. Usually accessed via the DecidableEq UInt64 instance.

This function is overridden at runtime with an efficient implementation.

Examples:

UInt64.decEq 123 123 = .isTrue rfl
(if (6 : UInt64) = 7 then "yes" else "no") = "no"
show (7 : UInt64) = 7 by decide

Equations

{ toBitVec := n }.decEq { toBitVec := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqUInt64 :

DecidableEq UInt64

Equations

instDecidableEqUInt64 = UInt64.decEq

instance instInhabitedUInt64 :

Inhabited UInt64

Equations

instInhabitedUInt64 = { default := UInt64.ofNatLT 0 instInhabitedUInt64._proof_20 }

@[reducible, inline]

abbrev USize.size :

The number of distinct values representable by USize, that is, 2^System.Platform.numBits.

Equations

USize.size = 2 ^ System.Platform.numBits

theorem USize.size_eq :

size = 4294967296 ∨ size = 18446744073709551616

theorem USize.size_pos :

structure USize :

Unsigned integers that are the size of a word on the platform's architecture.

On a 32-bit architecture, USize is equivalent to UInt32. On a 64-bit machine, it is equivalent to UInt64.

ofBitVec :: (

toBitVec : BitVec System.Platform.numBits
Unpacks a USize into a BitVec System.Platform.numBits. This function is overridden with a native implementation.

)

Instances For

@[extern lean_usize_of_nat]

def USize.ofNatLT (n : Nat) (h : n < size) :

Converts a natural number to a USize. Requires a proof that the number is small enough to be representable without overflow.

This function is overridden at runtime with an efficient implementation.

Equations

USize.ofNatLT n h = { toBitVec := n#'h }

@[extern lean_usize_dec_eq]

def USize.decEq (a b : USize) :

Decidable (a = b)

Decides whether two word-sized unsigned integers are equal. Usually accessed via the DecidableEq USize instance.

This function is overridden at runtime with an efficient implementation.

Examples:

USize.decEq 123 123 = .isTrue rfl
(if (6 : USize) = 7 then "yes" else "no") = "no"
show (7 : USize) = 7 by decide

Equations

{ toBitVec := n }.decEq { toBitVec := m } = if h : n = m then isTrue ⋯ else isFalse ⋯

instance instDecidableEqUSize :

DecidableEq USize

Equations

instDecidableEqUSize = USize.decEq

instance instInhabitedUSize :

Inhabited USize

Equations

instInhabitedUSize = { default := USize.ofNatLT 0 USize.size_pos }

@[reducible, inline]

abbrev Nat.isValidChar (n : Nat) :

A Nat denotes a valid Unicode code point if it is less than 0x110000 and it is also not a surrogate code point (the range 0xd800 to 0xdfff inclusive).

Equations

n.isValidChar = (n < 55296 ∨ 57343 < n ∧ n < 1114112)

@[reducible, inline]

abbrev UInt32.isValidChar (n : UInt32) :

A UInt32 denotes a valid Unicode code point if it is less than 0x110000 and it is also not a surrogate code point (the range 0xd800 to 0xdfff inclusive).

Equations

n.isValidChar = n.toNat.isValidChar

structure Char :

Characters are Unicode scalar values.

val : UInt32
The underlying Unicode scalar value as a UInt32.
valid : self.val.isValidChar
The value must be a legal scalar value.

Instances For

@[extern lean_uint32_of_nat]

def Char.ofNatAux (n : Nat) (h : n.isValidChar) :

Pack a Nat encoding a valid codepoint into a Char. This function is overridden with a native implementation.

Equations

Char.ofNatAux n h = { val := { toBitVec := n#'⋯ }, valid := h }

@[match_pattern, noinline]

def Char.ofNat (n : Nat) :

Converts a Nat into a Char. If the Nat does not encode a valid Unicode scalar value, '\0' is returned instead.

Equations

Char.ofNat n = if h : n.isValidChar then Char.ofNatAux n h else { val := { toBitVec := 0#'Char.ofNat._proof_23 }, valid := Char.ofNat._proof_24 }

theorem Char.eq_of_val_eq {c d : Char} :

c.val = d.val → c = d

theorem Char.val_eq_of_eq {c d : Char} :

c = d → c.val = d.val

theorem Char.ne_of_val_ne {c d : Char} (h : ¬c.val = d.val) :

¬c = d

theorem Char.val_ne_of_ne {c d : Char} (h : ¬c = d) :

¬c.val = d.val

instance instDecidableEqChar :

DecidableEq Char

Equations

instDecidableEqChar c d = match decEq c.val d.val with | isTrue h => isTrue ⋯ | isFalse h => isFalse ⋯

def Char.utf8Size (c : Char) :

Returns the number of bytes required to encode this Char in UTF-8.

Equations

One or more equations did not get rendered due to their size.

@[unbox]

inductive Option (α : Type u) :

Optional values, which are either some around a value from the underlying type or none.

Option can represent nullable types or computations that might fail. In the codomain of a function type, it can also represent partiality.

none {α : Type u} : Option α
No value.
some {α : Type u} (val : α) : Option α
Some value of type α.

Instances For

instance instInhabitedOption {α : Type u_1} :

Inhabited (Option α)

Equations

instInhabitedOption = { default := none }

@[macro_inline]

def Option.getD {α : Type u_1} (opt : Option α) (dflt : α) :

α

Gets an optional value, returning a given default on none.

This function is @[macro_inline], so dflt will not be evaluated unless opt turns out to be none.

Examples:

(some "hello").getD "goodbye" = "hello"
none.getD "goodbye" = "hello"

Equations

(some x).getD dflt = x
none.getD dflt = dflt

@[inline]

def Option.map {α : Type u_1} {β : Type u_2} (f : α → β) :

Option α → Option β

Apply a function to an optional value, if present.

From the perspective of Option as a container with at most one value, this is analogous to List.map. It can also be accessed via the Functor Option instance.

Examples:

(none : Option Nat).map (· + 1) = none
(some 3).map (· + 1) = some 4

Equations

Option.map f (some x) = some (f x)
Option.map f none = none

inductive List (α : Type u) :

Linked lists: ordered lists, in which each element has a reference to the next element.

Most operations on linked lists take time proportional to the length of the list, because each element must be traversed to find the next element.

List α is isomorphic to Array α, but they are useful for different things:

List α is easier for reasoning, and Array α is modeled as a wrapper around List α.
List α works well as a persistent data structure, when many copies of the tail are shared. When the value is not shared, Array α will have better performance because it can do destructive updates.

nil {α : Type u} : List α
The empty list, usually written [].
Conventions for notations in identifiers:
- The recommended spelling of [] in identifiers is nil.
cons {α : Type u} (head : α) (tail : List α) : List α
The list whose first element is head, where tail is the rest of the list. Usually written head :: tail.
Conventions for notations in identifiers:
- The recommended spelling of :: in identifiers is cons.
- The recommended spelling of [a] in identifiers is singleton.

Instances For

instance instInhabitedList {α : Type u_1} :

Inhabited (List α)

Equations

instInhabitedList = { default := [] }

def List.hasDecEq {α : Type u} [DecidableEq α] (a b : List α) :

Decidable (a = b)

Implements decidable equality for List α, assuming α has decidable equality.

Equations

[].hasDecEq [] = isTrue ⋯
(head :: tail).hasDecEq [] = isFalse ⋯
[].hasDecEq (head :: tail) = isFalse ⋯
(a :: as).hasDecEq (b :: bs) = match decEq a b with | isTrue hab => match as.hasDecEq bs with | isTrue habs => isTrue ⋯ | isFalse nabs => isFalse ⋯ | isFalse nab => isFalse ⋯

instance instDecidableEqList {α : Type u} [DecidableEq α] :

DecidableEq (List α)

Equations

instDecidableEqList = List.hasDecEq

def List.length {α : Type u_1} :

List α → Nat

The length of a list.

This function is overridden in the compiler to lengthTR, which uses constant stack space.

Examples:

([] : List String).length = 0
["green", "brown"].length = 2

Equations

[].length = 0
(head :: as).length = as.length + 1

def List.lengthTRAux {α : Type u_1} :

List α → Nat → Nat

Auxiliary function for List.lengthTR.

Equations

[].lengthTRAux x✝ = x✝
(head :: as).lengthTRAux x✝ = as.lengthTRAux x✝.succ

def List.lengthTR {α : Type u_1} (as : List α) :

The length of a list.

This is a tail-recursive version of List.length, used to implement List.length without running out of stack space.

Examples:

([] : List String).lengthTR = 0
["green", "brown"].lengthTR = 2

Equations

as.lengthTR = as.lengthTRAux 0

def List.get {α : Type u} (as : List α) :

Fin as.length → α

Returns the element at the provided index, counting from 0.

In other words, for i : Fin as.length, as.get i returns the i'th element of the list as. Because the index is a Fin bounded by the list's length, the index will never be out of bounds.

Examples:

["spring", "summer", "fall", "winter"].get (2 : Fin 4) = "fall"
["spring", "summer", "fall", "winter"].get (0 : Fin 4) = "spring"

Equations

(a :: tail).get ⟨0, isLt⟩ = a
(head :: as).get ⟨i.succ, h⟩ = as.get ⟨i, ⋯⟩

def List.set {α : Type u_1} (l : List α) (n : Nat) (a : α) :

Replaces the value at (zero-based) index n in l with a. If the index is out of bounds, then the list is returned unmodified.

Examples:

["water", "coffee", "soda", "juice"].set 1 "tea" = ["water", "tea", "soda", "juice"]
["water", "coffee", "soda", "juice"].set 4 "tea" = ["water", "coffee", "soda", "juice"]

Equations

(head :: as).set 0 x✝ = x✝ :: as
(a :: as).set n.succ x✝ = a :: as.set n x✝
[].set x✝¹ x✝ = []

@[specialize #[]]

def List.foldl {α : Type u} {β : Type v} (f : α → β → α) (init : α) :

List β → α

Folds a function over a list from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

Examples:

[a, b, c].foldl f z = f (f (f z a) b) c
[1, 2, 3].foldl (· ++ toString ·) "" = "123"
[1, 2, 3].foldl (s!"({·} {·})") "" = "((( 1) 2) 3)"

Equations

List.foldl f x✝ [] = x✝
List.foldl f x✝ (b :: l) = List.foldl f (f x✝ b) l

def List.concat {α : Type u} :

List α → α → List α

Adds an element to the end of a list.

The added element is the last element of the resulting list.

Examples:

List.concat ["red", "yellow"] "green" = ["red", "yellow", "green"]
List.concat [1, 2, 3] 4 = [1, 2, 3, 4]
List.concat [] () = [()]

Equations

[].concat x✝ = [x✝]
(a :: as).concat x✝ = a :: as.concat x✝

structure String :

A string is a sequence of Unicode code points.

At runtime, strings are represented by dynamic arrays of bytes using the UTF-8 encoding. Both the size in bytes (String.utf8ByteSize) and in characters (String.length) are cached and take constant time. Many operations on strings perform in-place modifications when the reference to the string is unique.

data : List Char
Unpack String into a List Char. This function is overridden by the compiler and is O(n) in the length of the list.

Instances For

@[extern lean_string_dec_eq]

def String.decEq (s₁ s₂ : String) :

Decidable (s₁ = s₂)

Decides whether two strings are equal. Normally used via the DecidableEq String instance and the = operator.

At runtime, this function is overridden with an efficient native implementation.

Equations

{ data := s₁_2 }.decEq { data := s₂_2 } = if h : s₁_2 = s₂_2 then isTrue ⋯ else isFalse ⋯

instance instDecidableEqString :

DecidableEq String

Equations

instDecidableEqString = String.decEq

structure String.Pos :

A byte position in a String, according to its UTF-8 encoding.

Character positions (counting the Unicode code points rather than bytes) are represented by plain Nats. Indexing a String by a String.Pos takes constant time, while character positions need to be translated internally to byte positions, which takes linear time.

A byte position p is valid for a string s if 0 ≤ p ≤ s.endPos and p lies on a UTF-8 character boundary.

byteIdx : Nat
Get the underlying byte index of a String.Pos

Instances For

instance instInhabitedPos :

Inhabited String.Pos

Equations

instInhabitedPos = { default := { } }

instance instDecidableEqPos :

DecidableEq String.Pos

Equations

instDecidableEqPos { byteIdx := b } { byteIdx := b_1 } = match decEq b b_1 with | isTrue h => isTrue ⋯ | isFalse h => isFalse ⋯

structure Substring :

A region or slice of some underlying string.

A substring contains an string together with the start and end byte positions of a region of interest. Actually extracting a substring requires copying and memory allocation, while many substrings of the same underlying string may exist with very little overhead, and they are more convenient than tracking the bounds by hand.

Using its constructor explicitly, it is possible to construct a Substring in which one or both of the positions is invalid for the string. Many operations will return unexpected or confusing results if the start and stop positions are not valid. Instead, it's better to use API functions that ensure the validity of the positions in a substring to create and manipulate them.

str : String
The underlying string.
startPos : String.Pos
The byte position of the start of the string slice.
stopPos : String.Pos
The byte position of the end of the string slice.

Instances For

instance instInhabitedSubstring :

Inhabited Substring

Equations

instInhabitedSubstring = { default := { str := "", startPos := { }, stopPos := { } } }

@[inline]

def Substring.bsize :

Substring → Nat

The number of bytes used by the string's UTF-8 encoding.

Equations

{ str := str, startPos := b, stopPos := e }.bsize = e.byteIdx.sub b.byteIdx

@[extern lean_string_utf8_byte_size]

def String.utf8ByteSize :

The number of bytes used by the string's UTF-8 encoding.

At runtime, this function takes constant time because the byte length of strings is cached.

Equations

{ data := s }.utf8ByteSize = String.utf8ByteSize.go s

def String.utf8ByteSize.go :

List Char → Nat

Equations

String.utf8ByteSize.go [] = 0
String.utf8ByteSize.go (c :: cs) = String.utf8ByteSize.go cs + c.utf8Size

instance instHAddPos :

HAdd String.Pos String.Pos String.Pos

Equations

instHAddPos = { hAdd := fun (p₁ p₂ : String.Pos) => { byteIdx := p₁.byteIdx + p₂.byteIdx } }

instance instHSubPos :

HSub String.Pos String.Pos String.Pos

Equations

instHSubPos = { hSub := fun (p₁ p₂ : String.Pos) => { byteIdx := p₁.byteIdx - p₂.byteIdx } }

instance instHAddPosChar :

HAdd String.Pos Char String.Pos

Equations

instHAddPosChar = { hAdd := fun (p : String.Pos) (c : Char) => { byteIdx := p.byteIdx + c.utf8Size } }

instance instHAddPosString :

HAdd String.Pos String String.Pos

Equations

instHAddPosString = { hAdd := fun (p : String.Pos) (s : String) => { byteIdx := p.byteIdx + s.utf8ByteSize } }

instance instLEPos :

Equations

instLEPos = { le := fun (p₁ p₂ : String.Pos) => p₁.byteIdx ≤ p₂.byteIdx }

instance instLTPos :

Equations

instLTPos = { lt := fun (p₁ p₂ : String.Pos) => p₁.byteIdx < p₂.byteIdx }

instance instDecidableLePos (p₁ p₂ : String.Pos) :

Decidable (p₁ ≤ p₂)

Equations

instDecidableLePos p₁ p₂ = inferInstanceAs (Decidable (p₁.byteIdx ≤ p₂.byteIdx))

instance instDecidableLtPos (p₁ p₂ : String.Pos) :

Decidable (p₁ < p₂)

Equations

instDecidableLtPos p₁ p₂ = inferInstanceAs (Decidable (p₁.byteIdx < p₂.byteIdx))

instance instMinPos :

Equations

instMinPos = minOfLe

instance instMaxPos :

Equations

instMaxPos = maxOfLe

@[inline]

def String.endPos (s : String) :

A UTF-8 byte position that points at the end of a string, just after the last character.

"abc".endPos = ⟨3⟩
"L∃∀N".endPos = ⟨8⟩

Equations

s.endPos = { byteIdx := s.utf8ByteSize }

@[inline]

def String.toSubstring (s : String) :

Converts a String into a Substring that denotes the entire string.

Equations

s.toSubstring = { str := s, startPos := { }, stopPos := s.endPos }

def String.toSubstring' (s : String) :

Converts a String into a Substring that denotes the entire string.

This is a version of String.toSubstring that doesn't have an @[inline] annotation.

Equations

s.toSubstring' = s.toSubstring

unsafe def unsafeCast {α : Sort u} {β : Sort v} (a : α) :

β

This function will cast a value of type α to type β, and is a no-op in the compiler. This function is extremely dangerous because there is no guarantee that types α and β have the same data representation, and this can lead to memory unsafety. It is also logically unsound, since you could just cast True to False. For all those reasons this function is marked as unsafe.

It is implemented by lifting both α and β into a common universe, and then using cast (lcProof : ULift (PLift α) = ULift (PLift β)) to actually perform the cast. All these operations are no-ops in the compiler.

Using this function correctly requires some knowledge of the data representation of the source and target types. Some general classes of casts which are safe in the current runtime:

Array α to Array β where α and β have compatible representations, or more generally for other inductive types.
Quot α r and α.
@Subtype α p and α, or generally any structure containing only one non-Prop field of type α.
Casting α to/from NonScalar when α is a boxed generic type (i.e. a function that accepts an arbitrary type α and is not specialized to a scalar type like UInt8).

Equations

unsafeCast a = (cast ⋯ { down := { down := a } }).down.down

@[never_extract, extern lean_panic_fn]

def panicCore {α : Sort u} [Inhabited α] (msg : String) :

α

Auxiliary definition for panic.

Equations

panicCore msg = default

@[never_extract, noinline]

def panic {α : Sort u} [Inhabited α] (msg : String) :

α

(panic "msg" : α) has a built-in implementation which prints msg to the error buffer. It does not terminate execution, and because it is a safe function, it still has to return an element of α, so it takes [Inhabited α] and returns default. It is primarily intended for debugging in pure contexts, and assertion failures.

Because this is a pure function with side effects, it is marked as @[never_extract] so that the compiler will not perform common sub-expression elimination and other optimizations that assume that the expression is pure.

Equations

panic msg = panicCore msg

structure Array (α : Type u) :

Array α is the type of dynamic arrays with elements from α. This type has special support in the runtime.

Arrays perform best when unshared. As long as there is never more than one reference to an array, all updates will be performed destructively. This results in performance comparable to mutable arrays in imperative programming languages.

An array has a size and a capacity. The size is the number of elements present in the array, while the capacity is the amount of memory currently allocated for elements. The size is accessible via Array.size, but the capacity is not observable from Lean code. Array.emptyWithCapacity n creates an array which is equal to #[], but internally allocates an array of capacity n. When the size exceeds the capacity, allocation is required to grow the array.

From the point of view of proofs, Array α is just a wrapper around List α.

toList : List α
Converts an Array α into a List α that contains the same elements in the same order.
At runtime, this is implemented by Array.toListImpl and is O(n) in the length of the array.

Instances For

@[reducible, match_pattern, inline]

abbrev List.toArray {α : Type u_1} (xs : List α) :

Converts a List α into an Array α.

O(|xs|). At runtime, this operation is implemented by List.toArrayImpl and takes time linear in the length of the list. List.toArray should be used instead of Array.mk.

Examples:

[1, 2, 3].toArray = #[1, 2, 3]
["monday", "wednesday", friday"].toArray = #["monday", "wednesday", friday"].

Equations

xs.toArray = { toList := xs }

@[extern lean_mk_empty_array_with_capacity]

def Array.mkEmpty {α : Type u} (c : Nat) :

Constructs a new empty array with initial capacity c.

This will be deprecated in favor of Array.emptyWithCapacity in the future.

Equations

Array.mkEmpty c = { toList := [] }

@[extern lean_mk_empty_array_with_capacity]

def Array.emptyWithCapacity {α : Type u} (c : Nat) :

Constructs a new empty array with initial capacity c.

Equations

Array.emptyWithCapacity c = { toList := [] }

def Array.empty {α : Type u} :

Constructs a new empty array with initial capacity 0.

Use Array.emptyWithCapacity to create an array with a greater initial capacity.

Equations

#[] = Array.emptyWithCapacity 0

@[reducible, extern lean_array_get_size]

def Array.size {α : Type u} (a : Array α) :

Gets the number of elements stored in an array.

This is a cached value, so it is O(1) to access. The space allocated for an array, referred to as its capacity, is at least as large as its size, but may be larger. The capacity of an array is an internal detail that's not observable by Lean code.

Equations

a.size = a.toList.length

@[extern lean_array_fget]

def Array.getInternal {α : Type u} (a : Array α) (i : Nat) (h : i < a.size) :

α

Use the indexing notation a[i] instead.

Access an element from an array without needing a runtime bounds checks, using a Nat index and a proof that it is in bounds.

This function does not use get_elem_tactic to automatically find the proof that the index is in bounds. This is because the tactic itself needs to look up values in arrays.

Equations

a.getInternal i h = a.toList.get ⟨i, h⟩

@[reducible, inline]

abbrev Array.getD {α : Type u_1} (a : Array α) (i : Nat) (v₀ : α) :

α

Returns the element at the provided index, counting from 0. Returns the fallback value v₀ if the index is out of bounds.

To return an Option depending on whether the index is in bounds, use a[i]?. To panic if the index is out of bounds, use a[i]!.

Examples:

#["spring", "summer", "fall", "winter"].getD 2 "never" = "fall"
#["spring", "summer", "fall", "winter"].getD 0 "never" = "spring"
#["spring", "summer", "fall", "winter"].getD 4 "never" = "never"

Equations

a.getD i v₀ = if h : i < a.size then a.getInternal i h else v₀

@[extern lean_array_get]

def Array.get!Internal {α : Type u} [Inhabited α] (a : Array α) (i : Nat) :

α

Use the indexing notation a[i]! instead.

Access an element from an array, or panic if the index is out of bounds.

Equations

a.get!Internal i = a.getD i default

@[extern lean_array_push]

def Array.push {α : Type u} (a : Array α) (v : α) :

Adds an element to the end of an array. The resulting array's size is one greater than the input array. If there are no other references to the array, then it is modified in-place.

This takes amortized O(1) time because Array α is represented by a dynamic array.

Examples:

#[].push "apple" = #["apple"]
#["apple"].push "orange" = #["apple", "orange"]

Equations

a.push v = { toList := a.toList.concat v }

def Array.mkArray0 {α : Type u} :

Create array #[]

Equations

#[] = Array.emptyWithCapacity 0

def Array.mkArray1 {α : Type u} (a₁ : α) :

Create array #[a₁]

Equations

#[a₁] = (Array.emptyWithCapacity 1).push a₁

def Array.mkArray2 {α : Type u} (a₁ a₂ : α) :

Create array #[a₁, a₂]

Equations

#[a₁, a₂] = ((Array.emptyWithCapacity 2).push a₁).push a₂

def Array.mkArray3 {α : Type u} (a₁ a₂ a₃ : α) :

Create array #[a₁, a₂, a₃]

Equations

#[a₁, a₂, a₃] = (((Array.emptyWithCapacity 3).push a₁).push a₂).push a₃

def Array.mkArray4 {α : Type u} (a₁ a₂ a₃ a₄ : α) :

Create array #[a₁, a₂, a₃, a₄]

Equations

#[a₁, a₂, a₃, a₄] = ((((Array.emptyWithCapacity 4).push a₁).push a₂).push a₃).push a₄

def Array.mkArray5 {α : Type u} (a₁ a₂ a₃ a₄ a₅ : α) :

Create array #[a₁, a₂, a₃, a₄, a₅]

Equations

#[a₁, a₂, a₃, a₄, a₅] = (((((Array.emptyWithCapacity 5).push a₁).push a₂).push a₃).push a₄).push a₅

def Array.mkArray6 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ : α) :

Create array #[a₁, a₂, a₃, a₄, a₅, a₆]

Equations

#[a₁, a₂, a₃, a₄, a₅, a₆] = ((((((Array.emptyWithCapacity 6).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆

def Array.mkArray7 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ a₇ : α) :

Create array #[a₁, a₂, a₃, a₄, a₅, a₆, a₇]

Equations

#[a₁, a₂, a₃, a₄, a₅, a₆, a₇] = (((((((Array.emptyWithCapacity 7).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆).push a₇

def Array.mkArray8 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : α) :

Create array #[a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈]

Equations

#[a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈] = ((((((((Array.emptyWithCapacity 8).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆).push a₇).push a₈

def Array.appendCore {α : Type u} (as bs : Array α) :

Slower Array.append used in quotations.

Equations

as.appendCore bs = Array.appendCore.loop bs bs.size 0 as

def Array.appendCore.loop {α : Type u} (bs : Array α) (i j : Nat) (as : Array α) :

Equations

Array.appendCore.loop bs i j as = if hlt : j < bs.size then match i with | 0 => as | i'.succ => Array.appendCore.loop bs i' (j + 1) (as.push (bs.getInternal j hlt)) else as

def Array.extract {α : Type u_1} (as : Array α) (start : Nat := 0) (stop : Nat := as.size) :

Returns the slice of as from indices start to stop (exclusive). The resulting array has size (min stop as.size) - start.

If start is greater or equal to stop, the result is empty. If stop is greater than the size of as, the size is used instead.

Examples:

#[0, 1, 2, 3, 4].extract 1 3 = #[1, 2]
#[0, 1, 2, 3, 4].extract 1 30 = #[1, 2, 3, 4]
#[0, 1, 2, 3, 4].extract 0 0 = #[]
#[0, 1, 2, 3, 4].extract 2 1 = #[]
#[0, 1, 2, 3, 4].extract 2 2 = #[]
#[0, 1, 2, 3, 4].extract 2 3 = #[2]
#[0, 1, 2, 3, 4].extract 2 4 = #[2, 3]

Equations

as.extract start stop = Array.extract.loop as ((min stop as.size).sub start) start (Array.emptyWithCapacity ((min stop as.size).sub start))

def Array.extract.loop {α : Type u_1} (as : Array α) (i j : Nat) (bs : Array α) :

Equations

Array.extract.loop as i j bs = if hlt : j < as.size then match i with | 0 => bs | i'.succ => Array.extract.loop as i' (j + 1) (bs.push (as.getInternal j hlt)) else bs

class Bind (m : Type u → Type v) :

Type (max (u + 1) v)

The >>= operator is overloaded via instances of bind.

Bind is typically used via Monad, which extends it.

bind {α β : Type u} : m α → (α → m β) → m β
Sequences two computations, allowing the second to depend on the value computed by the first.
If x : m α and f : α → m β, then x >>= f : m β represents the result of executing x to get a value of type α and then passing it to f.
Conventions for notations in identifiers:
- The recommended spelling of >>= in identifiers is bind.

Instances

class Pure (f : Type u → Type v) :

Type (max (u + 1) v)

The pure function is overloaded via Pure instances.

Pure is typically accessed via Monad or Applicative instances.

pure {α : Type u} : α → f α
Given a : α, then pure a : f α represents an action that does nothing and returns a.
Examples:
- (pure "hello" : Option String) = some "hello"
- (pure "hello" : Except (Array String) String) = Except.ok "hello"
- (pure "hello" : StateM Nat String).run 105 = ("hello", 105)

Instances

class Functor (f : Type u → Type v) :

Type (max (u + 1) v)

A functor in the sense used in functional programming, which means a function f : Type u → Type v has a way of mapping a function over its contents. This map operator is written <$>, and overloaded via Functor instances.

This map function should respect identity and function composition. In other words, for all terms v : f α, it should be the case that:

id <$> v = v
For all functions h : β → γ and g : α → β, (h ∘ g) <$> v = h <$> g <$> v

While all Functor instances should live up to these requirements, they are not required to prove that they do. Proofs may be required or provided via the LawfulFunctor class.

Assuming that instances are lawful, this definition corresponds to the category-theoretic notion of functor in the special case where the category is the category of types and functions between them.

map {α β : Type u} : (α → β) → f α → f β
Applies a function inside a functor. This is used to overload the <$> operator.
When mapping a constant function, use Functor.mapConst instead, because it may be more efficient.
Conventions for notations in identifiers:
- The recommended spelling of <$> in identifiers is map.
mapConst {α β : Type u} : α → f β → f α
Mapping a constant function.
Given a : α and v : f α, mapConst a v is equivalent to Function.const _ a <$> v. For some functors, this can be implemented more efficiently; for all other functors, the default implementation may be used.

Instances

class Seq (f : Type u → Type v) :

Type (max (u + 1) v)

The <*> operator is overloaded using the function Seq.seq.

While <$> from the class Functor allows an ordinary function to be mapped over its contents, <*> allows a function that's “inside” the functor to be applied. When thinking about f as possible side effects, this captures evaluation order: seq arranges for the effects that produce the function to occur prior to those that produce the argument value.

For most applications, Applicative or Monad should be used rather than Seq itself.

seq {α β : Type u} : f (α → β) → (Unit → f α) → f β
The implementation of the <*> operator.
In a monad, mf <*> mx is the same as do let f ← mf; x ← mx; pure (f x): it evaluates the function first, then the argument, and applies one to the other.
To avoid surprising evaluation semantics, mx is taken "lazily", using a Unit → f α function.
Conventions for notations in identifiers:
- The recommended spelling of <*> in identifiers is seq.

Instances

class SeqLeft (f : Type u → Type v) :

Type (max (u + 1) v)

The <* operator is overloaded using seqLeft.

When thinking about f as potential side effects, <* evaluates first the left and then the right argument for their side effects, discarding the value of the right argument and returning the value of the left argument.

For most applications, Applicative or Monad should be used rather than SeqLeft itself.

seqLeft {α β : Type u} : f α → (Unit → f β) → f α
Sequences the effects of two terms, discarding the value of the second. This function is usually invoked via the <* operator.
Given x : f α and y : f β, x <* y runs x, then runs y, and finally returns the result of x.
The evaluation of the second argument is delayed by wrapping it in a function, enabling “short-circuiting” behavior from f.
Conventions for notations in identifiers:
- The recommended spelling of <* in identifiers is seqLeft.

Instances

class SeqRight (f : Type u → Type v) :

Type (max (u + 1) v)

The *> operator is overloaded using seqRight.

When thinking about f as potential side effects, *> evaluates first the left and then the right argument for their side effects, discarding the value of the left argument and returning the value of the right argument.

For most applications, Applicative or Monad should be used rather than SeqLeft itself.

seqRight {α β : Type u} : f α → (Unit → f β) → f β
Sequences the effects of two terms, discarding the value of the first. This function is usually invoked via the *> operator.
Given x : f α and y : f β, x *> y runs x, then runs y, and finally returns the result of y.
The evaluation of the second argument is delayed by wrapping it in a function, enabling “short-circuiting” behavior from f.
Conventions for notations in identifiers:
- The recommended spelling of *> in identifiers is seqRight.

Instances

class Applicative (f : Type u → Type v) extends Functor f, Pure f, Seq f, SeqLeft f, SeqRight f :

Type (max (u + 1) v)

An applicative functor is more powerful than a Functor, but less powerful than a Monad.

Applicative functors capture sequencing of effects with the <*> operator, overloaded as seq, but not data-dependent effects. The results of earlier computations cannot be used to control later effects.

Applicative functors should satisfy four laws. Instances of Applicative are not required to prove that they satisfy these laws, which are part of the LawfulApplicative class.

map {α β : Type u} : (α → β) → f α → f β
mapConst {α β : Type u} : α → f β → f α
pure {α : Type u} : α → f α
seq {α β : Type u} : f (α → β) → (Unit → f α) → f β
seqLeft {α β : Type u} : f α → (Unit → f β) → f α
seqRight {α β : Type u} : f α → (Unit → f β) → f β

Instances

class Monad (m : Type u → Type v) extends Applicative m, Bind m :

Type (max (u + 1) v)

Monads are an abstraction of sequential control flow and side effects used in functional programming. Monads allow both sequencing of effects and data-dependent effects: the values that result from an early step may influence the effects carried out in a later step.

The Monad API may be used directly. However, it is most commonly accessed through do-notation.

Most Monad instances provide implementations of pure and bind, and use default implementations for the other methods inherited from Applicative. Monads should satisfy certain laws, but instances are not required to prove this. An instance of LawfulMonad expresses that a given monad's operations are lawful.

map {α β : Type u} : (α → β) → m α → m β
mapConst {α β : Type u} : α → m β → m α
pure {α : Type u} : α → m α
seq {α β : Type u} : m (α → β) → (Unit → m α) → m β
seqLeft {α β : Type u} : m α → (Unit → m β) → m α
seqRight {α β : Type u} : m α → (Unit → m β) → m β
bind {α β : Type u} : m α → (α → m β) → m β

Instances

instance instInhabitedForallOfMonad {α : Type u} {m : Type u → Type v} [Monad m] :

Inhabited (α → m α)

Equations

instInhabitedForallOfMonad = { default := pure }

instance instInhabitedOfMonad {α : Type u} {m : Type u → Type v} [Monad m] [Inhabited α] :

Inhabited (m α)

Equations

instInhabitedOfMonad = { default := pure default }

instance instNonemptyOfMonad {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] [Nonempty α] :

Nonempty (m α)

class MonadLift (m : semiOutParam (Type u → Type v)) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

Computations in the monad m can be run in the monad n. These translations are inserted automatically by the compiler.

Usually, n consists of some number of monad transformers applied to m, but this is not mandatory.

New instances should use this class, MonadLift. Clients that require one monad to be liftable into another should instead request MonadLiftT, which is the reflexive, transitive closure of MonadLift.

monadLift {α : Type u} : m α → n α
Translates an action from monad m into monad n.

Instances

class MonadLiftT (m : Type u → Type v) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

Computations in the monad m can be run in the monad n. These translations are inserted automatically by the compiler.

Usually, n consists of some number of monad transformers applied to m, but this is not mandatory.

This is the reflexive, transitive closure of MonadLift. Clients that require one monad to be liftable into another should request an instance of MonadLiftT. New instances should instead be defined for MonadLift itself.

monadLift {α : Type u} : m α → n α
Translates an action from monad m into monad n.

Instances

@[reducible, inline]

abbrev liftM {m : Type u_1 → Type u_2} {n : Type u_1 → Type u_3} [self : MonadLiftT m n] {α : Type u_1} :

m α → n α

Translates an action from monad m into monad n.

Equations

@liftM = @monadLift

@[always_inline]

instance instMonadLiftTOfMonadLift (m : Type u_1 → Type u_2) (n : Type u_1 → Type u_3) (o : Type u_1 → Type u_4) [MonadLift n o] [MonadLiftT m n] :

Equations

instMonadLiftTOfMonadLift m n o = { monadLift := fun {α : Type ?u.30} (x : m α) => MonadLift.monadLift (monadLift x) }

instance instMonadLiftT (m : Type u_1 → Type u_2) :

Equations

instMonadLiftT m = { monadLift := fun {α : Type ?u.10} (x : m α) => x }

class MonadEval (m : semiOutParam (Type u → Type v)) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

Typeclass used for adapting monads. This is similar to MonadLift, but instances are allowed to make use of default state for the purpose of synthesizing such an instance, if necessary. Every MonadLift instance gives a MonadEval instance.

The purpose of this class is for the #eval command, which looks for a MonadEval m CommandElabM or MonadEval m IO instance.

monadEval {α : Type u} : m α → n α
Evaluates a value from monad m into monad n.

Instances

instance instMonadEvalOfMonadLift {m : Type u_1 → Type u_2} {n : Type u_1 → Type u_3} [MonadLift m n] :

Equations

instMonadEvalOfMonadLift = { monadEval := fun {α : Type ?u.19} => MonadLift.monadLift }

class MonadEvalT (m : Type u → Type v) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

The transitive closure of MonadEval.

monadEval {α : Type u} : m α → n α
Evaluates a value from monad m into monad n.

Instances

instance instMonadEvalTOfMonadEval (m : Type u_1 → Type u_2) (n : Type u_1 → Type u_3) (o : Type u_1 → Type u_4) [MonadEval n o] [MonadEvalT m n] :

Equations

instMonadEvalTOfMonadEval m n o = { monadEval := fun {α : Type ?u.30} (x : m α) => MonadEval.monadEval (MonadEvalT.monadEval x) }

instance instMonadEvalT (m : Type u_1 → Type u_2) :

Equations

instMonadEvalT m = { monadEval := fun {α : Type ?u.10} (x : m α) => x }

class MonadFunctor (m : semiOutParam (Type u → Type v)) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

A way to interpret a fully-polymorphic function in m into n. Such a function can be thought of as one that may change the effects in m, but can't do so based on specific values that are provided.

Clients of MonadFunctor should typically use MonadFunctorT, which is the reflexive, transitive closure of MonadFunctor. New instances should be defined for MonadFunctor.

monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
Lifts a fully-polymorphic transformation of m into n.

Instances

class MonadFunctorT (m : Type u → Type v) (n : Type u → Type w) :

Type (max (max (u + 1) v) w)

A way to interpret a fully-polymorphic function in m into n. Such a function can be thought of as one that may change the effects in m, but can't do so based on specific values that are provided.

This is the reflexive, transitive closure of MonadFunctor. It automatically chains together MonadFunctor instances as needed. Clients of MonadFunctor should typically use MonadFunctorT, but new instances should be defined for MonadFunctor.

monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
Lifts a fully-polymorphic transformation of m into n.

Instances

@[always_inline]

instance instMonadFunctorTOfMonadFunctor (m : Type u_1 → Type u_2) (n : Type u_1 → Type u_3) (o : Type u_1 → Type u_4) [MonadFunctor n o] [MonadFunctorT m n] :

MonadFunctorT m o

Equations

One or more equations did not get rendered due to their size.

instance monadFunctorRefl (m : Type u_1 → Type u_2) :

MonadFunctorT m m

Equations

monadFunctorRefl m = { monadMap := fun {α : Type ?u.12} (f : {β : Type ?u.12} → m β → m β) => f }

@[unbox]

inductive Except (ε : Type u) (α : Type v) :

Type (max u v)

Except ε α is a type which represents either an error of type ε or a successful result with a value of type α.

Except ε : Type u → Type v is a Monad that represents computations that may throw exceptions: the pure operation is Except.ok and the bind operation returns the first encountered Except.error.

error {ε : Type u} {α : Type v} : ε → Except ε α
A failure value of type ε
ok {ε : Type u} {α : Type v} : α → Except ε α
A success value of type α

Instances For

instance instInhabitedExcept {ε : Type u} {α : Type v} [Inhabited ε] :

Inhabited (Except ε α)

Equations

instInhabitedExcept = { default := Except.error default }

class MonadExceptOf (ε : semiOutParam (Type u)) (m : Type v → Type w) :

Type (max (max u (v + 1)) w)

Exception monads provide the ability to throw errors and handle errors.

In this class, ε is a semiOutParam, which means that it can influence the choice of instance. MonadExcept ε provides the same operations, but requires that ε be inferrable from m.

tryCatchThe, which takes an explicit exception type, is used to desugar try ... catch ... steps inside do-blocks when the handlers have type annotations.

throw {α : Type v} : ε → m α
Throws an exception of type ε to the nearest enclosing catch.
tryCatch {α : Type v} (body : m α) (handler : ε → m α) : m α
Catches errors thrown in body, passing them to handler. Errors in handler are not caught.

Instances

@[reducible, inline]

abbrev throwThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (e : ε) :

m α

Throws an exception, with the exception type specified explicitly. This is useful when a monad supports throwing more than one type of exception.

Use throw for a version that expects the exception type to be inferred from m.

Equations

throwThe ε e = MonadExceptOf.throw e

@[reducible, inline]

abbrev tryCatchThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (x : m α) (handle : ε → m α) :

m α

Catches errors, recovering using handle. The exception type is specified explicitly. This is useful when a monad supports throwing or handling more than one type of exception.

Use tryCatch, for a version that expects the exception type to be inferred from m.

Equations

tryCatchThe ε x handle = MonadExceptOf.tryCatch x handle

class MonadExcept (ε : outParam (Type u)) (m : Type v → Type w) :

Type (max (max u (v + 1)) w)

Exception monads provide the ability to throw errors and handle errors.

In this class, ε is an outParam, which means that it is inferred from m. MonadExceptOf ε provides the same operations, but allows ε to influence instance synthesis.

MonadExcept.tryCatch is used to desugar try ... catch ... steps inside do-blocks when the handlers do not have exception type annotations.

throw {α : Type v} : ε → m α
Throws an exception of type ε to the nearest enclosing handler.
tryCatch {α : Type v} (body : m α) (handler : ε → m α) : m α
Catches errors thrown in body, passing them to handler. Errors in handler are not caught.

Instances

def MonadExcept.ofExcept {m : Type u_1 → Type u_2} {ε : Type u_3} {α : Type u_1} [Monad m] [MonadExcept ε m] :

Except ε α → m α

Re-interprets an Except ε action in an exception monad m, succeeding if it succeeds and throwing an exception if it throws an exception.

Equations

ofExcept (Except.ok a) = pure a
ofExcept (Except.error e) = throw e

instance instMonadExceptOfMonadExceptOf (ε : Type u) (m : Type v → Type w) [MonadExceptOf ε m] :

MonadExcept ε m

Equations

instMonadExceptOfMonadExceptOf ε m = { throw := fun {α : Type ?u.18} => throwThe ε, tryCatch := fun {α : Type ?u.18} => tryCatchThe ε }

@[inline]

def MonadExcept.orElse {ε : Type u} {m : Type v → Type w} [MonadExcept ε m] {α : Type v} (t₁ : m α) (t₂ : Unit → m α) :

m α

Unconditional error recovery that ignores which exception was thrown. Usually used via the <|> operator.

If both computations throw exceptions, then the result is the second exception.

Equations

MonadExcept.orElse t₁ t₂ = tryCatch t₁ fun (x : ε) => t₂ ()

instance MonadExcept.instOrElse {ε : Type u} {m : Type v → Type w} [MonadExcept ε m] {α : Type v} :

OrElse (m α)

Equations

MonadExcept.instOrElse = { orElse := MonadExcept.orElse }

def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) :

Type (max u v)

Adds the ability to access a read-only value of type ρ to a monad. The value can be locally overridden by withReader, but it cannot be mutated.

Actions in the resulting monad are functions that take the local value as a parameter, returning ordinary actions in m.

Equations

ReaderT ρ m α = (ρ → m α)

Instances For

instance instInhabitedReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) [Inhabited (m α)] :

Inhabited (ReaderT ρ m α)

Equations

instInhabitedReaderT ρ m α = { default := fun (x : ρ) => default }

@[inline]

def ReaderT.run {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : ReaderT ρ m α) (r : ρ) :

m α

Executes an action from a monad with a read-only value in the underlying monad m.

Equations

x.run r = x r

instance ReaderT.instMonadLift {ρ : Type u} {m : Type u → Type v} :

MonadLift m (ReaderT ρ m)

Equations

ReaderT.instMonadLift = { monadLift := fun {α : Type ?u.16} (x : m α) (x_1 : ρ) => x }

@[always_inline]

instance ReaderT.instMonadExceptOf {ρ : Type u} {m : Type u → Type v} (ε : Type u_1) [MonadExceptOf ε m] :

MonadExceptOf ε (ReaderT ρ m)

Equations

One or more equations did not get rendered due to their size.

@[inline]

def ReaderT.read {ρ : Type u} {m : Type u → Type v} [Monad m] :

ReaderT ρ m ρ

Retrieves the reader monad's local value. Typically accessed via read, or via readThe when more than one local value is available.

Equations

ReaderT.read = pure

@[inline]

def ReaderT.pure {ρ : Type u} {m : Type u → Type v} [Monad m] {α : Type u} (a : α) :

ReaderT ρ m α

Returns the provided value a, ignoring the reader monad's local value. Typically used via Pure.pure.

Equations

ReaderT.pure a x✝ = pure a

@[inline]

def ReaderT.bind {ρ : Type u} {m : Type u → Type v} [Monad m] {α β : Type u} (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) :

ReaderT ρ m β

Sequences two reader monad computations. Both are provided with the local value, and the second is passed the value of the first. Typically used via the >>= operator.

Equations

x.bind f r = do let a ← x r f a r

@[always_inline]

instance ReaderT.instFunctorOfMonad {ρ : Type u} {m : Type u → Type v} [Monad m] :

Functor (ReaderT ρ m)

Equations

One or more equations did not get rendered due to their size.

@[always_inline]

instance ReaderT.instApplicativeOfMonad {ρ : Type u} {m : Type u → Type v} [Monad m] :

Applicative (ReaderT ρ m)

Equations

One or more equations did not get rendered due to their size.

instance ReaderT.instMonad {ρ : Type u} {m : Type u → Type v} [Monad m] :

Monad (ReaderT ρ m)

Equations

ReaderT.instMonad = { toApplicative := ReaderT.instApplicativeOfMonad, bind := fun {α β : Type ?u.19} => ReaderT.bind }

instance ReaderT.instMonadFunctor (ρ : Type u_1) (m : Type u_1 → Type u_2) :

MonadFunctor m (ReaderT ρ m)

Equations

ReaderT.instMonadFunctor ρ m = { monadMap := fun {α : Type ?u.19} (f : {β : Type ?u.19} → m β → m β) (x : ReaderT ρ m α) (ctx : ρ) => f (x ctx) }

@[inline]

def ReaderT.adapt {ρ : Type u} {m : Type u → Type v} {ρ' α : Type u} (f : ρ' → ρ) :

ReaderT ρ m α → ReaderT ρ' m α

Modifies a reader monad's local value with f. The resulting computation applies f to the incoming local value and passes the result to the inner computation.

Equations

ReaderT.adapt f x r = x (f r)

class MonadReaderOf (ρ : semiOutParam (Type u)) (m : Type u → Type v) :

Reader monads provide the ability to implicitly thread a value through a computation. The value can be read, but not written. A MonadWithReader ρ instance additionally allows the value to be locally overridden for a sub-computation.

In this class, ρ is a semiOutParam, which means that it can influence the choice of instance. MonadReader ρ provides the same operations, but requires that ρ be inferrable from m.

read : m ρ
Retrieves the local value.

Instances

class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) :

Reader monads provide the ability to implicitly thread a value through a computation. The value can be read, but not written. A MonadWithReader ρ instance additionally allows the value to be locally overridden for a sub-computation.

In this class, ρ is an outParam, which means that it is inferred from m. MonadReaderOf ρ provides the same operations, but allows ρ to influence instance synthesis.

read : m ρ
Retrieves the local value.
Use readThe to explicitly specify a type when more than one value is available.

Instances

@[inline]

def readThe (ρ : Type u) {m : Type u → Type v} [MonadReaderOf ρ m] :

m ρ

Retrieves the local value whose type is ρ. This is useful when a monad supports reading more that one type of value.

Use read for a version that expects the type ρ to be inferred from m.

Equations

readThe ρ = MonadReaderOf.read

instance instMonadReaderOfMonadReaderOf (ρ : Type u) (m : Type u → Type v) [MonadReaderOf ρ m] :

MonadReader ρ m

Equations

instMonadReaderOfMonadReaderOf ρ m = { read := readThe ρ }

instance instMonadReaderOfOfMonadLift {ρ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadReaderOf ρ m] :

MonadReaderOf ρ n

Equations

instMonadReaderOfOfMonadLift = { read := liftM read }

instance instMonadReaderOfReaderTOfMonad {ρ : Type u} {m : Type u → Type v} [Monad m] :

MonadReaderOf ρ (ReaderT ρ m)

Equations

instMonadReaderOfReaderTOfMonad = { read := ReaderT.read }

class MonadWithReaderOf (ρ : semiOutParam (Type u)) (m : Type u → Type v) :

Type (max (u + 1) v)

A reader monad that additionally allows the value to be locally overridden.

In this class, ρ is a semiOutParam, which means that it can influence the choice of instance. MonadWithReader ρ provides the same operations, but requires that ρ be inferrable from m.

withReader {α : Type u} (f : ρ → ρ) (x : m α) : m α
Locally modifies the reader monad's value while running an action.
During the inner action x, reading the value returns f applied to the original value. After control returns from x, the reader monad's value is restored.

Instances

@[inline]

def withTheReader (ρ : Type u) {m : Type u → Type v} [MonadWithReaderOf ρ m] {α : Type u} (f : ρ → ρ) (x : m α) :

m α

Locally modifies the reader monad's value while running an action, with the reader monad's local value type specified explicitly. This is useful when a monad supports reading more than one type of value.

During the inner action x, reading the value returns f applied to the original value. After control returns from x, the reader monad's value is restored.

Use withReader for a version that expects the local value's type to be inferred from m.

Equations

withTheReader ρ f x = MonadWithReaderOf.withReader f x

class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) :

Type (max (u + 1) v)

A reader monad that additionally allows the value to be locally overridden.

In this class, ρ is an outParam, which means that it is inferred from m. MonadWithReaderOf ρ provides the same operations, but allows ρ to influence instance synthesis.

withReader {α : Type u} : (ρ → ρ) → m α → m α
Locally modifies the reader monad's value while running an action.
During the inner action x, reading the value returns f applied to the original value. After control returns from x, the reader monad's value is restored.

Instances

instMonadWithReaderOfMonadWithReaderOf

instance instMonadWithReaderOfMonadWithReaderOf (ρ : Type u) (m : Type u → Type v) [MonadWithReaderOf ρ m] :

MonadWithReader ρ m

Equations

instMonadWithReaderOfMonadWithReaderOf ρ m = { withReader := fun {α : Type ?u.17} => withTheReader ρ }

instance instMonadWithReaderOfOfMonadFunctor {ρ : Type u} {m n : Type u → Type v} [MonadFunctor m n] [MonadWithReaderOf ρ m] :

MonadWithReaderOf ρ n

Equations

instMonadWithReaderOfOfMonadFunctor = { withReader := fun {α : Type ?u.28} (f : ρ → ρ) => monadMap fun {β : Type ?u.28} => withTheReader ρ f }

instance instMonadWithReaderOfReaderT {ρ : Type u} {m : Type u → Type v} :

MonadWithReaderOf ρ (ReaderT ρ m)

Equations

instMonadWithReaderOfReaderT = { withReader := fun {α : Type ?u.18} (f : ρ → ρ) (x : ReaderT ρ m α) (ctx : ρ) => x (f ctx) }

class MonadStateOf (σ : semiOutParam (Type u)) (m : Type u → Type v) :

Type (max (u + 1) v)

State monads provide a value of a given type (the state) that can be retrieved or replaced. Instances may implement these operations by passing state values around, by using a mutable reference cell (e.g. ST.Ref σ), or in other ways.

In this class, σ is a semiOutParam, which means that it can influence the choice of instance. MonadState σ provides the same operations, but requires that σ be inferrable from m.

The mutable state of a state monad is visible between multiple do-blocks or functions, unlike local mutable state in do-notation.

get : m σ
Retrieves the current value of the monad's mutable state.
set : σ → m PUnit
Replaces the current value of the mutable state with a new one.
modifyGet {α : Type u} : (σ → α × σ) → m α
Applies a function to the current state that both computes a new state and a value. The new state replaces the current state, and the value is returned.
It is equivalent to do let (a, s) := f (← get); set s; pure a. However, using modifyGet may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Instances

@[reducible, inline]

abbrev getThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] :

m σ

Gets the current state that has the explicitly-provided type σ. When the current monad has multiple state types available, this function selects one of them.

Equations

getThe σ = MonadStateOf.get

@[reducible, inline]

abbrev modifyThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → σ) :

Mutates the current state that has the explicitly-provided type σ, replacing its value with the result of applying f to it. When the current monad has multiple state types available, this function selects one of them.

It is equivalent to do set (f (← get)). However, using modify may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Equations

modifyThe σ f = MonadStateOf.modifyGet fun (s : σ) => (PUnit.unit , f s)

@[reducible, inline]

abbrev modifyGetThe {α : Type u} (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → α × σ) :

m α

Applies a function to the current state that has the explicitly-provided type σ. The function both computes a new state and a value. The new state replaces the current state, and the value is returned.

It is equivalent to do let (a, s) := f (← getThe σ); set s; pure a. However, using modifyGetThe may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Equations

modifyGetThe σ f = MonadStateOf.modifyGet f

class MonadState (σ : outParam (Type u)) (m : Type u → Type v) :

Type (max (u + 1) v)

State monads provide a value of a given type (the state) that can be retrieved or replaced. Instances may implement these operations by passing state values around, by using a mutable reference cell (e.g. ST.Ref σ), or in other ways.

In this class, σ is an outParam, which means that it is inferred from m. MonadStateOf σ provides the same operations, but allows σ to influence instance synthesis.

The mutable state of a state monad is visible between multiple do-blocks or functions, unlike local mutable state in do-notation.

get : m σ
Retrieves the current value of the monad's mutable state.
set : σ → m PUnit
Replaces the current value of the mutable state with a new one.
modifyGet {α : Type u} : (σ → α × σ) → m α
Applies a function to the current state that both computes a new state and a value. The new state replaces the current state, and the value is returned.
It is equivalent to do let (a, s) := f (← get); set s; pure a. However, using modifyGet may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Instances

instance instMonadStateOfMonadStateOf (σ : Type u) (m : Type u → Type v) [MonadStateOf σ m] :

MonadState σ m

Equations

instMonadStateOfMonadStateOf σ m = { get := getThe σ, set := set, modifyGet := fun {α : Type ?u.19} (f : σ → α × σ) => MonadStateOf.modifyGet f }

@[inline]

def modify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) :

Mutates the current state, replacing its value with the result of applying f to it.

Use modifyThe to explicitly select a state type to modify.

It is equivalent to do set (f (← get)). However, using modify may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Equations

modify f = modifyGet fun (s : σ) => (PUnit.unit , f s)

@[inline]

def getModify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) :

m σ

Replaces the state with the result of applying f to it. Returns the old value of the state.

It is equivalent to get <* modify f but may be more efficient.

Equations

getModify f = modifyGet fun (s : σ) => (s, f s)

@[always_inline]

instance instMonadStateOfOfMonadLift {σ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadStateOf σ m] :

MonadStateOf σ n

Equations

instMonadStateOfOfMonadLift = { get := liftM MonadStateOf.get, set := fun (s : σ) => liftM (set s), modifyGet := fun {α : Type ?u.30} (f : σ → α × σ) => monadLift (modifyGet f) }

inductive EStateM.Result (ε σ α : Type u) :

The value returned from a combined state and exception monad in which exceptions do not automatically roll back the state.

Result ε σ α is equivalent to Except ε α × σ, but using a single combined inductive type yields a more efficient data representation.

ok {ε σ α : Type u} : α → σ → Result ε σ α
A success value of type α and a new state σ.
error {ε σ α : Type u} : ε → σ → Result ε σ α
An exception of type ε and a new state σ.

Instances For

instance EStateM.instInhabitedResult {ε σ α : Type u} [Inhabited ε] [Inhabited σ] :

Inhabited (Result ε σ α)

Equations

EStateM.instInhabitedResult = { default := EStateM.Result.error default default }

def EStateM (ε σ α : Type u) :

A combined state and exception monad in which exceptions do not automatically roll back the state.

Instances of EStateM.Backtrackable provide a way to roll back some part of the state if needed.

EStateM ε σ is equivalent to ExceptT ε (StateM σ), but it is more efficient.

Equations

EStateM ε σ α = (σ → EStateM.Result ε σ α)

Instances For

instance EStateM.instInhabited {ε σ α : Type u} [Inhabited ε] :

Inhabited (EStateM ε σ α)

Equations

EStateM.instInhabited = { default := fun (s : σ) => EStateM.Result.error default s }

@[inline]

def EStateM.pure {ε σ α : Type u} (a : α) :

EStateM ε σ α

Returns a value without modifying the state or throwing an exception.

Equations

EStateM.pure a s = EStateM.Result.ok a s

@[inline]

def EStateM.set {ε σ : Type u} (s : σ) :

EStateM ε σ PUnit

Replaces the current value of the mutable state with a new one.

Equations

EStateM.set s x✝ = EStateM.Result.ok PUnit.unit s

@[inline]

def EStateM.get {ε σ : Type u} :

EStateM ε σ σ

Retrieves the current value of the monad's mutable state.

Equations

EStateM.get s = EStateM.Result.ok s s

@[inline]

def EStateM.modifyGet {ε σ α : Type u} (f : σ → α × σ) :

EStateM ε σ α

Applies a function to the current state that both computes a new state and a value. The new state replaces the current state, and the value is returned.

It is equivalent to do let (a, s) := f (← get); set s; pure a. However, using modifyGet may lead to higher performance because it doesn't add a new reference to the state value. Additional references can inhibit in-place updates of data.

Equations

EStateM.modifyGet f s = match f s with | (a, s) => EStateM.Result.ok a s

@[inline]

def EStateM.throw {ε σ α : Type u} (e : ε) :

EStateM ε σ α

Throws an exception of type ε to the nearest enclosing handler.

Equations

EStateM.throw e s = EStateM.Result.error e s

class EStateM.Backtrackable (δ : outParam (Type u)) (σ : Type u) :

Exception handlers in EStateM save some part of the state, determined by δ, and restore it if an exception is caught. By default, δ is Unit, and no information is saved.

save : σ → δ
Extracts the information in the state that should be rolled back if an exception is handled.
restore : σ → δ → σ
Updates the current state with the saved information that should be rolled back. This updated state becomes the current state when an exception is handled.

Instances

EStateM.nonBacktrackable

@[inline]

def EStateM.tryCatch {ε σ δ : Type u} [Backtrackable δ σ] {α : Type u} (x : EStateM ε σ α) (handle : ε → EStateM ε σ α) :

EStateM ε σ α

Handles exceptions thrown in the combined error and state monad.

The Backtrackable δ σ instance is used to save a snapshot of part of the state prior to running x. If an exception is caught, the state is updated with the saved snapshot, rolling back part of the state. If no instance of Backtrackable is provided, a fallback instance in which δ is Unit is used, and no information is rolled back.

Equations

x.tryCatch handle s = match x s with | EStateM.Result.error e s_1 => handle e (EStateM.Backtrackable.restore s_1 (EStateM.Backtrackable.save s)) | ok => ok

@[inline]

def EStateM.orElse {ε σ α δ : Type u} [Backtrackable δ σ] (x₁ : EStateM ε σ α) (x₂ : Unit → EStateM ε σ α) :

EStateM ε σ α

Failure handling that does not depend on specific exception values.

The Backtrackable δ σ instance is used to save a snapshot of part of the state prior to running x₁. If an exception is caught, the state is updated with the saved snapshot, rolling back part of the state. If no instance of Backtrackable is provided, a fallback instance in which δ is Unit is used, and no information is rolled back.

Equations

x₁.orElse x₂ s = match x₁ s with | EStateM.Result.error a s_1 => x₂ () (EStateM.Backtrackable.restore s_1 (EStateM.Backtrackable.save s)) | ok => ok

@[inline]

def EStateM.adaptExcept {ε σ α ε' : Type u} (f : ε → ε') (x : EStateM ε σ α) :

EStateM ε' σ α

Transforms exceptions with a function, doing nothing on successful results.

Equations

EStateM.adaptExcept f x s = match x s with | EStateM.Result.error e s => EStateM.Result.error (f e) s | EStateM.Result.ok a s => EStateM.Result.ok a s

@[inline]

def EStateM.bind {ε σ α β : Type u} (x : EStateM ε σ α) (f : α → EStateM ε σ β) :

EStateM ε σ β

Sequences two EStateM ε σ actions, passing the returned value from the first into the second.

Equations

x.bind f s = match x s with | EStateM.Result.ok a s => f a s | EStateM.Result.error e s => EStateM.Result.error e s

@[inline]

def EStateM.map {ε σ α β : Type u} (f : α → β) (x : EStateM ε σ α) :

EStateM ε σ β

Transforms the value returned from an EStateM ε σ action using a function.

Equations

EStateM.map f x s = match x s with | EStateM.Result.ok a s => EStateM.Result.ok (f a) s | EStateM.Result.error e s => EStateM.Result.error e s

@[inline]

def EStateM.seqRight {ε σ α β : Type u} (x : EStateM ε σ α) (y : Unit → EStateM ε σ β) :

EStateM ε σ β

Sequences two EStateM ε σ actions, running x before y. The first action's return value is ignored.

Equations

x.seqRight y s = match x s with | EStateM.Result.ok a s => y () s | EStateM.Result.error e s => EStateM.Result.error e s

@[always_inline]

instance EStateM.instMonad {ε σ : Type u} :

Monad (EStateM ε σ)

Equations

One or more equations did not get rendered due to their size.

instance EStateM.instOrElseOfBacktrackable {ε σ α δ : Type u} [Backtrackable δ σ] :

OrElse (EStateM ε σ α)

Equations

EStateM.instOrElseOfBacktrackable = { orElse := EStateM.orElse }

instance EStateM.instMonadStateOf {ε σ : Type u} :

MonadStateOf σ (EStateM ε σ)

Equations

EStateM.instMonadStateOf = { get := EStateM.get, set := EStateM.set, modifyGet := fun {α : Type ?u.12} => EStateM.modifyGet }

instance EStateM.instMonadExceptOfOfBacktrackable {ε σ δ : Type u} [Backtrackable δ σ] :

MonadExceptOf ε (EStateM ε σ)

Equations

EStateM.instMonadExceptOfOfBacktrackable = { throw := fun {α : Type ?u.23} => EStateM.throw, tryCatch := fun {α : Type ?u.23} => EStateM.tryCatch }

@[inline]

def EStateM.run {ε σ α : Type u} (x : EStateM ε σ α) (s : σ) :

Result ε σ α

Executes an EStateM action with the initial state s. The returned value includes the final state and indicates whether an exception was thrown or a value was returned.

Equations

x.run s = x s

@[inline]

def EStateM.run' {ε σ α : Type u} (x : EStateM ε σ α) (s : σ) :

Executes an EStateM with the initial state s for the returned value α, discarding the final state. Returns none if an unhandled exception was thrown.

Equations

x.run' s = match x.run s with | EStateM.Result.ok v a => some v | EStateM.Result.error a a_1 => none

@[inline]

def EStateM.dummySave {σ : Type u} :

σ → PUnit

The save implementation for Backtrackable PUnit σ.

Equations

EStateM.dummySave x✝ = PUnit.unit

@[inline]

def EStateM.dummyRestore {σ : Type u} :

σ → PUnit → σ

The restore implementation for Backtrackable PUnit σ.

Equations

EStateM.dummyRestore s x✝ = s

instance EStateM.nonBacktrackable {σ : Type u} :

Backtrackable PUnit σ

A fallback Backtrackable instance that saves no information from a state. This allows every type to be used as a state in EStateM, with no rollback.

Because this is the first declared instance of Backtrackable _ σ, it will be picked only if there are no other Backtrackable _ σ instances registered.

Equations

EStateM.nonBacktrackable = { save := EStateM.dummySave, restore := EStateM.dummyRestore }

class Hashable (α : Sort u) :

Sort (max 1 u)

Types that can be hashed into a UInt64.

hash : α → UInt64
Hashes a value into a UInt64.

Instances

@[extern lean_uint64_mix_hash]

opaque mixHash (u₁ u₂ : UInt64) :

An opaque hash mixing operation, used to implement hashing for products.

instance instHashableSubtype {α : Sort u_1} [Hashable α] {p : α → Prop} :

Hashable (Subtype p)

Equations

instHashableSubtype = { hash := fun (a : Subtype p) => hash a.val }

@[extern lean_string_hash]

opaque String.hash (s : String) :

Computes a hash for strings.

instance instHashableString :

Hashable String

Equations

instHashableString = { hash := String.hash }

inductive Lean.Name :

Hierarchical names consist of a sequence of components, each of which is either a string or numeric, that are written separated by dots (.).

Hierarchical names are used to name declarations and for creating unique identifiers for free variables and metavariables.

You can create hierarchical names using a backtick:

`Lean.Meta.whnf

It is short for .str (.str (.str .anonymous "Lean") "Meta") "whnf".

You can use double backticks to request Lean to statically check whether the name corresponds to a Lean declaration in scope.

``Lean.Meta.whnf

If the name is not in scope, Lean will report an error.

There are two ways to convert a String to a Name:

Name.mkSimple creates a name with a single string component.
String.toName first splits the string into its dot-separated components, and then creates a hierarchical name.

anonymous : Name
The "anonymous" name.
str (pre : Name) (str : String) : Name
A string name. The name Lean.Meta.run is represented at
```
.str (.str (.str .anonymous "Lean") "Meta") "run"
```
num (pre : Name) (i : Nat) : Name
A numerical name. This kind of name is used, for example, to create hierarchical names for free variables and metavariables. The identifier _uniq.231 is represented as
```
.num (.str .anonymous "_uniq") 231
```

Instances For

@[implemented_by Lean.Name.hash._override]

noncomputable def Lean.Name.hash :

Name → UInt64

A hash function for names, which is stored inside the name itself as a computed field.

Equations

Lean.Name.anonymous.hash = UInt64.ofNatLT 1723 Lean.Name.hash._proof_40
(p.str s).hash = mixHash p.hash s.hash
(p.num v).hash = mixHash p.hash (if h : v < UInt64.size then UInt64.ofNatLT v h else UInt64.ofNatLT 17 Lean.Name.hash._proof_41)

instance Lean.instInhabitedName :

Equations

Lean.instInhabitedName = { default := Lean.Name.anonymous }

instance Lean.instHashableName :

Equations

Lean.instHashableName = { hash := Lean.Name.hash }

@[reducible, inline, export lean_name_mk_string]

abbrev Lean.Name.mkStr (p : Name) (s : String) :

.str p s is now the preferred form.

Equations

p.mkStr s = p.str s

@[reducible, inline, export lean_name_mk_numeral]

abbrev Lean.Name.mkNum (p : Name) (v : Nat) :

.num p v is now the preferred form.

Equations

p.mkNum v = p.num v

@[reducible, inline]

abbrev Lean.Name.mkSimple (s : String) :

Converts a String to a Name without performing any parsing. mkSimple s is short for .str .anonymous s.

This means that mkSimple "a.b" is the name «a.b», not a.b.

Equations

Lean.Name.mkSimple s = Lean.Name.anonymous.str s

@[reducible]

def Lean.Name.mkStr1 (s₁ : String) :

Make name s₁

Equations

Lean.Name.mkStr1 s₁ = Lean.Name.anonymous.str s₁

Instances For

@[reducible]

def Lean.Name.mkStr2 (s₁ s₂ : String) :

Make name s₁.s₂

Equations

Lean.Name.mkStr2 s₁ s₂ = (Lean.Name.anonymous.str s₁).str s₂

Instances For

@[reducible]

def Lean.Name.mkStr3 (s₁ s₂ s₃ : String) :

Make name s₁.s₂.s₃

Equations

Lean.Name.mkStr3 s₁ s₂ s₃ = ((Lean.Name.anonymous.str s₁).str s₂).str s₃

Instances For

@[reducible]

def Lean.Name.mkStr4 (s₁ s₂ s₃ s₄ : String) :

Make name s₁.s₂.s₃.s₄

Equations

Lean.Name.mkStr4 s₁ s₂ s₃ s₄ = (((Lean.Name.anonymous.str s₁).str s₂).str s₃).str s₄

Instances For

@[reducible]

def Lean.Name.mkStr5 (s₁ s₂ s₃ s₄ s₅ : String) :

Make name s₁.s₂.s₃.s₄.s₅

Equations

Lean.Name.mkStr5 s₁ s₂ s₃ s₄ s₅ = ((((Lean.Name.anonymous.str s₁).str s₂).str s₃).str s₄).str s₅

@[reducible]

def Lean.Name.mkStr6 (s₁ s₂ s₃ s₄ s₅ s₆ : String) :

Make name s₁.s₂.s₃.s₄.s₅.s₆

Equations

Lean.Name.mkStr6 s₁ s₂ s₃ s₄ s₅ s₆ = (((((Lean.Name.anonymous.str s₁).str s₂).str s₃).str s₄).str s₅).str s₆

@[reducible]

def Lean.Name.mkStr7 (s₁ s₂ s₃ s₄ s₅ s₆ s₇ : String) :

Make name s₁.s₂.s₃.s₄.s₅.s₆.s₇

Equations

Lean.Name.mkStr7 s₁ s₂ s₃ s₄ s₅ s₆ s₇ = ((((((Lean.Name.anonymous.str s₁).str s₂).str s₃).str s₄).str s₅).str s₆).str s₇

@[reducible]

def Lean.Name.mkStr8 (s₁ s₂ s₃ s₄ s₅ s₆ s₇ s₈ : String) :

Make name s₁.s₂.s₃.s₄.s₅.s₆.s₇.s₈

Equations

Lean.Name.mkStr8 s₁ s₂ s₃ s₄ s₅ s₆ s₇ s₈ = (((((((Lean.Name.anonymous.str s₁).str s₂).str s₃).str s₄).str s₅).str s₆).str s₇).str s₈

@[extern lean_name_eq]

def Lean.Name.beq :

Name → Name → Bool

(Boolean) equality comparator for names.

Equations

Lean.Name.anonymous.beq Lean.Name.anonymous = true
(p₁.str s₁).beq (p₂.str s₂) = (s₁ == s₂ && p₁.beq p₂)
(p₁.num n₁).beq (p₂.num n₂) = (n₁ == n₂ && p₁.beq p₂)
x✝¹.beq x✝ = false

instance Lean.Name.instBEq :

Equations

Lean.Name.instBEq = { beq := Lean.Name.beq }

def Lean.Name.appendCore :

Name → Name → Name

This function does not have special support for macro scopes. See Name.append.

Equations

x✝.appendCore Lean.Name.anonymous = x✝
x✝.appendCore (p.str s) = (x✝.appendCore p).str s
x✝.appendCore (p.num d) = (x✝.appendCore p).num d

def Lean.defaultMaxRecDepth :

The default maximum recursion depth. This is adjustable using the maxRecDepth option.

Equations

Lean.defaultMaxRecDepth = 512

def Lean.maxRecDepthErrorMessage :

The message to display on stack overflow.

Equations

Lean.maxRecDepthErrorMessage = "maximum recursion depth has been reached\nuse `set_option maxRecDepth <num>` to increase limit\nuse `set_option diagnostics true` to get diagnostic information"

Syntax #

inductive Lean.SourceInfo :

Source information that relates syntax to the context that it came from.

The primary purpose of SourceInfo is to relate the output of the parser and the macro expander to the original source file. When produced by the parser, Syntax.node does not carry source info; the parser associates it only with atoms and identifiers. If a Syntax.node is introduced by a quotation, then it has synthetic source info that both associates it with an original reference position and indicates that the original atoms in it may not originate from the Lean file under elaboration.

Source info is also used to relate Lean's output to the internal data that it represents; this is the basis for many interactive features. When used this way, it can occur on Syntax.node as well.

original (leading : Substring) (pos : String.Pos) (trailing : Substring) (endPos : String.Pos) : SourceInfo
A token produced by the parser from original input that includes both leading and trailing whitespace as well as position information.
The leading whitespace is inferred after parsing by Syntax.updateLeading. This is because the “preceding token” is not well-defined during parsing, especially in the presence of backtracking.
synthetic (pos endPos : String.Pos) (canonical : Bool := false) : SourceInfo
Synthetic syntax is syntax that was produced by a metaprogram or by Lean itself (e.g. by a quotation). Synthetic syntax is annotated with a source span from the original syntax, which relates it to the source file.
The delaborator uses this constructor to store an encoded indicator of which core language expression gave rise to the syntax.
The canonical flag on synthetic syntax is enabled for syntax that is not literally part of the original input syntax but should be treated “as if” the user really wrote it for the purpose of hovers and error messages. This is usually used on identifiers in order to connect the binding site to the user's original syntax even if the name of the identifier changes during expansion, as well as on tokens that should receive targeted messages.
Generally speaking, a macro expansion should only use a given piece of input syntax in a single canonical token. An exception to this rule is when the same identifier is used to declare two binders, as in the macro expansion for dependent if:
```
`(if $h : $cond then $t else $e) ~>
`(dite $cond (fun $h => $t) (fun $h => $t))
```
In these cases, if the user hovers over h they will see information about both binding sites.
none : SourceInfo
A synthesized token without position information.

Instances For

instance Lean.instInhabitedSourceInfo :

Inhabited SourceInfo

Equations

Lean.instInhabitedSourceInfo = { default := Lean.SourceInfo.none }

def Lean.SourceInfo.getPos? (info : SourceInfo) (canonicalOnly : Bool := false) :

Option String.Pos

Gets the position information from a SourceInfo, if available. If canonicalOnly is true, then .synthetic syntax with canonical := false will also return none.

Equations

(Lean.SourceInfo.original leading pos trailing endPos).getPos? canonicalOnly = some pos
(Lean.SourceInfo.synthetic pos endPos true).getPos? canonicalOnly = some pos
(Lean.SourceInfo.synthetic pos endPos canonical).getPos? = some pos
info.getPos? canonicalOnly = none

def Lean.SourceInfo.getTailPos? (info : SourceInfo) (canonicalOnly : Bool := false) :

Option String.Pos

Gets the end position information from a SourceInfo, if available. If canonicalOnly is true, then .synthetic syntax with canonical := false will also return none.

Equations

(Lean.SourceInfo.original leading pos trailing endPos).getTailPos? canonicalOnly = some endPos
(Lean.SourceInfo.synthetic pos endPos true).getTailPos? canonicalOnly = some endPos
(Lean.SourceInfo.synthetic pos endPos canonical).getTailPos? = some endPos
info.getTailPos? canonicalOnly = none

def Lean.SourceInfo.getTrailing? (info : SourceInfo) :

Option Substring

Gets the substring representing the trailing whitespace of a SourceInfo, if available.

Equations

(Lean.SourceInfo.original leading pos trailing endPos).getTrailing? = some trailing
info.getTrailing? = none

def Lean.SourceInfo.getTrailingTailPos? (info : SourceInfo) (canonicalOnly : Bool := false) :

Option String.Pos

Gets the end position information of the trailing whitespace of a SourceInfo, if available. If canonicalOnly is true, then .synthetic syntax with canonical := false will also return none.

Equations

info.getTrailingTailPos? canonicalOnly = match info.getTrailing? with | some trailing => some trailing.stopPos | none => info.getTailPos? canonicalOnly

@[reducible, inline]

abbrev Lean.SyntaxNodeKind :

Specifies the interpretation of a Syntax.node value. An abbreviation for Name.

Node kinds may be any name, and do not need to refer to declarations in the environment. Conventionally, however, a node's kind corresponds to the Parser or ParserDesc declaration that produces it. There are also a number of built-in node kinds that are used by the parsing infrastructure, such as nullKind and choiceKind; these do not correspond to parser declarations.

Equations

Lean.SyntaxNodeKind = Lean.Name

Instances For

Lean.instCoeSyntaxNodeKindSyntaxNodeKinds

Syntax AST #

inductive Lean.Syntax.Preresolved :

A possible binding of an identifier in the context in which it was quoted.

Identifiers in quotations may refer to either global declarations or to namespaces that are in scope at the site of the quotation. These are saved in the Syntax.ident constructor and are part of the implementation of hygienic macros.

namespace (ns : Name) : Preresolved
A potential namespace reference
decl (n : Name) (fields : List String) : Preresolved
A potential global constant or section variable reference, with additional field accesses

Instances For

inductive Lean.Syntax :

Lean syntax trees.

Syntax trees are used pervasively throughout Lean: they are produced by the parser, transformed by the macro expander, and elaborated. They are also produced by the delaborator and presented to users.

missing : Syntax
A portion of the syntax tree that is missing because of a parse error.
The indexing operator on Syntax also returns Syntax.missing when the index is out of bounds.
node (info : SourceInfo) (kind : SyntaxNodeKind) (args : Array Syntax) : Syntax
A node in the syntax tree that may have further syntax as child nodes. The node's kind determines its interpretation.
For nodes produced by the parser, the info field is typically Lean.SourceInfo.none, and source information is stored in the corresponding fields of identifiers and atoms. This field is used in two ways:
1. The delaborator uses it to associate nodes with metadata that are used to implement interactive features.
2. Nodes created by quotations use the field to mark the syntax as synthetic (storing the result of Lean.SourceInfo.fromRef) even when its leading or trailing tokens are not.
atom (info : SourceInfo) (val : String) : Syntax
A non-identifier atomic component of syntax.
All of the following are atoms:
- keywords, such as def, fun, and inductive
- literals, such as numeric or string literals
- punctuation and delimiters, such as (, ), and =>.
Identifiers are represented by the Lean.Syntax.ident constructor. Atoms also correspond to quoted strings inside syntax declarations.
ident (info : SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List Preresolved) : Syntax
An identifier.
In addition to source information, identifiers have the following fields:
- rawVal is the literal substring from the input file
- val is the parsed Lean name, potentially including macro scopes.
- preresolved is the list of possible declarations this could refer to, populated by quotations.

Instances For

def Lean.Syntax.node1 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ : Syntax) :

Create syntax node with 1 child

Equations

Lean.Syntax.node1 info kind a₁ = Lean.Syntax.node info kind #[a₁]

def Lean.Syntax.node2 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ : Syntax) :

Create syntax node with 2 children

Equations

Lean.Syntax.node2 info kind a₁ a₂ = Lean.Syntax.node info kind #[a₁, a₂]

def Lean.Syntax.node3 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ : Syntax) :

Create syntax node with 3 children

Equations

Lean.Syntax.node3 info kind a₁ a₂ a₃ = Lean.Syntax.node info kind #[a₁, a₂, a₃]

def Lean.Syntax.node4 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ : Syntax) :

Create syntax node with 4 children

Equations

Lean.Syntax.node4 info kind a₁ a₂ a₃ a₄ = Lean.Syntax.node info kind #[a₁, a₂, a₃, a₄]

def Lean.Syntax.node5 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ : Syntax) :

Create syntax node with 5 children

Equations

Lean.Syntax.node5 info kind a₁ a₂ a₃ a₄ a₅ = Lean.Syntax.node info kind #[a₁, a₂, a₃, a₄, a₅]

def Lean.Syntax.node6 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ : Syntax) :

Create syntax node with 6 children

Equations

Lean.Syntax.node6 info kind a₁ a₂ a₃ a₄ a₅ a₆ = Lean.Syntax.node info kind #[a₁, a₂, a₃, a₄, a₅, a₆]

def Lean.Syntax.node7 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ a₇ : Syntax) :

Create syntax node with 7 children

Equations

Lean.Syntax.node7 info kind a₁ a₂ a₃ a₄ a₅ a₆ a₇ = Lean.Syntax.node info kind #[a₁, a₂, a₃, a₄, a₅, a₆, a₇]

def Lean.Syntax.node8 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : Syntax) :

Create syntax node with 8 children

Equations

Lean.Syntax.node8 info kind a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ = Lean.Syntax.node info kind #[a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈]

def Lean.SyntaxNodeKinds :

SyntaxNodeKinds is a set of SyntaxNodeKind, implemented as a list.

Singleton SyntaxNodeKinds are extremely common. They are written as name literals, rather than as lists; list syntax is required only for empty or non-singleton sets of kinds.

Equations

Lean.SyntaxNodeKinds = List Lean.SyntaxNodeKind

Instances For

Lean.instCoeSyntaxNodeKindSyntaxNodeKinds

structure Lean.TSyntax (ks : SyntaxNodeKinds) :

Typed syntax, which tracks the potential kinds of the Syntax it contains.

While syntax quotations produce or expect TSyntax values of the correct kinds, this is not otherwise enforced; it can easily be circumvented by direct use of the constructor.

raw : Syntax
The underlying Syntax value.

Instances For

instance Lean.instInhabitedSyntax :

Inhabited Syntax

Equations

Lean.instInhabitedSyntax = { default := Lean.Syntax.missing }

instance Lean.instInhabitedTSyntax {ks : SyntaxNodeKinds} :

Inhabited (TSyntax ks)

Equations

Lean.instInhabitedTSyntax = { default := { raw := default } }

Builtin kinds #

@[reducible, inline]

abbrev Lean.choiceKind :

The `choice kind is used to represent ambiguous parse results.

The parser prioritizes longer matches over shorter ones, but there is not always a unique longest match. All the parse results are saved, and the determination of which to use is deferred until typing information is available.

Equations

Lean.choiceKind = `choice

@[reducible, inline]

abbrev Lean.nullKind :

`null is the “fallback” kind, used when no other kind applies. Null nodes result from repetition operators, and empty null nodes represent the failure of an optional parse.

The null kind is used for raw list parsers like many.

Equations

Lean.nullKind = `null

@[reducible, inline]

abbrev Lean.groupKind :

The `group kind is used for nodes that result from Lean.Parser.group. This avoids confusion with the null kind when used inside optional.

Equations

Lean.groupKind = `group

@[reducible, inline]

abbrev Lean.identKind :

The pseudo-kind assigned to identifiers: `ident.

The name `ident is not actually used as a kind for Syntax.node values. It is used by convention as the kind of Syntax.ident values.

Equations

Lean.identKind = `ident

@[reducible, inline]

abbrev Lean.strLitKind :

`str is the node kind of string literals like "foo".

Equations

Lean.strLitKind = `str

Instances For

Lean.instQuoteStringStrLitKind

@[reducible, inline]

abbrev Lean.charLitKind :

`char is the node kind of character literals like 'A'.

Equations

Lean.charLitKind = `char

Instances For

Lean.instQuoteCharCharLitKind

@[reducible, inline]

abbrev Lean.numLitKind :

`num is the node kind of number literals like 42.

Equations

Lean.numLitKind = `num

Instances For

Lean.instQuoteNatNumLitKind

@[reducible, inline]

abbrev Lean.scientificLitKind :

`scientific is the node kind of floating point literals like 1.23e-3.

Equations

Lean.scientificLitKind = `scientific

@[reducible, inline]

abbrev Lean.nameLitKind :

`name is the node kind of name literals like `foo.

Equations

Lean.nameLitKind = `name

@[reducible, inline]

abbrev Lean.fieldIdxKind :

`` fieldIdx is the node kind of projection indices like the 2 in x.2.

Equations

Lean.fieldIdxKind = `fieldIdx

@[reducible, inline]

abbrev Lean.hygieneInfoKind :

`hygieneInfo is the node kind of the Lean.Parser.hygieneInfo parser, which produces an “invisible token” that captures the hygiene information at the current point without parsing anything.

They can be used to generate identifiers (with Lean.HygieneInfo.mkIdent) as if they were introduced in a macro's input, rather than by its implementation.

Equations

Lean.hygieneInfoKind = `hygieneInfo

@[reducible, inline]

abbrev Lean.interpolatedStrLitKind :

`interpolatedStrLitKind is the node kind of interpolated string literal fragments like "value = { and }" in s!"value = {x}".

Equations

Lean.interpolatedStrLitKind = `interpolatedStrLitKind

@[reducible, inline]

abbrev Lean.interpolatedStrKind :

`interpolatedStrKind is the node kind of an interpolated string literal like "value = {x}" in s!"value = {x}".

Equations

Lean.interpolatedStrKind = `interpolatedStrKind

@[inline]

def Lean.mkNode (k : SyntaxNodeKind) (args : Array Syntax) :

Creates an info-less node of the given kind and children.

Equations

Lean.mkNode k args = { raw := Lean.Syntax.node Lean.SourceInfo.none k args }

@[inline]

def Lean.mkNullNode (args : Array Syntax := #[]) :

Creates an info-less nullKind node with the given children, if any.

Equations

Lean.mkNullNode args = (Lean.mkNode Lean.nullKind args).raw

def Lean.Syntax.getKind (stx : Syntax) :

Gets the kind of a Syntax.node value, or the pseudo-kind of any other Syntax value.

“Pseudo-kinds” are kinds that are assigned by convention to non-Syntax.node values: identKind for Syntax.ident, `missing for Syntax.missing, and the atom's string literal for atoms.

Equations

(Lean.Syntax.node info k args).getKind = k
Lean.Syntax.missing.getKind = `missing
(Lean.Syntax.atom info v).getKind = Lean.Name.mkSimple v
(Lean.Syntax.ident info rawVal val preresolved).getKind = Lean.identKind

def Lean.Syntax.setKind (stx : Syntax) (k : SyntaxNodeKind) :

Changes the kind at the root of a Syntax.node to k.

Returns all other Syntax values unchanged.

Equations

(Lean.Syntax.node info kind args).setKind k = Lean.Syntax.node info k args
stx.setKind k = stx

def Lean.Syntax.isOfKind (stx : Syntax) (k : SyntaxNodeKind) :

Checks whether syntax has the given kind or pseudo-kind.

“Pseudo-kinds” are kinds that are assigned by convention to non-Syntax.node values: identKind for Syntax.ident, `missing for Syntax.missing, and the atom's string literal for atoms.

Equations

stx.isOfKind k = (stx.getKind == k)

def Lean.Syntax.getArg (stx : Syntax) (i : Nat) :

Gets the i'th argument of the syntax node. This can also be written stx[i]. Returns missing if i is out of range.

Equations

(Lean.Syntax.node info kind args).getArg i = args.getD i Lean.Syntax.missing
stx.getArg i = Lean.Syntax.missing

def Lean.Syntax.getArgs (stx : Syntax) :

Gets the list of arguments of the syntax node, or #[] if it's not a node.

Equations

(Lean.Syntax.node info kind args).getArgs = args
stx.getArgs = #[]

def Lean.Syntax.getNumArgs (stx : Syntax) :

Gets the number of arguments of the syntax node, or 0 if it's not a node.

Equations

(Lean.Syntax.node info kind args).getNumArgs = args.size
stx.getNumArgs = 0

def Lean.Syntax.getOptional? (stx : Syntax) :

Assuming stx was parsed by optional, returns the enclosed syntax if it parsed something and none otherwise.

Equations

(Lean.Syntax.node info kind args).getOptional? = match kind == Lean.nullKind && args.size == 1 with | true => some (args.get!Internal 0) | false => none
stx.getOptional? = none

def Lean.Syntax.isMissing :

Syntax → Bool

Is this syntax .missing?

Equations

Lean.Syntax.missing.isMissing = true
x✝.isMissing = false

def Lean.Syntax.isNodeOf (stx : Syntax) (k : SyntaxNodeKind) (n : Nat) :

Is this syntax a node with kind k?

Equations

stx.isNodeOf k n = (stx.isOfKind k && stx.getNumArgs == n)

def Lean.Syntax.isIdent :

Syntax → Bool

stx.isIdent is true iff stx is an identifier.

Equations

(Lean.Syntax.ident info rawVal val preresolved).isIdent = true
x✝.isIdent = false

def Lean.Syntax.getId :

Syntax → Name

If this is an ident, return the parsed value, else .anonymous.

Equations

(Lean.Syntax.ident info rawVal val preresolved).getId = val
x✝.getId = Lean.Name.anonymous

def Lean.Syntax.getInfo? :

Syntax → Option SourceInfo

Retrieve the immediate info from the Syntax node.

Equations

(Lean.Syntax.atom info val).getInfo? = some info
(Lean.Syntax.ident info rawVal val preresolved).getInfo? = some info
(Lean.Syntax.node info kind args).getInfo? = some info
Lean.Syntax.missing.getInfo? = none

partial def Lean.Syntax.getHeadInfo? :

Syntax → Option SourceInfo

Retrieve the left-most node or leaf's info in the Syntax tree.

partial def Lean.Syntax.getHeadInfo?.loop (args : Array Syntax) (i : Nat) :

Option SourceInfo

def Lean.Syntax.getHeadInfo (stx : Syntax) :

Retrieve the left-most leaf's info in the Syntax tree, or none if there is no token.

Equations

stx.getHeadInfo = match stx.getHeadInfo? with | some info => info | none => Lean.SourceInfo.none

def Lean.Syntax.getPos? (stx : Syntax) (canonicalOnly : Bool := false) :

Option String.Pos

Get the starting position of the syntax, if possible. If canonicalOnly is true, non-canonical synthetic nodes are treated as not carrying position information.

Equations

stx.getPos? canonicalOnly = stx.getHeadInfo.getPos? canonicalOnly

partial def Lean.Syntax.getTailPos? (stx : Syntax) (canonicalOnly : Bool := false) :

Option String.Pos

Get the ending position of the syntax, if possible. If canonicalOnly is true, non-canonical synthetic nodes are treated as not carrying position information.

partial def Lean.Syntax.getTailPos?.loop (canonicalOnly : Bool := false) (args : Array Syntax) (i : Nat) :

Option String.Pos

structure Lean.Syntax.SepArray (sep : String) :

An array of syntax elements interspersed with the given separators.

Separator arrays result from repetition operators such as ,*. Coercions to and from Array Syntax insert or remove separators as required.

The typed equivalent is Lean.Syntax.TSepArray.

elemsAndSeps : Array Syntax
The array of elements and separators, ordered like #[el1, sep1, el2, sep2, el3].

Instances For

structure Lean.Syntax.TSepArray (ks : SyntaxNodeKinds) (sep : String) :

An array of syntax elements that alternate with the given separator. Each syntax element has a kind drawn from ks.

Separator arrays result from repetition operators such as ,*. Coercions to and from Array (TSyntax ks) insert or remove separators as required. The untyped equivalent is Lean.Syntax.SepArray.

elemsAndSeps : Array Syntax
The array of elements and separators, ordered like #[el1, sep1, el2, sep2, el3].

Instances For

@[reducible, inline]

abbrev Lean.TSyntaxArray (ks : SyntaxNodeKinds) :

An array of syntaxes of kind ks.

Equations

Lean.TSyntaxArray ks = Array (Lean.TSyntax ks)

Instances For

unsafe def Lean.TSyntaxArray.rawImpl {ks : SyntaxNodeKinds} :

TSyntaxArray ks → Array Syntax

Implementation of TSyntaxArray.raw.

Equations

Lean.TSyntaxArray.rawImpl = unsafeCast

@[implemented_by Lean.TSyntaxArray.rawImpl]

opaque Lean.TSyntaxArray.raw {ks : SyntaxNodeKinds} (as : TSyntaxArray ks) :

Converts a TSyntaxArray to an Array Syntax, without reallocation.

unsafe def Lean.TSyntaxArray.mkImpl {ks : SyntaxNodeKinds} :

Array Syntax → TSyntaxArray ks

Implementation of TSyntaxArray.mk.

Equations

Lean.TSyntaxArray.mkImpl = unsafeCast

@[implemented_by Lean.TSyntaxArray.mkImpl]

opaque Lean.TSyntaxArray.mk {ks : SyntaxNodeKinds} (as : Array Syntax) :

TSyntaxArray ks

Converts an Array Syntax to a TSyntaxArray, without reallocation.

def Lean.SourceInfo.fromRef (ref : Syntax) (canonical : Bool := false) :

Constructs a synthetic SourceInfo using a ref : Syntax for the span.

Equations

One or more equations did not get rendered due to their size.

def Lean.mkAtom (val : String) :

Constructs a synthetic atom with no source info.

Equations

Lean.mkAtom val = Lean.Syntax.atom Lean.SourceInfo.none val

def Lean.mkAtomFrom (src : Syntax) (val : String) (canonical : Bool := false) :

Constructs a synthetic atom with source info coming from src.

Equations

Lean.mkAtomFrom src val canonical = Lean.Syntax.atom (Lean.SourceInfo.fromRef src canonical) val

Parser descriptions #

inductive Lean.ParserDescr :

A ParserDescr is a grammar for parsers. This is used by the syntax command to produce parsers without having to import Lean.

const (name : Name) : ParserDescr
A (named) nullary parser, like ppSpace
unary (name : Name) (p : ParserDescr) : ParserDescr
A (named) unary parser, like group(p)
binary (name : Name) (p₁ p₂ : ParserDescr) : ParserDescr
A (named) binary parser, like orelse or andthen (written as p1 <|> p2 and p1 p2 respectively in syntax)
node (kind : SyntaxNodeKind) (prec : Nat) (p : ParserDescr) : ParserDescr
Parses using p, then pops the stack to create a new node with kind kind. The precedence prec is used to determine whether the parser should apply given the current precedence level.
trailingNode (kind : SyntaxNodeKind) (prec lhsPrec : Nat) (p : ParserDescr) : ParserDescr
Like node but for trailing parsers (which start with a nonterminal). Assumes the lhs is already on the stack, and parses using p, then pops the stack including the lhs to create a new node with kind kind. The precedence prec and lhsPrec are used to determine whether the parser should apply.
symbol (val : String) : ParserDescr
A literal symbol parser: parses val as a literal. This parser does not work on identifiers, so symbol arguments are declared as "keywords" and cannot be used as identifiers anywhere in the file.
nonReservedSymbol (val : String) (includeIdent : Bool) : ParserDescr
Like symbol, but without reserving val as a keyword. If includeIdent is true then ident will be reinterpreted as atom if it matches.
cat (catName : Name) (rbp : Nat) : ParserDescr
Parses using the category parser catName with right binding power (i.e. precedence) rbp.
parser (declName : Name) : ParserDescr
Parses using another parser declName, which can be either a Parser or ParserDescr.
nodeWithAntiquot (name : String) (kind : SyntaxNodeKind) (p : ParserDescr) : ParserDescr
Like node, but also declares that the body can be matched using an antiquotation with name name. For example, def $id:declId := 1 uses an antiquotation with name declId in the place where a declId is expected.
sepBy (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false) : ParserDescr
A sepBy(p, sep) parses 0 or more occurrences of p separated by sep. psep is usually the same as symbol sep, but it can be overridden. sep is only used in the antiquot syntax: $x;* would match if sep is ";". allowTrailingSep is true if e.g. a, b, is also allowed to match.
sepBy1 (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false) : ParserDescr
sepBy1 is just like sepBy, except it takes 1 or more instead of 0 or more occurrences of p.

Instances For

Lean.instInhabitedParserDescr

instance Lean.instInhabitedParserDescr :

Inhabited ParserDescr

Equations

Lean.instInhabitedParserDescr = { default := Lean.ParserDescr.symbol "" }

@[reducible, inline]

abbrev Lean.TrailingParserDescr :

Although TrailingParserDescr is an abbreviation for ParserDescr, Lean will look at the declared type in order to determine whether to add the parser to the leading or trailing parser table. The determination is done automatically by the syntax command.

Equations

Lean.TrailingParserDescr = Lean.ParserDescr

Runtime support for making quotation terms auto-hygienic, by mangling identifiers introduced by them with a "macro scope" supplied by the context. Details to appear in a paper soon.

@[reducible, inline]

abbrev Lean.MacroScope :

A macro scope identifier is just a Nat that gets bumped every time we enter a new macro scope. Within a macro scope, all occurrences of identifier x parse to the same thing, but x parsed from different macro scopes will produce different identifiers.

Equations

Lean.MacroScope = Nat

def Lean.reservedMacroScope :

Macro scope used internally. It is not available for our frontend.

Equations

Lean.reservedMacroScope = 0

def Lean.firstFrontendMacroScope :

First macro scope available for our frontend

Equations

Lean.firstFrontendMacroScope = Lean.reservedMacroScope + 1

class Lean.MonadRef (m : Type → Type) :

A MonadRef is a monad that has a ref : Syntax in the read-only state. This is used to keep track of the location where we are working; if an exception is thrown, the ref gives the location where the error will be reported, assuming no more specific location is provided.

getRef : m Syntax
Get the current value of the ref
withRef {α : Type} : Syntax → m α → m α
Run x : m α with a modified value for the ref

Instances

instance Lean.instMonadRefOfMonadLiftOfMonadFunctor (m n : Type → Type) [MonadLift m n] [MonadFunctor m n] [MonadRef m] :

Equations

One or more equations did not get rendered due to their size.

def Lean.replaceRef (ref oldRef : Syntax) :

Replaces oldRef with ref, unless ref has no position info. This biases us to having a valid span to report an error on.

Equations

Lean.replaceRef ref oldRef = match ref.getPos? with | some val => ref | x => oldRef

@[inline]

def Lean.withRef {m : Type → Type} [Monad m] [MonadRef m] {α : Type} (ref : Syntax) (x : m α) :

m α

Run x : m α with a modified value for the ref. This is not exactly the same as MonadRef.withRef, because it uses replaceRef to avoid putting syntax with bad spans in the state.

Equations

Lean.withRef ref x = do let oldRef ← Lean.getRef let ref : Lean.Syntax := Lean.replaceRef ref oldRef Lean.MonadRef.withRef ref x

@[inline]

def Lean.withRef? {m : Type → Type} [Monad m] [MonadRef m] {α : Type} (ref? : Option Syntax) (x : m α) :

m α

If ref? = some ref, run x : m α with a modified value for the ref by calling withRef. Otherwise, run x directly.

Equations

Lean.withRef? (some ref) x = Lean.withRef ref x
Lean.withRef? ref? x = x

class Lean.MonadQuotation (m : Type → Type) extends Lean.MonadRef m :

A monad that supports syntax quotations. Syntax quotations (in term position) are monadic values that when executed retrieve the current "macro scope" from the monad and apply it to every identifier they introduce (independent of whether this identifier turns out to be a reference to an existing declaration, or an actually fresh binding during further elaboration). We also apply the position of the result of getRef to each introduced symbol, which results in better error positions than not applying any position.

getRef : m Syntax
withRef {α : Type} : Syntax → m α → m α
getCurrMacroScope : m MacroScope
Get the fresh scope of the current macro invocation
getMainModule : m Name
Get the module name of the current file. This is used to ensure that hygienic names don't clash across multiple files.
withFreshMacroScope {α : Type} : m α → m α
Execute action in a new macro invocation context. This transformer should be used at all places that morally qualify as the beginning of a "macro call", e.g. elabCommand and elabTerm in the case of the elaborator. However, it can also be used internally inside a "macro" if identifiers introduced by e.g. different recursive calls should be independent and not collide. While returning an intermediate syntax tree that will recursively be expanded by the elaborator can be used for the same effect, doing direct recursion inside the macro guarded by this transformer is often easier because one is not restricted to passing a single syntax tree. Modelling this helper as a transformer and not just a monadic action ensures that the current macro scope before the recursive call is restored after it, as expected.

Instances

@[inline]

def Lean.MonadRef.mkInfoFromRefPos {m : Type → Type} [Monad m] [MonadRef m] :

Construct a synthetic SourceInfo from the ref in the monad state.

Equations

Lean.MonadRef.mkInfoFromRefPos = do let __do_lift ← Lean.getRef pure (Lean.SourceInfo.fromRef __do_lift)

instance Lean.instMonadQuotationOfMonadFunctorOfMonadLift {m n : Type → Type} [MonadFunctor m n] [MonadLift m n] [MonadQuotation m] :

MonadQuotation n

Equations

One or more equations did not get rendered due to their size.

We represent a name with macro scopes as

<actual name>._@.(<module_name>.<scopes>)*.<module_name>._hyg.<scopes>

Example: suppose the module name is Init.Data.List.Basic, and name is foo.bla, and macroscopes [2, 5]

foo.bla._@.Init.Data.List.Basic._hyg.2.5

We may have to combine scopes from different files/modules. The main modules being processed is always the right-most one. This situation may happen when we execute a macro generated in an imported file in the current file.

foo.bla._@.Init.Data.List.Basic.2.1.Init.Lean.Expr._hyg.4

The delimiter _hyg is used just to improve the hasMacroScopes performance.

def Lean.Name.hasMacroScopes :

Does this name have hygienic macro scopes?

Equations

(pre.str s).hasMacroScopes = (s == "_hyg")
(p.num i).hasMacroScopes = p.hasMacroScopes
x✝.hasMacroScopes = false

@[export lean_erase_macro_scopes]

def Lean.Name.eraseMacroScopes (n : Name) :

Remove the macro scopes from the name.

Equations

n.eraseMacroScopes = match n.hasMacroScopes with | true => Lean.eraseMacroScopesAux✝ n | false => n

@[export lean_simp_macro_scopes]

def Lean.Name.simpMacroScopes (n : Name) :

Helper function we use to create binder names that do not need to be unique.

Equations

n.simpMacroScopes = match n.hasMacroScopes with | true => Lean.simpMacroScopesAux✝ n | false => n

structure Lean.MacroScopesView :

A MacroScopesView represents a parsed hygienic name. extractMacroScopes will decode it from a Name, and .review will re-encode it. The grammar of a hygienic name is:

<name>._@.(<module_name>.<scopes>)*.<mainModule>._hyg.<scopes>

name : Name
The original (unhygienic) name.
imported : Name
All the name components (<module_name>.<scopes>)* from the imports concatenated together.
mainModule : Name
The main module in which this identifier was parsed.
scopes : List MacroScope
The list of macro scopes.

Instances For

Lean.instInhabitedMacroScopesView

instance Lean.instInhabitedMacroScopesView :

Inhabited MacroScopesView

Equations

Lean.instInhabitedMacroScopesView = { default := { name := default, imported := default, mainModule := default, scopes := default } }

def Lean.MacroScopesView.review (view : MacroScopesView) :

Encode a hygienic name from the parsed pieces.

Equations

One or more equations did not get rendered due to their size.

def Lean.extractMacroScopes (n : Name) :

MacroScopesView

Revert all addMacroScope calls. v = extractMacroScopes n → n = v.review. This operation is useful for analyzing/transforming the original identifiers, then adding back the scopes (via MacroScopesView.review).

Equations

One or more equations did not get rendered due to their size.

def Lean.addMacroScope (mainModule n : Name) (scp : MacroScope) :

Add a new macro scope onto the name n, in the given mainModule.

Equations

One or more equations did not get rendered due to their size.

def Lean.Name.append (a b : Name) :

Appends two names a and b, propagating macro scopes from a or b, if any, to the result. Panics if both a and b have macro scopes.

This function is used for the Append Name instance.

See also Lean.Name.appendCore, which appends names without any consideration for macro scopes. Also consider Lean.Name.eraseMacroScopes to erase macro scopes before appending, if appropriate.

Equations

One or more equations did not get rendered due to their size.

instance Lean.instAppendName :

Equations

Lean.instAppendName = { append := Lean.Name.append }

@[inline]

def Lean.MonadQuotation.addMacroScope {m : Type → Type} [MonadQuotation m] [Monad m] (n : Name) :

Add a new macro scope onto the name n, using the monad state to supply the main module and current macro scope.

Equations

Lean.MonadQuotation.addMacroScope n = do let mainModule ← Lean.getMainModule let scp ← Lean.getCurrMacroScope pure (Lean.addMacroScope mainModule n scp)

def Lean.Syntax.matchesNull (stx : Syntax) (n : Nat) :

Is this syntax a null node?

Equations

stx.matchesNull n = stx.isNodeOf Lean.nullKind n

def Lean.Syntax.matchesIdent (stx : Syntax) (id : Name) :

Function used for determining whether a syntax pattern `(id) is matched. There are various conceivable notions of when two syntactic identifiers should be regarded as identical, but semantic definitions like whether they refer to the same global name cannot be implemented without context information (i.e. MonadResolveName). Thus in patterns we default to the structural solution of comparing the identifiers' Name values, though we at least do so modulo macro scopes so that identifiers that "look" the same match. This is particularly useful when dealing with identifiers that do not actually refer to Lean bindings, e.g. in the stx pattern `(many($p)).

Equations

stx.matchesIdent id = (stx.isIdent && stx.getId.eraseMacroScopes == id.eraseMacroScopes)

def Lean.Syntax.matchesLit (stx : Syntax) (k : SyntaxNodeKind) (val : String) :

Is this syntax a node kind k wrapping an atom _ val?

Equations

(Lean.Syntax.node info kind args).matchesLit k val = (k == kind && match args.getD 0 Lean.Syntax.missing with | Lean.Syntax.atom info val' => val == val' | x => false)
stx.matchesLit k val = false

instance Lean.Macro.instNonemptyMethodsRef :

Nonempty Lean.Macro.MethodsRef✝

structure Lean.Macro.Context :

The read-only context for the MacroM monad.

methods : Lean.Macro.MethodsRef✝
An opaque reference to the Methods object. This is done to break a dependency cycle: the Methods involve MacroM which has not been defined yet.
mainModule : Name
The currently parsing module.
currMacroScope : MacroScope
The current macro scope.
currRecDepth : Nat
The current recursion depth.
maxRecDepth : Nat
The maximum recursion depth.
ref : Syntax
The syntax which supplies the position of error messages.

inductive Lean.Macro.Exception :

An exception in the MacroM monad.

error : Syntax → String → Exception
A general error, given a message and a span (expressed as a Syntax).
unsupportedSyntax : Exception
An unsupported syntax exception. We keep this separate because it is used for control flow: if one macro does not support a syntax then we try the next one.

structure Lean.Macro.State :

The mutable state for the MacroM monad.

macroScope : MacroScope
The global macro scope counter, used for producing fresh scope names.
traceMsgs : List (Name × String)
The list of trace messages that have been produced, each with a trace class and a message.

Instances For

Lean.Macro.instInhabitedState

instance Lean.Macro.instInhabitedState :

Inhabited State

Equations

Lean.Macro.instInhabitedState = { default := { macroScope := default, traceMsgs := default } }

@[reducible, inline]

abbrev Lean.MacroM (α : Type) :

The MacroM monad is the main monad for macro expansion. It has the information needed to handle hygienic name generation, and is the monad that macro definitions live in.

Notably, this is a (relatively) pure monad: there is no IO and no access to the Environment. That means that things like declaration lookup are impossible here, as well as IO.Ref or other side-effecting operations. For more capabilities, macros can instead be written as elab using adaptExpander.

Equations

Lean.MacroM = ReaderT Lean.Macro.Context (EStateM Lean.Macro.Exception Lean.Macro.State)

Instances For

@[reducible, inline]

abbrev Lean.Macro :

A macro has type Macro, which is a Syntax → MacroM Syntax: it receives an input syntax and is supposed to "expand" it into another piece of syntax.

Equations

Lean.Macro = (Lean.Syntax → Lean.MacroM Lean.Syntax)

instance Lean.Macro.instMonadRefMacroM :

MonadRef MacroM

Equations

One or more equations did not get rendered due to their size.

def Lean.Macro.addMacroScope (n : Name) :

Add a new macro scope to the name n.

Equations

Lean.Macro.addMacroScope n = do let ctx ← read pure (Lean.addMacroScope ctx.mainModule n ctx.currMacroScope)

def Lean.Macro.throwUnsupported {α : Type} :

Throw an unsupportedSyntax exception.

Equations

Lean.Macro.throwUnsupported = throw Lean.Macro.Exception.unsupportedSyntax

def Lean.Macro.throwError {α : Type} (msg : String) :

Throw an error with the given message, using the ref for the location information.

Equations

Lean.Macro.throwError msg = do let ref ← Lean.getRef throw (Lean.Macro.Exception.error ref msg)

def Lean.Macro.throwErrorAt {α : Type} (ref : Syntax) (msg : String) :

Throw an error with the given message and location information.

Equations

Lean.Macro.throwErrorAt ref msg = Lean.withRef ref (Lean.Macro.throwError msg)

@[inline]

def Lean.Macro.withFreshMacroScope {α : Type} (x : MacroM α) :

Increments the macro scope counter so that inside the body of x the macro scope is fresh.

Equations

One or more equations did not get rendered due to their size.

@[inline]

def Lean.Macro.withIncRecDepth {α : Type} (ref : Syntax) (x : MacroM α) :

Run x with an incremented recursion depth counter.

Equations

One or more equations did not get rendered due to their size.

instance Lean.Macro.instMonadQuotationMacroM :

MonadQuotation MacroM

Equations

One or more equations did not get rendered due to their size.

structure Lean.Macro.Methods :

The opaque methods that are available to MacroM.

expandMacro? : Syntax → MacroM (Option Syntax)
Expands macros in the given syntax. A return value of none means there was nothing to expand.
getCurrNamespace : MacroM Name
Get the current namespace in the file.
hasDecl : Name → MacroM Bool
Check if a given name refers to a declaration.
resolveNamespace : Name → MacroM (List Name)
Resolves the given name to an overload list of namespaces.
resolveGlobalName : Name → MacroM (List (Name × List String))
Resolves the given name to an overload list of global definitions. The List String in each alternative is the deduced list of projections (which are ambiguous with name components).

Instances For

Lean.Macro.instInhabitedMethods

instance Lean.Macro.instInhabitedMethods :

Inhabited Methods

Equations

One or more equations did not get rendered due to their size.

unsafe def Lean.Macro.mkMethodsImp (methods : Methods) :

Lean.Macro.MethodsRef✝

Implementation of mkMethods.

Equations

Lean.Macro.mkMethodsImp methods = unsafeCast methods

@[implemented_by Lean.Macro.mkMethodsImp]

opaque Lean.Macro.mkMethods (methods : Methods) :

Lean.Macro.MethodsRef✝

Make an opaque reference to a Methods.

instance Lean.Macro.instInhabitedMethodsRef :

Inhabited Lean.Macro.MethodsRef✝

Equations

Lean.Macro.instInhabitedMethodsRef = { default := Lean.Macro.mkMethods default }

unsafe def Lean.Macro.getMethodsImp :

Implementation of getMethods.

Equations

Lean.Macro.getMethodsImp = do let ctx ← read pure (unsafeCast ctx.methods)

@[implemented_by Lean.Macro.getMethodsImp]

opaque Lean.Macro.getMethods :

Extract the methods list from the MacroM state.

def Lean.Macro.expandMacro? (stx : Syntax) :

MacroM (Option Syntax)

expandMacro? stx returns some stxNew if stx is a macro, and stxNew is its expansion.

Equations

Lean.expandMacro? stx = do let __do_lift ← Lean.Macro.getMethods __do_lift.expandMacro? stx

def Lean.Macro.hasDecl (declName : Name) :

Returns true if the environment contains a declaration with name declName

Equations

Lean.Macro.hasDecl declName = do let __do_lift ← Lean.Macro.getMethods __do_lift.hasDecl declName

def Lean.Macro.getCurrNamespace :

Gets the current namespace given the position in the file.

Equations

Lean.Macro.getCurrNamespace = do let __do_lift ← Lean.Macro.getMethods __do_lift.getCurrNamespace

def Lean.Macro.resolveNamespace (n : Name) :

MacroM (List Name)

Resolves the given name to an overload list of namespaces.

Equations

Lean.Macro.resolveNamespace n = do let __do_lift ← Lean.Macro.getMethods __do_lift.resolveNamespace n

def Lean.Macro.resolveGlobalName (n : Name) :

MacroM (List (Name × List String))

Resolves the given name to an overload list of global definitions. The List String in each alternative is the deduced list of projections (which are ambiguous with name components).

Remark: it will not trigger actions associated with reserved names. Recall that Lean has reserved names. For example, a definition foo has a reserved name foo.def for theorem containing stating that foo is equal to its definition. The action associated with foo.def automatically proves the theorem. At the macro level, the name is resolved, but the action is not executed. The actions are executed by the elaborator when converting Syntax into Expr.

Equations

Lean.Macro.resolveGlobalName n = do let __do_lift ← Lean.Macro.getMethods __do_lift.resolveGlobalName n

def Lean.Macro.trace (clsName : Name) (msg : String) :

Add a new trace message, with the given trace class and message.

Equations

Lean.Macro.trace clsName msg = modify fun (s : Lean.Macro.State) => { macroScope := s.macroScope, traceMsgs := (clsName, msg) :: s.traceMsgs }

@[reducible, inline]

abbrev Lean.PrettyPrinter.UnexpandM (α : Type) :

The unexpander monad, essentially Syntax → Option α. The Syntax is the ref, and it has the possibility of failure without an error message.

Equations

Lean.PrettyPrinter.UnexpandM = ReaderT Lean.Syntax (EStateM Unit Unit)

Instances For

Lean.PrettyPrinter.instMonadQuotationUnexpandM

@[reducible, inline]

abbrev Lean.PrettyPrinter.Unexpander :

Function that tries to reverse macro expansions as a post-processing step of delaboration. While less general than an arbitrary delaborator, it can be declared without importing Lean. Used by the [app_unexpander] attribute.

Equations

Lean.PrettyPrinter.Unexpander = (Lean.Syntax → Lean.PrettyPrinter.UnexpandM Lean.Syntax)

instance Lean.PrettyPrinter.instMonadQuotationUnexpandM :

MonadQuotation UnexpandM

Equations

One or more equations did not get rendered due to their size.