inductive Lean.Literal : Type

Literal values for Expr.

• natVal: Nat → Lean.Literal

  Natural number literal

• strVal: String → Lean.Literal

  String literal

instance Lean.instInhabitedLiteral : Inhabited Lean.Literal

Equations

• Lean.instInhabitedLiteral = { default := Lean.Literal.natVal default }

instance Lean.instBEqLiteral : BEq Lean.Literal

Equations

• Lean.instBEqLiteral = { beq := Lean.beqLiteral✝ }

instance Lean.instReprLiteral : Repr Lean.Literal

Equations

• Lean.instReprLiteral = { reprPrec := Lean.reprLiteral✝ }

def Lean.Literal.hash : Lean.Literal → UInt64

Equations

• Lean.Literal.hash x = match x with | Lean.Literal.natVal v => hash v | Lean.Literal.strVal v => hash v

instance Lean.instHashableLiteral : Hashable Lean.Literal

Equations

• Lean.instHashableLiteral = { hash := Lean.Literal.hash }

def Lean.Literal.lt : Lean.Literal → Lean.Literal → Bool

Total order on Expr literal values. Natural number values are smaller than string literal values.

Equations

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

instance Lean.instLTLiteral : LT Lean.Literal

Equations

• Lean.instLTLiteral = { lt := fun (a b : Lean.Literal) => Lean.Literal.lt a b = true }

instance Lean.instDecidableLtLiteralInstLTLiteral (a : Lean.Literal) (b : Lean.Literal) : Decidable (a < b)

Equations

• Lean.instDecidableLtLiteralInstLTLiteral a b = inferInstanceAs (Decidable (Lean.Literal.lt a b = true))

inductive Lean.BinderInfo : Type

Arguments in forallE binders can be labelled as implicit or explicit.

Each lam or forallE binder comes with a binderInfo argument (stored in ExprData). This can be set to

• default -- (x : α)
• implicit -- {x : α}
• strict_implicit -- ⦃x : α⦄
• inst_implicit -- [x : α].
• aux_decl -- Auxiliary definitions are helper methods that Lean generates. aux_decl is used for _match, _fun_match, _let_match and the self reference that appears in recursive pattern matching.

The difference between implicit {} and strict-implicit ⦃⦄ is how implicit arguments are treated that are not followed by explicit arguments. {} arguments are applied eagerly, while ⦃⦄ arguments are left partially applied:

def foo {x : Nat} : Nat := x
def bar ⦃x : Nat⦄ : Nat := x
#check foo -- foo : Nat
#check bar -- bar : ⦃x : Nat⦄ → Nat

See also the Lean manual.

• default: Lean.BinderInfo

  Default binder annotation, e.g. (x : α)

• implicit: Lean.BinderInfo

  Implicit binder annotation, e.g., {x : α}

• strictImplicit: Lean.BinderInfo

  Strict implicit binder annotation, e.g., {{ x : α }}

• instImplicit: Lean.BinderInfo

  Local instance binder annotataion, e.g., [Decidable α]

instance Lean.instInhabitedBinderInfo : Inhabited Lean.BinderInfo

Equations

• Lean.instInhabitedBinderInfo = { default := Lean.BinderInfo.default }

instance Lean.instBEqBinderInfo : BEq Lean.BinderInfo

Equations

• Lean.instBEqBinderInfo = { beq := Lean.beqBinderInfo✝ }

instance Lean.instReprBinderInfo : Repr Lean.BinderInfo

Equations

• Lean.instReprBinderInfo = { reprPrec := Lean.reprBinderInfo✝ }

def Lean.BinderInfo.hash : Lean.BinderInfo → UInt64

Equations

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

def Lean.BinderInfo.isExplicit : Lean.BinderInfo → Bool

Return true if the given BinderInfo does not correspond to an implicit binder annotation (i.e., implicit, strictImplicit, or instImplicit).

Equations

• Lean.BinderInfo.isExplicit x = match x with | Lean.BinderInfo.implicit => false | Lean.BinderInfo.strictImplicit => false | Lean.BinderInfo.instImplicit => false | x => true

instance Lean.instHashableBinderInfo : Hashable Lean.BinderInfo

Equations

• Lean.instHashableBinderInfo = { hash := Lean.BinderInfo.hash }

def Lean.BinderInfo.isInstImplicit : Lean.BinderInfo → Bool

Return true if the given BinderInfo is an instance implicit annotation (e.g., [Decidable α])

Equations

• Lean.BinderInfo.isInstImplicit x = match x with | Lean.BinderInfo.instImplicit => true | x => false

def Lean.BinderInfo.isImplicit : Lean.BinderInfo → Bool

Return true if the given BinderInfo is a regular implicit annotation (e.g., {α : Type u})

Equations

• Lean.BinderInfo.isImplicit x = match x with | Lean.BinderInfo.implicit => true | x => false

def Lean.BinderInfo.isStrictImplicit : Lean.BinderInfo → Bool

Return true if the given BinderInfo is a strict implicit annotation (e.g., {{α : Type u}})

Equations

• Lean.BinderInfo.isStrictImplicit x = match x with | Lean.BinderInfo.strictImplicit => true | x => false

@[inline, reducible]

abbrev Lean.MData : Type

Expression metadata. Used with the Expr.mdata constructor.

Equations

• Lean.MData = Lean.KVMap

@[inline, reducible]

abbrev Lean.MData.empty : Lean.MData

Equations

• Lean.MData.empty = { entries := [] }

def Lean.Expr.Data : Type

Cached hash code, cached results, and other data for Expr.

• hash : 32-bits
• approxDepth : 8-bits -- the approximate depth is used to minimize the number of hash collisions
• hasFVar : 1-bit -- does it contain free variables?
• hasExprMVar : 1-bit -- does it contain metavariables?
• hasLevelMVar : 1-bit -- does it contain level metavariables?
• hasLevelParam : 1-bit -- does it contain level parameters?
• looseBVarRange : 20-bits

Remark: this is mostly an internal datastructure used to implement Expr, most will never have to use it.

Equations

• Lean.Expr.Data = UInt64

instance Lean.instInhabitedData_1 : Inhabited Lean.Expr.Data

Equations

• Lean.instInhabitedData_1 = inferInstanceAs (Inhabited UInt64)

def Lean.Expr.Data.hash (c : Lean.Expr.Data) : UInt64

Equations

• Lean.Expr.Data.hash c = UInt32.toUInt64 (UInt64.toUInt32 c)

instance Lean.instBEqData_1 : BEq Lean.Expr.Data

Equations

• Lean.instBEqData_1 = { beq := fun (a b : UInt64) => a == b }

def Lean.Expr.Data.approxDepth (c : Lean.Expr.Data) : UInt8

Equations

• Lean.Expr.Data.approxDepth c = UInt64.toUInt8 (UInt64.land (UInt64.shiftRight c 32) 255)

def Lean.Expr.Data.looseBVarRange (c : Lean.Expr.Data) : UInt32

Equations

• Lean.Expr.Data.looseBVarRange c = UInt64.toUInt32 (UInt64.shiftRight c 44)

def Lean.Expr.Data.hasFVar (c : Lean.Expr.Data) : Bool

Equations

• Lean.Expr.Data.hasFVar c = (UInt64.land (UInt64.shiftRight c 40) 1 == 1)

def Lean.Expr.Data.hasExprMVar (c : Lean.Expr.Data) : Bool

Equations

• Lean.Expr.Data.hasExprMVar c = (UInt64.land (UInt64.shiftRight c 41) 1 == 1)

def Lean.Expr.Data.hasLevelMVar (c : Lean.Expr.Data) : Bool

Equations

• Lean.Expr.Data.hasLevelMVar c = (UInt64.land (UInt64.shiftRight c 42) 1 == 1)

def Lean.Expr.Data.hasLevelParam (c : Lean.Expr.Data) : Bool

Equations

• Lean.Expr.Data.hasLevelParam c = (UInt64.land (UInt64.shiftRight c 43) 1 == 1)

@[extern lean_uint8_to_uint64]

def Lean.BinderInfo.toUInt64 : Lean.BinderInfo → UInt64

Equations

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

def Lean.Expr.mkData (h : UInt64) (looseBVarRange : optParam Nat 0) (approxDepth : optParam UInt32 0) (hasFVar : optParam Bool false) (hasExprMVar : optParam Bool false) (hasLevelMVar : optParam Bool false) (hasLevelParam : optParam Bool false) : Lean.Expr.Data

Equations

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

@[inline]

def Lean.Expr.mkAppData (fData : Lean.Expr.Data) (aData : Lean.Expr.Data) : Lean.Expr.Data

Optimized version of Expr.mkData for applications.

Equations

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

@[inline]

def Lean.Expr.mkDataForBinder (h : UInt64) (looseBVarRange : Nat) (approxDepth : UInt32) (hasFVar : Bool) (hasExprMVar : Bool) (hasLevelMVar : Bool) (hasLevelParam : Bool) : Lean.Expr.Data

Equations

• Lean.Expr.mkDataForBinder h looseBVarRange approxDepth hasFVar hasExprMVar hasLevelMVar hasLevelParam = Lean.Expr.mkData h looseBVarRange approxDepth hasFVar hasExprMVar hasLevelMVar hasLevelParam

@[inline]

def Lean.Expr.mkDataForLet (h : UInt64) (looseBVarRange : Nat) (approxDepth : UInt32) (hasFVar : Bool) (hasExprMVar : Bool) (hasLevelMVar : Bool) (hasLevelParam : Bool) : Lean.Expr.Data

Equations

• Lean.Expr.mkDataForLet h looseBVarRange approxDepth hasFVar hasExprMVar hasLevelMVar hasLevelParam = Lean.Expr.mkData h looseBVarRange approxDepth hasFVar hasExprMVar hasLevelMVar hasLevelParam

instance Lean.instReprData_1 : Repr Lean.Expr.Data

Equations

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

structure Lean.FVarId : Type

The unique free variable identifier. It is just a hierarchical name, but we wrap it in FVarId to make sure they don't get mixed up with MVarId.

This is not the user-facing name for a free variable. This information is stored in the local context (LocalContext). The unique identifiers are generated using a NameGenerator.

• name : Lake.Name

instance Lean.instInhabitedFVarId : Inhabited Lean.FVarId

Equations

• Lean.instInhabitedFVarId = { default := { name := default } }

instance Lean.instBEqFVarId : BEq Lean.FVarId

Equations

• Lean.instBEqFVarId = { beq := Lean.beqFVarId✝ }

instance Lean.instHashableFVarId : Hashable Lean.FVarId

Equations

• Lean.instHashableFVarId = { hash := Lean.hashFVarId✝ }

instance Lean.instReprFVarId : Repr Lean.FVarId

Equations

• Lean.instReprFVarId = { reprPrec := fun (n : Lean.FVarId) (p : Nat) => reprPrec n.name p }

def Lean.FVarIdSet : Type

A set of unique free variable identifiers. This is a persistent data structure implemented using red-black trees.

Equations

• Lean.FVarIdSet = Lean.RBTree Lean.FVarId fun (x x_1 : Lean.FVarId) => Lean.Name.quickCmp x.name x_1.name

instance Lean.instForInFVarIdSetFVarId {m : Type u_1 → Type u_2} : ForIn m Lean.FVarIdSet Lean.FVarId

Equations

• Lean.instForInFVarIdSetFVarId = inferInstanceAs (ForIn m (Lean.RBTree Lean.FVarId fun (x x_1 : Lean.FVarId) => Lean.Name.quickCmp x.name x_1.name) Lean.FVarId)

def Lean.FVarIdSet.insert (s : Lean.FVarIdSet) (fvarId : Lean.FVarId) : Lean.FVarIdSet

Equations

• Lean.FVarIdSet.insert s fvarId = Lean.RBTree.insert s fvarId

def Lean.FVarIdHashSet : Type

A set of unique free variable identifiers implemented using hashtables. Hashtables are faster than red-black trees if they are used linearly. They are not persistent data-structures.

Equations

• Lean.FVarIdHashSet = Lean.HashSet Lean.FVarId

def Lean.FVarIdMap (α : Type) : Type

A mapping from free variable identifiers to values of type α. This is a persistent data structure implemented using red-black trees.

Equations

• Lean.FVarIdMap α = Lean.RBMap Lean.FVarId α fun (x x_1 : Lean.FVarId) => Lean.Name.quickCmp x.name x_1.name

def Lean.FVarIdMap.insert {α : Type} (s : Lean.FVarIdMap α) (fvarId : Lean.FVarId) (a : α) : Lean.FVarIdMap α

Equations

• Lean.FVarIdMap.insert s fvarId a = Lean.RBMap.insert s fvarId a

instance Lean.instEmptyCollectionFVarIdMap {α : Type} : EmptyCollection (Lean.FVarIdMap α)

Equations

• Lean.instEmptyCollectionFVarIdMap = inferInstanceAs (EmptyCollection (Lean.RBMap Lean.FVarId α fun (x x_1 : Lean.FVarId) => Lean.Name.quickCmp x.name x_1.name))

instance Lean.instInhabitedFVarIdMap {α : Type} : Inhabited (Lean.FVarIdMap α)

Equations

• Lean.instInhabitedFVarIdMap = { default := ∅ }

structure Lean.MVarId : Type

Universe metavariable Id

• name : Lake.Name

instance Lean.instInhabitedMVarId : Inhabited Lean.MVarId

Equations

• Lean.instInhabitedMVarId = { default := { name := default } }

instance Lean.instBEqMVarId : BEq Lean.MVarId

Equations

• Lean.instBEqMVarId = { beq := Lean.beqMVarId✝ }

instance Lean.instHashableMVarId : Hashable Lean.MVarId

Equations

• Lean.instHashableMVarId = { hash := Lean.hashMVarId✝ }

instance Lean.instReprMVarId : Repr Lean.MVarId

Equations

• Lean.instReprMVarId = { reprPrec := Lean.reprMVarId✝ }

instance Lean.instReprMVarId_1 : Repr Lean.MVarId

Equations

• Lean.instReprMVarId_1 = { reprPrec := fun (n : Lean.MVarId) (p : Nat) => reprPrec n.name p }

def Lean.MVarIdSet : Type

Equations

• Lean.MVarIdSet = Lean.RBTree Lean.MVarId fun (x x_1 : Lean.MVarId) => Lean.Name.quickCmp x.name x_1.name

def Lean.MVarIdSet.insert (s : Lean.MVarIdSet) (mvarId : Lean.MVarId) : Lean.MVarIdSet

Equations

• Lean.MVarIdSet.insert s mvarId = Lean.RBTree.insert s mvarId

instance Lean.instForInMVarIdSetMVarId {m : Type u_1 → Type u_2} : ForIn m Lean.MVarIdSet Lean.MVarId

Equations

• Lean.instForInMVarIdSetMVarId = inferInstanceAs (ForIn m (Lean.RBTree Lean.MVarId fun (x x_1 : Lean.MVarId) => Lean.Name.quickCmp x.name x_1.name) Lean.MVarId)

def Lean.MVarIdMap (α : Type) : Type

Equations

• Lean.MVarIdMap α = Lean.RBMap Lean.MVarId α fun (x x_1 : Lean.MVarId) => Lean.Name.quickCmp x.name x_1.name

def Lean.MVarIdMap.insert {α : Type} (s : Lean.MVarIdMap α) (mvarId : Lean.MVarId) (a : α) : Lean.MVarIdMap α

Equations

• Lean.MVarIdMap.insert s mvarId a = Lean.RBMap.insert s mvarId a

instance Lean.instEmptyCollectionMVarIdMap {α : Type} : EmptyCollection (Lean.MVarIdMap α)

Equations

• Lean.instEmptyCollectionMVarIdMap = inferInstanceAs (EmptyCollection (Lean.RBMap Lean.MVarId α fun (x x_1 : Lean.MVarId) => Lean.Name.quickCmp x.name x_1.name))

instance Lean.instForInMVarIdMapProdMVarId {m : Type u_1 → Type u_2} {α : Type} : ForIn m (Lean.MVarIdMap α) (Lean.MVarId × α)

Equations

• Lean.instForInMVarIdMapProdMVarId = inferInstanceAs (ForIn m (Lean.RBMap Lean.MVarId α fun (x x_1 : Lean.MVarId) => Lean.Name.quickCmp x.name x_1.name) (Lean.MVarId × α))

instance Lean.instInhabitedMVarIdMap {α : Type} : Inhabited (Lean.MVarIdMap α)

Equations

• Lean.instInhabitedMVarIdMap = { default := ∅ }

@[extern lean_expr_data]

noncomputable def Lean.Expr.data : Lean.Expr → Lean.Expr.Data

inductive Lean.Expr : Type

Lean expressions. This data structure is used in the kernel and elaborator. However, expressions sent to the kernel should not contain metavariables.

Remark: we use the E suffix (short for Expr) to avoid collision with keywords. We considered using «...», but it is too inconvenient to use.

• bvar: Nat → Lean.Expr

  The bvar constructor represents bound variables, i.e. occurrences of a variable in the expression where there is a variable binder above it (i.e. introduced by a lam, forallE, or letE).

  The deBruijnIndex parameter is the de-Bruijn index for the bound variable. See the Wikipedia page on de-Bruijn indices for additional information.

  For example, consider the expression fun x : Nat => forall y : Nat, x = y. The x and y variables in the equality expression are constructed using bvar and bound to the binders introduced by the earlier lam and forallE constructors. Here is the corresponding Expr representation for the same expression:

  .lam `x (.const `Nat [])
    (.forallE `y (.const `Nat [])
      (.app (.app (.app (.const `Eq [.succ .zero]) (.const `Nat [])) (.bvar 1)) (.bvar 0))
      .default)
    .default
  

• fvar: Lean.FVarId → Lean.Expr

  The fvar constructor represent free variables. These free variable occurrences are not bound by an earlier lam, forallE, or letE constructor and its binder exists in a local context only.

  Note that Lean uses the locally nameless approach. See McBride and McKinna for additional details.

  When "visiting" the body of a binding expression (i.e. lam, forallE, or letE), bound variables are converted into free variables using a unique identifier, and their user-facing name, type, value (for LetE), and binder annotation are stored in the LocalContext.

• mvar: Lean.MVarId → Lean.Expr

  Metavariables are used to represent "holes" in expressions, and goals in the tactic framework. Metavariable declarations are stored in the MetavarContext. Metavariables are used during elaboration, and are not allowed in the kernel, or in the code generator.

• sort: Lean.Level → Lean.Expr

  Used for Type u, Sort u, and Prop:

  • Prop is represented as .sort .zero,
  • Sort u as .sort (.param `u), and
  • Type u as .sort (.succ (.param `u))
• const: Lake.Name → List Lean.Level → Lean.Expr

  A (universe polymorphic) constant that has been defined earlier in the module or by another imported module. For example, @Eq.{1} is represented as Expr.const `Eq [.succ .zero], and @Array.map.{0, 0} is represented as Expr.const `Array.map [.zero, .zero].

• app: Lean.Expr → Lean.Expr → Lean.Expr

  A function application.

  For example, the natural number one, i.e. Nat.succ Nat.zero is represented as Expr.app (.const `Nat.succ []) (.const .zero []). Note that multiple arguments are represented using partial application.

  For example, the two argument application f x y is represented as Expr.app (.app f x) y.

• lam: Lake.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → Lean.Expr

  A lambda abstraction (aka anonymous functions). It introduces a new binder for variable x in scope for the lambda body.

  For example, the expression fun x : Nat => x is represented as

  Expr.lam `x (.const `Nat []) (.bvar 0) .default
  

• forallE: Lake.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → Lean.Expr

  A dependent arrow (a : α) → β) (aka forall-expression) where β may dependent on a. Note that this constructor is also used to represent non-dependent arrows where β does not depend on a.

  For example:

  • forall x : Prop, x ∧ x:

    Expr.forallE `x (.sort .zero)
      (.app (.app (.const `And []) (.bvar 0)) (.bvar 0)) .default
    

  • Nat → Bool:

    Expr.forallE `a (.const `Nat [])
      (.const `Bool []) .default
    

• letE: Lake.Name → Lean.Expr → Lean.Expr → Lean.Expr → Bool → Lean.Expr

  Let-expressions.

  IMPORTANT: The nonDep flag is for "local" use only. That is, a module should not "trust" its value for any purpose. In the intended use-case, the compiler will set this flag, and be responsible for maintaining it. Other modules may not preserve its value while applying transformations.

  Given an environment, a metavariable context, and a local context, we say a let-expression let x : t := v; e is non-dependent when it is equivalent to (fun x : t => e) v. In contrast, the dependent let-expression let n : Nat := 2; fun (a : Array Nat n) (b : Array Nat 2) => a = b is type correct, but (fun (n : Nat) (a : Array Nat n) (b : Array Nat 2) => a = b) 2 is not.

  The let-expression let x : Nat := 2; Nat.succ x is represented as

  Expr.letE `x (.const `Nat []) (.lit (.natVal 2)) (.app (.const `Nat.succ []) (.bvar 0)) true
  

• lit: Lean.Literal → Lean.Expr

  Natural number and string literal values.

  They are not really needed, but provide a more compact representation in memory for these two kinds of literals, and are used to implement efficient reduction in the elaborator and kernel. The "raw" natural number 2 can be represented as Expr.lit (.natVal 2). Note that, it is definitionally equal to:

  Expr.app (.const `Nat.succ []) (.app (.const `Nat.succ []) (.const `Nat.zero []))
  

• mdata: Lean.MData → Lean.Expr → Lean.Expr

  Metadata (aka annotations).

  We use annotations to provide hints to the pretty-printer, store references to Syntax nodes, position information, and save information for elaboration procedures (e.g., we use the inaccessible annotation during elaboration to mark Exprs that correspond to inaccessible patterns).

  Note that Expr.mdata data e is definitionally equal to e.

• proj: Lake.Name → Nat → Lean.Expr → Lean.Expr

  Projection-expressions. They are redundant, but are used to create more compact terms, speedup reduction, and implement eta for structures. The type of struct must be an structure-like inductive type. That is, it has only one constructor, is not recursive, and it is not an inductive predicate. The kernel and elaborators check whether the typeName matches the type of struct, and whether the (zero-based) index is valid (i.e., it is smaller than the number of constructor fields). When exporting Lean developments to other systems, proj can be replaced with typeName.rec applications.

  Example, given a : Nat × Bool, a.1 is represented as

  .proj `Prod 0 a
  

instance Lean.instInhabitedExpr : Inhabited Lean.Expr

Equations

• Lean.instInhabitedExpr = { default := Lean.Expr.bvar default }

instance Lean.instReprExpr : Repr Lean.Expr

Equations

• Lean.instReprExpr = { reprPrec := Lean.reprExpr✝ }

def Lean.Expr.ctorName : Lean.Expr → String

The constructor name for the given expression. This is used for debugging purposes.

Equations

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

def Lean.Expr.hash (e : Lean.Expr) : UInt64

Equations

• Lean.Expr.hash e = Lean.Expr.Data.hash (Lean.Expr.data e)

instance Lean.Expr.instHashableExpr : Hashable Lean.Expr

Equations

• Lean.Expr.instHashableExpr = { hash := Lean.Expr.hash }

def Lean.Expr.hasFVar (e : Lean.Expr) : Bool

Return true if e contains free variables. This is a constant time operation.

Equations

• Lean.Expr.hasFVar e = Lean.Expr.Data.hasFVar (Lean.Expr.data e)

def Lean.Expr.hasExprMVar (e : Lean.Expr) : Bool

Return true if e contains expression metavariables. This is a constant time operation.

Equations

• Lean.Expr.hasExprMVar e = Lean.Expr.Data.hasExprMVar (Lean.Expr.data e)

def Lean.Expr.hasLevelMVar (e : Lean.Expr) : Bool

Return true if e contains universe (aka Level) metavariables. This is a constant time operation.

Equations

• Lean.Expr.hasLevelMVar e = Lean.Expr.Data.hasLevelMVar (Lean.Expr.data e)

def Lean.Expr.hasMVar (e : Lean.Expr) : Bool

Does the expression contain level (aka universe) or expression metavariables? This is a constant time operation.

Equations

• Lean.Expr.hasMVar e = let d := Lean.Expr.data e; Lean.Expr.Data.hasExprMVar d || Lean.Expr.Data.hasLevelMVar d

def Lean.Expr.hasLevelParam (e : Lean.Expr) : Bool

Return true if e contains universe level parameters. This is a constant time operation.

Equations

• Lean.Expr.hasLevelParam e = Lean.Expr.Data.hasLevelParam (Lean.Expr.data e)

def Lean.Expr.approxDepth (e : Lean.Expr) : UInt32

Return the approximated depth of an expression. This information is used to compute the expression hash code, and speedup comparisons. This is a constant time operation. We say it is approximate because it maxes out at 255.

Equations

• Lean.Expr.approxDepth e = UInt8.toUInt32 (Lean.Expr.Data.approxDepth (Lean.Expr.data e))

def Lean.Expr.looseBVarRange (e : Lean.Expr) : Nat

The range of de-Bruijn variables that are loose. That is, bvars that are not bound by a binder. For example, bvar i has range i + 1 and an expression with no loose bvars has range 0.

Equations

• Lean.Expr.looseBVarRange e = UInt32.toNat (Lean.Expr.Data.looseBVarRange (Lean.Expr.data e))

def Lean.Expr.binderInfo (e : Lean.Expr) : Lean.BinderInfo

Return the binder information if e is a lambda or forall expression, and .default otherwise.

Equations

• Lean.Expr.binderInfo e = match e with | Lean.Expr.forallE binderName binderType body bi => bi | Lean.Expr.lam binderName binderType body bi => bi | x => Lean.BinderInfo.default

Export functions.

@[export lean_expr_hash]

def Lean.Expr.hashEx : Lean.Expr → UInt64

Equations

• Lean.Expr.hashEx = hash

@[export lean_expr_has_fvar]

def Lean.Expr.hasFVarEx : Lean.Expr → Bool

Equations

• Lean.Expr.hasFVarEx = Lean.Expr.hasFVar

@[export lean_expr_has_expr_mvar]

def Lean.Expr.hasExprMVarEx : Lean.Expr → Bool

Equations

• Lean.Expr.hasExprMVarEx = Lean.Expr.hasExprMVar

@[export lean_expr_has_level_mvar]

def Lean.Expr.hasLevelMVarEx : Lean.Expr → Bool

Equations

• Lean.Expr.hasLevelMVarEx = Lean.Expr.hasLevelMVar

@[export lean_expr_has_mvar]

def Lean.Expr.hasMVarEx : Lean.Expr → Bool

Equations

• Lean.Expr.hasMVarEx = Lean.Expr.hasMVar

@[export lean_expr_has_level_param]

def Lean.Expr.hasLevelParamEx : Lean.Expr → Bool

Equations

• Lean.Expr.hasLevelParamEx = Lean.Expr.hasLevelParam

@[export lean_expr_loose_bvar_range]

def Lean.Expr.looseBVarRangeEx (e : Lean.Expr) : UInt32

Equations

• Lean.Expr.looseBVarRangeEx e = Lean.Expr.Data.looseBVarRange (Lean.Expr.data e)

@[export lean_expr_binder_info]

def Lean.Expr.binderInfoEx : Lean.Expr → Lean.BinderInfo

Equations

• Lean.Expr.binderInfoEx = Lean.Expr.binderInfo

def Lean.mkConst (declName : Lake.Name) (us : optParam (List Lean.Level) []) : Lean.Expr

mkConst declName us return .const declName us.

Equations

• Lean.mkConst declName us = Lean.Expr.const declName us

def Lean.Literal.type : Lean.Literal → Lean.Expr

Return the type of a literal value.

Equations

• Lean.Literal.type x = match x with | Lean.Literal.natVal val => Lean.mkConst `Nat | Lean.Literal.strVal val => Lean.mkConst `String

@[export lean_lit_type]

def Lean.Literal.typeEx : Lean.Literal → Lean.Expr

Equations

• Lean.Literal.typeEx = Lean.Literal.type

def Lean.mkBVar (idx : Nat) : Lean.Expr

.bvar idx is now the preferred form.

Equations

• Lean.mkBVar idx = Lean.Expr.bvar idx

def Lean.mkSort (u : Lean.Level) : Lean.Expr

.sort u is now the preferred form.

Equations

• Lean.mkSort u = Lean.Expr.sort u

def Lean.mkFVar (fvarId : Lean.FVarId) : Lean.Expr

.fvar fvarId is now the preferred form. This function is seldom used, free variables are often automatically created using the telescope functions (e.g., forallTelescope and lambdaTelescope) at MetaM.

Equations

• Lean.mkFVar fvarId = Lean.Expr.fvar fvarId

def Lean.mkMVar (mvarId : Lean.MVarId) : Lean.Expr

.mvar mvarId is now the preferred form. This function is seldom used, metavariables are often created using functions such as mkFresheExprMVar at MetaM.

Equations

• Lean.mkMVar mvarId = Lean.Expr.mvar mvarId

def Lean.mkMData (m : Lean.MData) (e : Lean.Expr) : Lean.Expr

.mdata m e is now the preferred form.

Equations

• Lean.mkMData m e = Lean.Expr.mdata m e

def Lean.mkProj (structName : Lake.Name) (idx : Nat) (struct : Lean.Expr) : Lean.Expr

.proj structName idx struct is now the preferred form.

Equations

• Lean.mkProj structName idx struct = Lean.Expr.proj structName idx struct

@[match_pattern]

def Lean.mkApp (f : Lean.Expr) (a : Lean.Expr) : Lean.Expr

.app f a is now the preferred form.

Equations

• Lean.mkApp f a = Lean.Expr.app f a

def Lean.mkLambda (x : Lake.Name) (bi : Lean.BinderInfo) (t : Lean.Expr) (b : Lean.Expr) : Lean.Expr

.lam x t b bi is now the preferred form.

Equations

• Lean.mkLambda x bi t b = Lean.Expr.lam x t b bi

def Lean.mkForall (x : Lake.Name) (bi : Lean.BinderInfo) (t : Lean.Expr) (b : Lean.Expr) : Lean.Expr

.forallE x t b bi is now the preferred form.

Equations

• Lean.mkForall x bi t b = Lean.Expr.forallE x t b bi

def Lean.mkSimpleThunkType (type : Lean.Expr) : Lean.Expr

Return Unit -> type. Do not confuse with Thunk type

Equations

• Lean.mkSimpleThunkType type = Lean.mkForall Lean.Name.anonymous Lean.BinderInfo.default (Lean.mkConst `Unit) type

def Lean.mkSimpleThunk (type : Lean.Expr) : Lean.Expr

Return fun (_ : Unit), e

Equations

• Lean.mkSimpleThunk type = Lean.mkLambda `_ Lean.BinderInfo.default (Lean.mkConst `Unit) type

def Lean.mkLet (x : Lake.Name) (t : Lean.Expr) (v : Lean.Expr) (b : Lean.Expr) (nonDep : optParam Bool false) : Lean.Expr

.letE x t v b nonDep is now the preferred form.

Equations

• Lean.mkLet x t v b nonDep = Lean.Expr.letE x t v b nonDep

@[match_pattern]

def Lean.mkAppB (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) : Lean.Expr

Equations

• Lean.mkAppB f a b = Lean.mkApp (Lean.mkApp f a) b

@[match_pattern]

def Lean.mkApp2 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp2 f a b = Lean.mkAppB f a b

@[match_pattern]

def Lean.mkApp3 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp3 f a b c = Lean.mkApp (Lean.mkAppB f a b) c

@[match_pattern]

def Lean.mkApp4 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp4 f a b c d = Lean.mkAppB (Lean.mkAppB f a b) c d

@[match_pattern]

def Lean.mkApp5 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp5 f a b c d e = Lean.mkApp (Lean.mkApp4 f a b c d) e

@[match_pattern]

def Lean.mkApp6 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e₁ : Lean.Expr) (e₂ : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp6 f a b c d e₁ e₂ = Lean.mkAppB (Lean.mkApp4 f a b c d) e₁ e₂

@[match_pattern]

def Lean.mkApp7 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e₁ : Lean.Expr) (e₂ : Lean.Expr) (e₃ : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp7 f a b c d e₁ e₂ e₃ = Lean.mkApp3 (Lean.mkApp4 f a b c d) e₁ e₂ e₃

@[match_pattern]

def Lean.mkApp8 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e₁ : Lean.Expr) (e₂ : Lean.Expr) (e₃ : Lean.Expr) (e₄ : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp8 f a b c d e₁ e₂ e₃ e₄ = Lean.mkApp4 (Lean.mkApp4 f a b c d) e₁ e₂ e₃ e₄

@[match_pattern]

def Lean.mkApp9 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e₁ : Lean.Expr) (e₂ : Lean.Expr) (e₃ : Lean.Expr) (e₄ : Lean.Expr) (e₅ : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp9 f a b c d e₁ e₂ e₃ e₄ e₅ = Lean.mkApp5 (Lean.mkApp4 f a b c d) e₁ e₂ e₃ e₄ e₅

@[match_pattern]

def Lean.mkApp10 (f : Lean.Expr) (a : Lean.Expr) (b : Lean.Expr) (c : Lean.Expr) (d : Lean.Expr) (e₁ : Lean.Expr) (e₂ : Lean.Expr) (e₃ : Lean.Expr) (e₄ : Lean.Expr) (e₅ : Lean.Expr) (e₆ : Lean.Expr) : Lean.Expr

Equations

• Lean.mkApp10 f a b c d e₁ e₂ e₃ e₄ e₅ e₆ = Lean.mkApp6 (Lean.mkApp4 f a b c d) e₁ e₂ e₃ e₄ e₅ e₆

def Lean.mkLit (l : Lean.Literal) : Lean.Expr

.lit l is now the preferred form.

Equations

• Lean.mkLit l = Lean.Expr.lit l

def Lean.mkRawNatLit (n : Nat) : Lean.Expr

Return the "raw" natural number .lit (.natVal n). This is not the default representation used by the Lean frontend. See mkNatLit.

Equations

• Lean.mkRawNatLit n = Lean.mkLit (Lean.Literal.natVal n)

def Lean.mkNatLit (n : Nat) : Lean.Expr

Return a natural number literal used in the frontend. It is a OfNat.ofNat application. Recall that all theorems and definitions containing numeric literals are encoded using OfNat.ofNat applications in the frontend.

Equations

• Lean.mkNatLit n = let r := Lean.mkRawNatLit n; Lean.mkApp3 (Lean.mkConst `OfNat.ofNat [Lean.levelZero]) (Lean.mkConst `Nat) r (Lean.mkApp (Lean.mkConst `instOfNatNat) r)

def Lean.mkStrLit (s : String) : Lean.Expr

Return the string literal .lit (.strVal s)

Equations

• Lean.mkStrLit s = Lean.mkLit (Lean.Literal.strVal s)

@[export lean_expr_mk_bvar]

def Lean.mkBVarEx : Nat → Lean.Expr

Equations

• Lean.mkBVarEx = Lean.mkBVar

@[export lean_expr_mk_fvar]

def Lean.mkFVarEx : Lean.FVarId → Lean.Expr

Equations

• Lean.mkFVarEx = Lean.mkFVar

@[export lean_expr_mk_mvar]

def Lean.mkMVarEx : Lean.MVarId → Lean.Expr

Equations

• Lean.mkMVarEx = Lean.mkMVar

@[export lean_expr_mk_sort]

def Lean.mkSortEx : Lean.Level → Lean.Expr

Equations

• Lean.mkSortEx = Lean.mkSort

@[export lean_expr_mk_const]

def Lean.mkConstEx (c : Lake.Name) (lvls : List Lean.Level) : Lean.Expr

Equations

• Lean.mkConstEx c lvls = Lean.mkConst c lvls

@[export lean_expr_mk_app]

def Lean.mkAppEx : Lean.Expr → Lean.Expr → Lean.Expr

Equations

• Lean.mkAppEx = Lean.mkApp

@[export lean_expr_mk_lambda]

def Lean.mkLambdaEx (n : Lake.Name) (d : Lean.Expr) (b : Lean.Expr) (bi : Lean.BinderInfo) : Lean.Expr

Equations

• Lean.mkLambdaEx n d b bi = Lean.mkLambda n bi d b

@[export lean_expr_mk_forall]

def Lean.mkForallEx (n : Lake.Name) (d : Lean.Expr) (b : Lean.Expr) (bi : Lean.BinderInfo) : Lean.Expr

Equations

• Lean.mkForallEx n d b bi = Lean.mkForall n bi d b

@[export lean_expr_mk_let]

def Lean.mkLetEx (n : Lake.Name) (t : Lean.Expr) (v : Lean.Expr) (b : Lean.Expr) : Lean.Expr

Equations

• Lean.mkLetEx n t v b = Lean.mkLet n t v b

@[export lean_expr_mk_lit]

def Lean.mkLitEx : Lean.Literal → Lean.Expr

Equations

• Lean.mkLitEx = Lean.mkLit

@[export lean_expr_mk_mdata]

def Lean.mkMDataEx : Lean.MData → Lean.Expr → Lean.Expr

Equations

• Lean.mkMDataEx = Lean.mkMData

@[export lean_expr_mk_proj]

def Lean.mkProjEx : Lake.Name → Nat → Lean.Expr → Lean.Expr

Equations

• Lean.mkProjEx = Lean.mkProj

def Lean.mkAppN (f : Lean.Expr) (args : Array Lean.Expr) : Lean.Expr

mkAppN f #[a₀, ..., aₙ] constructs the application f a₀ a₁ ... aₙ.

Equations

• Lean.mkAppN f args = Array.foldl Lean.mkApp f args 0

def Lean.mkAppRange (f : Lean.Expr) (i : Nat) (j : Nat) (args : Array Lean.Expr) : Lean.Expr

mkAppRange f i j #[a_1, ..., a_i, ..., a_j, ... ] ==> the expression f a_i ... a_{j-1}

Equations

• Lean.mkAppRange f i j args = Lean.mkAppRangeAux j args i f

def Lean.mkAppRev (fn : Lean.Expr) (revArgs : Array Lean.Expr) : Lean.Expr

Same as mkApp f args but reversing args.

Equations

• Lean.mkAppRev fn revArgs = Array.foldr (fun (a r : Lean.Expr) => Lean.mkApp r a) fn revArgs (Array.size revArgs)

@[extern lean_expr_dbg_to_string]

opaque Lean.Expr.dbgToString (e : Lean.Expr) : String

@[extern lean_expr_quick_lt]

opaque Lean.Expr.quickLt (a : Lean.Expr) (b : Lean.Expr) : Bool

A total order for expressions. We say it is quick because it first compares the hashcodes.

@[extern lean_expr_lt]

opaque Lean.Expr.lt (a : Lean.Expr) (b : Lean.Expr) : Bool

A total order for expressions that takes the structure into account (e.g., variable names).

@[extern lean_expr_eqv]

opaque Lean.Expr.eqv (a : Lean.Expr) (b : Lean.Expr) : Bool

Return true iff a and b are alpha equivalent. Binder annotations are ignored.

instance Lean.Expr.instBEqExpr : BEq Lean.Expr

Equations

• Lean.Expr.instBEqExpr = { beq := Lean.Expr.eqv }

@[extern lean_expr_equal]

opaque Lean.Expr.equal (a : Lean.Expr) (b : Lean.Expr) : Bool

Return true iff a and b are equal. Binder names and annotations are taken into account.

def Lean.Expr.isSort : Lean.Expr → Bool

Return true if the given expression is a .sort ..

Equations

• Lean.Expr.isSort x = match x with | Lean.Expr.sort u => true | x => false

def Lean.Expr.isType : Lean.Expr → Bool

Return true if the given expression is of the form .sort (.succ ..).

Equations

• Lean.Expr.isType x = match x with | Lean.Expr.sort (Lean.Level.succ a) => true | x => false

def Lean.Expr.isType0 : Lean.Expr → Bool

Return true if the given expression is of the form .sort (.succ .zero).

Equations

• Lean.Expr.isType0 x = match x with | Lean.Expr.sort (Lean.Level.succ Lean.Level.zero) => true | x => false

def Lean.Expr.isProp : Lean.Expr → Bool

Return true if the given expression is a .sort .zero

Equations

• Lean.Expr.isProp x = match x with | Lean.Expr.sort Lean.Level.zero => true | x => false

def Lean.Expr.isBVar : Lean.Expr → Bool

Return true if the given expression is a bound variable.

Equations

• Lean.Expr.isBVar x = match x with | Lean.Expr.bvar deBruijnIndex => true | x => false

def Lean.Expr.isMVar : Lean.Expr → Bool

Return true if the given expression is a metavariable.

Equations

• Lean.Expr.isMVar x = match x with | Lean.Expr.mvar mvarId => true | x => false

def Lean.Expr.isFVar : Lean.Expr → Bool

Return true if the given expression is a free variable.

Equations

• Lean.Expr.isFVar x = match x with | Lean.Expr.fvar fvarId => true | x => false

def Lean.Expr.isApp : Lean.Expr → Bool

Return true if the given expression is an application.

Equations

• Lean.Expr.isApp x = match x with | Lean.Expr.app fn arg => true | x => false

def Lean.Expr.isProj : Lean.Expr → Bool

Return true if the given expression is a projection .proj ..

Equations

• Lean.Expr.isProj x = match x with | Lean.Expr.proj typeName idx struct => true | x => false

def Lean.Expr.isConst : Lean.Expr → Bool

Return true if the given expression is a constant.

Equations

• Lean.Expr.isConst x = match x with | Lean.Expr.const declName us => true | x => false

def Lean.Expr.isConstOf : Lean.Expr → Lake.Name → Bool

Return true if the given expression is a constant of the given name. Examples:

• (.const `Nat []).isConstOf `Nat is true
• (.const `Nat []).isConstOf `False is false

Equations

• Lean.Expr.isConstOf x✝ x = match x✝, x with | Lean.Expr.const n us, m => n == m | x, x_1 => false

def Lean.Expr.isFVarOf : Lean.Expr → Lean.FVarId → Bool

Return true if the given expression is a free variable with the given id. Examples:

• isFVarOf (.fvar id) id is true
• isFVarOf (.fvar id) id' is false
• isFVarOf (.sort levelZero) id is false

Equations

• Lean.Expr.isFVarOf x✝ x = match x✝, x with | Lean.Expr.fvar fvarId, fvarId' => fvarId == fvarId' | x, x_1 => false

def Lean.Expr.isForall : Lean.Expr → Bool

Return true if the given expression is a forall-expression aka (dependent) arrow.

Equations

• Lean.Expr.isForall x = match x with | Lean.Expr.forallE binderName binderType body binderInfo => true | x => false

def Lean.Expr.isLambda : Lean.Expr → Bool

Return true if the given expression is a lambda abstraction aka anonymous function.

Equations

• Lean.Expr.isLambda x = match x with | Lean.Expr.lam binderName binderType body binderInfo => true | x => false

def Lean.Expr.isBinding : Lean.Expr → Bool

Return true if the given expression is a forall or lambda expression.

Equations

• Lean.Expr.isBinding x = match x with | Lean.Expr.lam binderName binderType body binderInfo => true | Lean.Expr.forallE binderName binderType body binderInfo => true | x => false

def Lean.Expr.isLet : Lean.Expr → Bool

Return true if the given expression is a let-expression.

Equations

• Lean.Expr.isLet x = match x with | Lean.Expr.letE declName type value body nonDep => true | x => false

def Lean.Expr.isMData : Lean.Expr → Bool

Return true if the given expression is a metadata.

Equations

• Lean.Expr.isMData x = match x with | Lean.Expr.mdata data expr => true | x => false

def Lean.Expr.isLit : Lean.Expr → Bool

Return true if the given expression is a literal value.

Equations

• Lean.Expr.isLit x = match x with | Lean.Expr.lit a => true | x => false

def Lean.Expr.getForallBody : Lean.Expr → Lean.Expr

Return the "body" of a forall expression. Example: let e be the representation for forall (p : Prop) (q : Prop), p ∧ q, then getForallBody e returns .app (.app (.const `And []) (.bvar 1)) (.bvar 0)

Equations

• Lean.Expr.getForallBody (Lean.Expr.forallE binderName binderType body binderInfo) = Lean.Expr.getForallBody body
• Lean.Expr.getForallBody x = x

def Lean.Expr.getForallBodyMaxDepth (maxDepth : Nat) : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.getForallBodyMaxDepth (Nat.succ n) (Lean.Expr.forallE binderName binderType b binderInfo) = Lean.Expr.getForallBodyMaxDepth n b
• Lean.Expr.getForallBodyMaxDepth 0 x = x
• Lean.Expr.getForallBodyMaxDepth x✝ x = x

def Lean.Expr.getForallBinderNames : Lean.Expr → List Lake.Name

Given a sequence of nested foralls (a₁ : α₁) → ... → (aₙ : αₙ) → _, returns the names [a₁, ... aₙ].

Equations

• Lean.Expr.getForallBinderNames (Lean.Expr.forallE binderName binderType body binderInfo) = binderName :: Lean.Expr.getForallBinderNames body
• Lean.Expr.getForallBinderNames x = []

def Lean.Expr.getAppFn : Lean.Expr → Lean.Expr

If the given expression is a sequence of function applications f a₁ .. aₙ, return f. Otherwise return the input expression.

Equations

• Lean.Expr.getAppFn (Lean.Expr.app fn arg) = Lean.Expr.getAppFn fn
• Lean.Expr.getAppFn x = x

def Lean.Expr.getAppNumArgs (e : Lean.Expr) : Nat

Counts the number n of arguments for an expression f a₁ .. aₙ.

Equations

• Lean.Expr.getAppNumArgs e = Lean.Expr.getAppNumArgsAux e 0

def Lean.Expr.getBoundedAppFn (maxArgs : Nat) : Lean.Expr → Lean.Expr

Like Lean.Expr.getAppFn but assumes the application has up to maxArgs arguments. If there are any more arguments than this, then they are returned by getAppFn as part of the function.

In particular, if the given expression is a sequence of function applications f a₁ .. aₙ, returns f a₁ .. aₖ where k is minimal such that n - k ≤ maxArgs.

Equations

• Lean.Expr.getBoundedAppFn (Nat.succ maxArgs') (Lean.Expr.app f arg) = Lean.Expr.getBoundedAppFn maxArgs' f
• Lean.Expr.getBoundedAppFn x✝ x = x

@[inline]

def Lean.Expr.getAppArgs (e : Lean.Expr) : Array Lean.Expr

Given f a₁ a₂ ... aₙ, returns #[a₁, ..., aₙ]

Equations

• Lean.Expr.getAppArgs e = let dummy := Lean.mkSort Lean.levelZero; let nargs := Lean.Expr.getAppNumArgs e; Lean.Expr.getAppArgsAux e (mkArray nargs dummy) (nargs - 1)

@[inline]

def Lean.Expr.getBoundedAppArgs (maxArgs : Nat) (e : Lean.Expr) : Array Lean.Expr

Like Lean.Expr.getAppArgs but returns up to maxArgs arguments.

In particular, given f a₁ a₂ ... aₙ, returns #[aₖ₊₁, ..., aₙ] where k is minimal such that the size of this array is at most maxArgs.

Equations

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

@[inline]

def Lean.Expr.getAppRevArgs (e : Lean.Expr) : Array Lean.Expr

Same as getAppArgs but reverse the output array.

Equations

• Lean.Expr.getAppRevArgs e = Lean.Expr.getAppRevArgsAux e (Array.mkEmpty (Lean.Expr.getAppNumArgs e))

@[specialize #[]]

def Lean.Expr.withAppAux {α : Sort u_1} (k : Lean.Expr → Array Lean.Expr → α) : Lean.Expr → Array Lean.Expr → Nat → α

Equations

• Lean.Expr.withAppAux k (Lean.Expr.app f a) x✝ x = Lean.Expr.withAppAux k f (Array.set! x✝ x a) (x - 1)
• Lean.Expr.withAppAux k x✝¹ x✝ x = k x✝¹ x✝

@[inline]

def Lean.Expr.withApp {α : Sort u_1} (e : Lean.Expr) (k : Lean.Expr → Array Lean.Expr → α) : α

Given e = f a₁ a₂ ... aₙ, returns k f #[a₁, ..., aₙ].

Equations

• Lean.Expr.withApp e k = let dummy := Lean.mkSort Lean.levelZero; let nargs := Lean.Expr.getAppNumArgs e; Lean.Expr.withAppAux k e (mkArray nargs dummy) (nargs - 1)

@[inline]

def Lean.Expr.getAppArgsN (e : Lean.Expr) (n : Nat) : Array Lean.Expr

Given f a_1 ... a_n, returns #[a_1, ..., a_n]. Note that f may be an application. The resulting array has size n even if f.getAppNumArgs < n.

Equations

• Lean.Expr.getAppArgsN e n = let dummy := Lean.mkSort Lean.levelZero; Lean.Expr.getAppArgsN.loop n e (mkArray n dummy)

def Lean.Expr.getAppArgsN.loop : Nat → Lean.Expr → Array Lean.Expr → Array Lean.Expr

Equations

• Lean.Expr.getAppArgsN.loop 0 x✝ x = x
• Lean.Expr.getAppArgsN.loop (Nat.succ i) (Lean.Expr.app f a) x = Lean.Expr.getAppArgsN.loop i f (Array.set! x i a)
• Lean.Expr.getAppArgsN.loop x✝¹ x✝ x = panicWithPosWithDecl "Lean.Expr" "Lean.Expr.getAppArgsN.loop" 990 27 "too few arguments at"

def Lean.Expr.stripArgsN (e : Lean.Expr) (n : Nat) : Lean.Expr

Given e of the form f a_1 ... a_n, return f. If n is greater than the number of arguments, then return e.getAppFn.

def Lean.Expr.getAppPrefix (e : Lean.Expr) (n : Nat) : Lean.Expr

Given e of the form f a_1 ... a_n ... a_m, return f a_1 ... a_n. If n is greater than the arity, then return e.

Equations

• Lean.Expr.getAppPrefix e n = Lean.Expr.stripArgsN e (Lean.Expr.getAppNumArgs e - n)

def Lean.Expr.traverseApp {M : Type → Type u_1} [Monad M] (f : Lean.Expr → M Lean.Expr) (e : Lean.Expr) : M Lean.Expr

Given e = fn a₁ ... aₙ, runs f on fn and each of the arguments aᵢ and makes a new function application with the results.

Equations

• Lean.Expr.traverseApp f e = Lean.Expr.withApp e fun (fn : Lean.Expr) (args : Array Lean.Expr) => Seq.seq (Lean.mkAppN <$> f fn) fun (x : Unit) => Array.mapM f args

@[inline]

def Lean.Expr.withAppRev {α : Sort u_1} (e : Lean.Expr) (k : Lean.Expr → Array Lean.Expr → α) : α

Same as withApp but with arguments reversed.

Equations

• Lean.Expr.withAppRev e k = Lean.Expr.withAppRevAux k e (Array.mkEmpty (Lean.Expr.getAppNumArgs e))

def Lean.Expr.getRevArgD : Lean.Expr → Nat → Lean.Expr → Lean.Expr

Equations

• Lean.Expr.getRevArgD (Lean.Expr.app fn a) 0 x = a
• Lean.Expr.getRevArgD (Lean.Expr.app f arg) (Nat.succ i) x = Lean.Expr.getRevArgD f i x
• Lean.Expr.getRevArgD x✝¹ x✝ x = x

def Lean.Expr.getRevArg! : Lean.Expr → Nat → Lean.Expr

Equations

• Lean.Expr.getRevArg! (Lean.Expr.app fn a) 0 = a
• Lean.Expr.getRevArg! (Lean.Expr.app f arg) (Nat.succ i) = Lean.Expr.getRevArg! f i
• Lean.Expr.getRevArg! x✝ x = panicWithPosWithDecl "Lean.Expr" "Lean.Expr.getRevArg!" 1031 20 "invalid index"

@[inline]

def Lean.Expr.getArg! (e : Lean.Expr) (i : Nat) (n : optParam Nat (Lean.Expr.getAppNumArgs e)) : Lean.Expr

Given f a₀ a₁ ... aₙ, returns the ith argument or panics if out of bounds.

Equations

• Lean.Expr.getArg! e i n = Lean.Expr.getRevArg! e (n - i - 1)

@[inline]

def Lean.Expr.getArgD (e : Lean.Expr) (i : Nat) (v₀ : Lean.Expr) (n : optParam Nat (Lean.Expr.getAppNumArgs e)) : Lean.Expr

Given f a₀ a₁ ... aₙ, returns the ith argument or returns v₀ if out of bounds.

Equations

• Lean.Expr.getArgD e i v₀ n = Lean.Expr.getRevArgD e (n - i - 1) v₀

def Lean.Expr.isAppOf (e : Lean.Expr) (n : Lake.Name) : Bool

Given f a₀ a₁ ... aₙ, returns true if f is a constant with name n.

Equations

• Lean.Expr.isAppOf e n = match Lean.Expr.getAppFn e with | Lean.Expr.const c us => c == n | x => false

def Lean.Expr.isAppOfArity : Lean.Expr → Lake.Name → Nat → Bool

Given f a₁ ... aᵢ, returns true if f is a constant with name n and has the correct number of arguments.

Equations

• Lean.Expr.isAppOfArity (Lean.Expr.const c us) x 0 = (c == x)
• Lean.Expr.isAppOfArity (Lean.Expr.app f arg) x (Nat.succ a) = Lean.Expr.isAppOfArity f x a
• Lean.Expr.isAppOfArity x✝¹ x✝ x = false

def Lean.Expr.isAppOfArity' : Lean.Expr → Lake.Name → Nat → Bool

Similar to isAppOfArity but skips Expr.mdata.

Equations

• Lean.Expr.isAppOfArity' (Lean.Expr.mdata data b) x✝ x = Lean.Expr.isAppOfArity' b x✝ x
• Lean.Expr.isAppOfArity' (Lean.Expr.const c us) x 0 = (c == x)
• Lean.Expr.isAppOfArity' (Lean.Expr.app f arg) x (Nat.succ a) = Lean.Expr.isAppOfArity' f x a
• Lean.Expr.isAppOfArity' x✝¹ x✝ x = false

def Lean.Expr.appFn! : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.appFn! x = match x with | Lean.Expr.app f arg => f | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.appFn!" 1065 15 "application expected"

def Lean.Expr.appArg! : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.appArg! x = match x with | Lean.Expr.app fn a => a | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.appArg!" 1069 15 "application expected"

def Lean.Expr.appFn!' : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.appFn!' (Lean.Expr.mdata data b) = Lean.Expr.appFn!' b
• Lean.Expr.appFn!' (Lean.Expr.app f arg) = f
• Lean.Expr.appFn!' x = panicWithPosWithDecl "Lean.Expr" "Lean.Expr.appFn!'" 1074 17 "application expected"

def Lean.Expr.appArg!' : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.appArg!' (Lean.Expr.mdata data b) = Lean.Expr.appArg!' b
• Lean.Expr.appArg!' (Lean.Expr.app f arg) = arg
• Lean.Expr.appArg!' x = panicWithPosWithDecl "Lean.Expr" "Lean.Expr.appArg!'" 1079 17 "application expected"

def Lean.Expr.sortLevel! : Lean.Expr → Lean.Level

Equations

• Lean.Expr.sortLevel! x = match x with | Lean.Expr.sort u => u | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.sortLevel!" 1083 14 "sort expected"

def Lean.Expr.litValue! : Lean.Expr → Lean.Literal

Equations

• Lean.Expr.litValue! x = match x with | Lean.Expr.lit v => v | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.litValue!" 1087 13 "literal expected"

def Lean.Expr.isNatLit : Lean.Expr → Bool

Equations

• Lean.Expr.isNatLit x = match x with | Lean.Expr.lit (Lean.Literal.natVal val) => true | x => false

def Lean.Expr.natLit? : Lean.Expr → Option Nat

Equations

• Lean.Expr.natLit? x = match x with | Lean.Expr.lit (Lean.Literal.natVal v) => some v | x => none

def Lean.Expr.isStringLit : Lean.Expr → Bool

Equations

• Lean.Expr.isStringLit x = match x with | Lean.Expr.lit (Lean.Literal.strVal val) => true | x => false

def Lean.Expr.isCharLit (e : Lean.Expr) : Bool

Equations

• Lean.Expr.isCharLit e = (Lean.Expr.isAppOfArity e `Char.ofNat 1 && Lean.Expr.isNatLit (Lean.Expr.appArg! e))

def Lean.Expr.constName! : Lean.Expr → Lake.Name

Equations

• Lean.Expr.constName! x = match x with | Lean.Expr.const n us => n | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.constName!" 1106 17 "constant expected"

def Lean.Expr.constName? : Lean.Expr → Option Lake.Name

Equations

• Lean.Expr.constName? x = match x with | Lean.Expr.const n us => some n | x => none

def Lean.Expr.constLevels! : Lean.Expr → List Lean.Level

Equations

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

def Lean.Expr.bvarIdx! : Lean.Expr → Nat

Equations

• Lean.Expr.bvarIdx! x = match x with | Lean.Expr.bvar idx => idx | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.bvarIdx!" 1118 16 "bvar expected"

def Lean.Expr.fvarId! : Lean.Expr → Lean.FVarId

Equations

• Lean.Expr.fvarId! x = match x with | Lean.Expr.fvar n => n | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.fvarId!" 1122 14 "fvar expected"

def Lean.Expr.mvarId! : Lean.Expr → Lean.MVarId

Equations

• Lean.Expr.mvarId! x = match x with | Lean.Expr.mvar n => n | x => panicWithPosWithDecl "Lean.Expr" "Lean.Expr.mvarId!" 1126 14 "mvar expected"

def Lean.Expr.bindingName! : Lean.Expr → Lake.Name

Equations

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

def Lean.Expr.bindingDomain! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.bindingBody! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.bindingInfo! : Lean.Expr → Lean.BinderInfo

Equations

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

def Lean.Expr.letName! : Lean.Expr → Lake.Name

Equations

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

def Lean.Expr.letType! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.letValue! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.letBody! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.consumeMData : Lean.Expr → Lean.Expr

Equations

• Lean.Expr.consumeMData (Lean.Expr.mdata data expr) = Lean.Expr.consumeMData expr
• Lean.Expr.consumeMData x = x

def Lean.Expr.mdataExpr! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.projExpr! : Lean.Expr → Lean.Expr

Equations

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

def Lean.Expr.projIdx! : Lean.Expr → Nat

Equations

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

def Lean.Expr.hasLooseBVars (e : Lean.Expr) : Bool

Equations

• Lean.Expr.hasLooseBVars e = decide (Lean.Expr.looseBVarRange e > 0)

def Lean.Expr.isArrow (e : Lean.Expr) : Bool

Return true if e is a non-dependent arrow. Remark: the following function assumes e does not have loose bound variables.

Equations

• Lean.Expr.isArrow e = match e with | Lean.Expr.forallE binderName binderType b binderInfo => !Lean.Expr.hasLooseBVars b | x => false

@[extern lean_expr_has_loose_bvar]

opaque Lean.Expr.hasLooseBVar (e : Lean.Expr) (bvarIdx : Nat) : Bool

def Lean.Expr.hasLooseBVarInExplicitDomain : Lean.Expr → Nat → Bool → Bool

Return true if e contains the loose bound variable bvarIdx in an explicit parameter, or in the range if tryRange == true.

Equations

• One or more equations did not get rendered due to their size.
• Lean.Expr.hasLooseBVarInExplicitDomain x✝¹ x✝ x = (x && Lean.Expr.hasLooseBVar x✝¹ x✝)

@[extern lean_expr_lower_loose_bvars]

opaque Lean.Expr.lowerLooseBVars (e : Lean.Expr) (s : Nat) (d : Nat) : Lean.Expr

Lower the loose bound variables >= s in e by d. That is, a loose bound variable bvar i with i >= s is mapped to bvar (i-d).

Remark: if s < d, then the result is e.

@[extern lean_expr_lift_loose_bvars]

opaque Lean.Expr.liftLooseBVars (e : Lean.Expr) (s : Nat) (d : Nat) : Lean.Expr

Lift loose bound variables >= s in e by d.

def Lean.Expr.inferImplicit : Lean.Expr → Nat → Bool → Lean.Expr

inferImplicit e numParams considerRange updates the first numParams parameter binder annotations of the e forall type. It marks any parameter with an explicit binder annotation if there is another explicit arguments that depends on it or the resulting type if considerRange == true.

Remark: we use this function to infer the bind annotations of inductive datatype constructors, and structure projections. When the {} annotation is used in these commands, we set considerRange == false.

@[extern lean_expr_instantiate]

opaque Lean.Expr.instantiate (e : Lean.Expr) (subst : Array Lean.Expr) : Lean.Expr

Instantiates the loose bound variables in e using the subst array, where a loose Expr.bvar i at "binding depth" d is instantiated with subst[i - d] if 0 <= i - d < subst.size, and otherwise it is replaced with Expr.bvar (i - subst.size); non-loose bound variables are not touched.

If we imagine all expressions as being able to refer to the infinite list of loose bound variables ..., 3, 2, 1, 0 in that order, then conceptually instantiate is instantiating the last n of these and reindexing the remaining ones. Warning: instantiate uses the de Bruijn indexing to index the subst array, which might be the reverse order from what you might expect. See also Lean.Expr.instantiateRev.

Terminology. The "binding depth" of a subexpression is the number of bound variables available to that subexpression by virtue of being in the bodies of Expr.forallE, Expr.lam, and Expr.letE expressions. A bound variable Expr.bvar i is "loose" if its de Bruijn index i is not less than its binding depth.)

About instantiation. Instantiation isn't mere substitution. When an expression from subst is being instantiated, its internal loose bound variables have their de Bruijn indices incremented by the binding depth of the replaced loose bound variable. This is necessary for the substituted expression to still refer to the correct binders after instantiation. Similarly, the reason loose bound variables not instantiated using subst have their de Bruijn indices decremented like Expr.bvar (i - subst.size) is that instantiate can be used to eliminate binding expressions internal to a larger expression, and this adjustment keeps these bound variables referring to the same binders.

@[extern lean_expr_instantiate1]

opaque Lean.Expr.instantiate1 (e : Lean.Expr) (subst : Lean.Expr) : Lean.Expr

Instantiates loose bound variable 0 in e using the expression subst, where in particular a loose Expr.bvar i at binding depth d is instantiated with subst if i = d, and otherwise it is replaced with Expr.bvar (i - 1); non-loose bound variables are not touched.

If we imagine all expressions as being able to refer to the infinite list of loose bound variables ..., 3, 2, 1, 0 in that order, then conceptually instantiate1 is instantiating the last one of these and reindexing the remaining ones.

This function is equivalent to instantiate e #[subst], but it avoids allocating an array.

See the documentation for Lean.Expr.instantiate for a description of instantiation. In short, during instantiation the loose bound variables in subst have their own de Bruijn indices updated to account for the binding depth of the replaced loose bound variable.

@[extern lean_expr_instantiate_rev]

opaque Lean.Expr.instantiateRev (e : Lean.Expr) (subst : Array Lean.Expr) : Lean.Expr

Instantiates the loose bound variables in e using the subst array. This is equivalent to Lean.Expr.instantiate e subst.reverse, but it avoids reversing the array. In particular, rather than instantiating Expr.bvar i with subst[i - d] it instantiates with subst[subst.size - 1 - (i - d)], where d is the binding depth.

This function instantiates with the "forwards" indexing scheme. For example, if e represents the expression fun x y => x + y, then instantiateRev e.bindingBody!.bindingBody! #[a, b] yields a + b. The instantiate function on the other hand would yield b + a, since de Bruijn indices count outwards.

@[extern lean_expr_instantiate_range]

opaque Lean.Expr.instantiateRange (e : Lean.Expr) (beginIdx : Nat) (endIdx : Nat) (subst : Array Lean.Expr) : Lean.Expr

Similar to Lean.Expr.instantiate, but considers only the substitutions subst in the range [beginIdx, endIdx). Function panics if beginIdx <= endIdx <= subst.size does not hold.

This function is equivalent to instantiate e (subst.extract beginIdx endIdx), but it does not allocate a new array.

This instantiates with the "backwards" indexing scheme. See also Lean.Expr.instantiateRevRange, which instantiates with the "forwards" indexing scheme.

@[extern lean_expr_instantiate_rev_range]

opaque Lean.Expr.instantiateRevRange (e : Lean.Expr) (beginIdx : Nat) (endIdx : Nat) (subst : Array Lean.Expr) : Lean.Expr

Similar to Lean.Expr.instantiateRev, but considers only the substitutions subst in the range [beginIdx, endIdx). Function panics if beginIdx <= endIdx <= subst.size does not hold.

This function is equivalent to instantiateRev e (subst.extract beginIdx endIdx), but it does not allocate a new array.

This instantiates with the "forwards" indexing scheme (see the docstring for Lean.Expr.instantiateRev for an example). See also Lean.Expr.instantiateRange, which instantiates with the "backwards" indexing scheme.

@[extern lean_expr_abstract]

opaque Lean.Expr.abstract (e : Lean.Expr) (xs : Array Lean.Expr) : Lean.Expr

Replace free (or meta) variables xs with loose bound variables.

@[extern lean_expr_abstract_range]

opaque Lean.Expr.abstractRange (e : Lean.Expr) (n : Nat) (xs : Array Lean.Expr) : Lean.Expr

Similar to abstract, but consider only the first min n xs.size entries in xs.

def Lean.Expr.replaceFVar (e : Lean.Expr) (fvar : Lean.Expr) (v : Lean.Expr) : Lean.Expr

Replace occurrences of the free variable fvar in e with v

Equations

• Lean.Expr.replaceFVar e fvar v = Lean.Expr.instantiate1 (Lean.Expr.abstract e #[fvar]) v

def Lean.Expr.replaceFVarId (e : Lean.Expr) (fvarId : Lean.FVarId) (v : Lean.Expr) : Lean.Expr

Replace occurrences of the free variable fvarId in e with v

Equations

• Lean.Expr.replaceFVarId e fvarId v = Lean.Expr.replaceFVar e (Lean.mkFVar fvarId) v

def Lean.Expr.replaceFVars (e : Lean.Expr) (fvars : Array Lean.Expr) (vs : Array Lean.Expr) : Lean.Expr

Replace occurrences of the free variables fvars in e with vs

Equations

• Lean.Expr.replaceFVars e fvars vs = Lean.Expr.instantiateRev (Lean.Expr.abstract e fvars) vs

instance Lean.Expr.instToStringExpr : ToString Lean.Expr

Equations

• Lean.Expr.instToStringExpr = { toString := Lean.Expr.dbgToString }

def Lean.Expr.isAtomic : Lean.Expr → Bool

Returns true when the expression does not have any sub-expressions.

Equations

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

def Lean.mkDecIsTrue (pred : Lean.Expr) (proof : Lean.Expr) : Lean.Expr

Equations

• Lean.mkDecIsTrue pred proof = Lean.mkAppB (Lean.mkConst `Decidable.isTrue) pred proof

def Lean.mkDecIsFalse (pred : Lean.Expr) (proof : Lean.Expr) : Lean.Expr

Equations

• Lean.mkDecIsFalse pred proof = Lean.mkAppB (Lean.mkConst `Decidable.isFalse) pred proof

@[inline, reducible]

abbrev Lean.ExprMap (α : Type) : Type

Equations

• Lean.ExprMap α = Lean.HashMap Lean.Expr α

@[inline, reducible]

abbrev Lean.PersistentExprMap (α : Type) : Type

Equations

• Lean.PersistentExprMap α = Lean.PHashMap Lean.Expr α

@[inline, reducible]

abbrev Lean.ExprSet : Type

Equations

• Lean.ExprSet = Lean.HashSet Lean.Expr

@[inline, reducible]

abbrev Lean.PersistentExprSet : Type

Equations

• Lean.PersistentExprSet = Lean.PHashSet Lean.Expr

@[inline, reducible]

abbrev Lean.PExprSet : Type

Equations

• Lean.PExprSet = Lean.PersistentExprSet

structure Lean.ExprStructEq : Type

Auxiliary type for forcing == to be structural equality for Expr

• val : Lean.Expr

instance Lean.instInhabitedExprStructEq : Inhabited Lean.ExprStructEq

Equations

• Lean.instInhabitedExprStructEq = { default := { val := default } }

instance Lean.instCoeExprExprStructEq : Coe Lean.Expr Lean.ExprStructEq

Equations

• Lean.instCoeExprExprStructEq = { coe := Lean.ExprStructEq.mk }

def Lean.ExprStructEq.beq : Lean.ExprStructEq → Lean.ExprStructEq → Bool

Equations

• Lean.ExprStructEq.beq x✝ x = match x✝, x with | { val := e₁ }, { val := e₂ } => Lean.Expr.equal e₁ e₂

def Lean.ExprStructEq.hash : Lean.ExprStructEq → UInt64

Equations

• Lean.ExprStructEq.hash x = match x with | { val := e } => Lean.Expr.hash e

instance Lean.ExprStructEq.instBEqExprStructEq : BEq Lean.ExprStructEq

Equations

• Lean.ExprStructEq.instBEqExprStructEq = { beq := Lean.ExprStructEq.beq }

instance Lean.ExprStructEq.instHashableExprStructEq : Hashable Lean.ExprStructEq

Equations

• Lean.ExprStructEq.instHashableExprStructEq = { hash := Lean.ExprStructEq.hash }

instance Lean.ExprStructEq.instToStringExprStructEq : ToString Lean.ExprStructEq

Equations

• Lean.ExprStructEq.instToStringExprStructEq = { toString := fun (e : Lean.ExprStructEq) => toString e.val }

@[inline, reducible]

abbrev Lean.ExprStructMap (α : Type) : Type

Equations

• Lean.ExprStructMap α = Lean.HashMap Lean.ExprStructEq α

@[inline, reducible]

abbrev Lean.PersistentExprStructMap (α : Type) : Type

Equations

• Lean.PersistentExprStructMap α = Lean.PHashMap Lean.ExprStructEq α

def Lean.Expr.mkAppRevRange (f : Lean.Expr) (beginIdx : Nat) (endIdx : Nat) (revArgs : Array Lean.Expr) : Lean.Expr

mkAppRevRange f b e args == mkAppRev f (revArgs.extract b e)

Equations

• Lean.Expr.mkAppRevRange f beginIdx endIdx revArgs = Lean.Expr.mkAppRevRangeAux revArgs beginIdx f endIdx

def Lean.Expr.betaRev (f : Lean.Expr) (revArgs : Array Lean.Expr) (useZeta : optParam Bool false) (preserveMData : optParam Bool false) : Lean.Expr

If f is a lambda expression, than "beta-reduce" it using revArgs. This function is often used with getAppRev or withAppRev. Examples:

• betaRev (fun x y => t x y) #[] ==> fun x y => t x y
• betaRev (fun x y => t x y) #[a] ==> fun y => t a y
• betaRev (fun x y => t x y) #[a, b] ==> t b a
• betaRev (fun x y => t x y) #[a, b, c, d] ==> t d c b a Suppose t is (fun x y => t x y) a b c d, then args := t.getAppRev is #[d, c, b, a], and betaRev (fun x y => t x y) #[d, c, b, a] is t a b c d.

If useZeta is true, the function also performs zeta-reduction (reduction of let binders) to create further opportunities for beta reduction.

Equations

• Lean.Expr.betaRev f revArgs useZeta preserveMData = if (Array.size revArgs == 0) = true then f else let sz := Array.size revArgs; Lean.Expr.betaRev.go revArgs useZeta preserveMData sz f 0

partial def Lean.Expr.betaRev.go (revArgs : Array Lean.Expr) (useZeta : optParam Bool false) (preserveMData : optParam Bool false) (sz : Nat) (e : Lean.Expr) (i : Nat) : Lean.Expr

def Lean.Expr.beta (f : Lean.Expr) (args : Array Lean.Expr) : Lean.Expr

Apply the given arguments to f, beta-reducing if f is a lambda expression. See docstring for betaRev for examples.

Equations

• Lean.Expr.beta f args = Lean.Expr.betaRev f (Array.reverse args) false

def Lean.Expr.getNumHeadLambdas : Lean.Expr → Nat

Count the number of lambdas at the head of the given expression.

Equations

• Lean.Expr.getNumHeadLambdas (Lean.Expr.lam binderName binderType b binderInfo) = Lean.Expr.getNumHeadLambdas b + 1
• Lean.Expr.getNumHeadLambdas (Lean.Expr.mdata data b) = Lean.Expr.getNumHeadLambdas b
• Lean.Expr.getNumHeadLambdas x = 0

def Lean.Expr.isHeadBetaTargetFn (useZeta : Bool) : Lean.Expr → Bool

Return true if the given expression is the function of an expression that is target for (head) beta reduction. If useZeta = true, then let-expressions are visited. That is, it assumes that zeta-reduction (aka let-expansion) is going to be used.

See isHeadBetaTarget.

Equations

• Lean.Expr.isHeadBetaTargetFn useZeta (Lean.Expr.lam binderName binderType b binderInfo) = true
• Lean.Expr.isHeadBetaTargetFn useZeta (Lean.Expr.letE declName type v b nonDep) = (useZeta && Lean.Expr.isHeadBetaTargetFn useZeta b)
• Lean.Expr.isHeadBetaTargetFn useZeta (Lean.Expr.mdata k b) = Lean.Expr.isHeadBetaTargetFn useZeta b
• Lean.Expr.isHeadBetaTargetFn useZeta x✝ = false

def Lean.Expr.headBeta (e : Lean.Expr) : Lean.Expr

(fun x => e) a ==> e[x/a].

Equations

• Lean.Expr.headBeta e = let f := Lean.Expr.getAppFn e; if Lean.Expr.isHeadBetaTargetFn false f = true then Lean.Expr.betaRev f (Lean.Expr.getAppRevArgs e) false else e

def Lean.Expr.isHeadBetaTarget (e : Lean.Expr) (useZeta : optParam Bool false) : Bool

Return true if the given expression is a target for (head) beta reduction. If useZeta = true, then let-expressions are visited. That is, it assumes that zeta-reduction (aka let-expansion) is going to be used.

Equations

• Lean.Expr.isHeadBetaTarget e useZeta = (Lean.Expr.isApp e && Lean.Expr.isHeadBetaTargetFn useZeta (Lean.Expr.getAppFn e))

def Lean.Expr.etaExpanded? (e : Lean.Expr) : Option Lean.Expr

If e is of the form (fun x₁ ... xₙ => f x₁ ... xₙ) and f does not contain x₁, ..., xₙ, then return some f. Otherwise, return none.

It assumes e does not have loose bound variables.

Remark: ₙ may be 0

Equations

• Lean.Expr.etaExpanded? e = Lean.Expr.etaExpandedAux e 0

def Lean.Expr.etaExpandedStrict? : Lean.Expr → Option Lean.Expr

Similar to etaExpanded?, but only succeeds if ₙ ≥ 1.

Equations

• Lean.Expr.etaExpandedStrict? x = match x with | Lean.Expr.lam binderName binderType b binderInfo => Lean.Expr.etaExpandedAux b 1 | x => none

def Lean.Expr.getOptParamDefault? (e : Lean.Expr) : Option Lean.Expr

Return some e' if e is of the form optParam _ e'

Equations

• Lean.Expr.getOptParamDefault? e = if Lean.Expr.isAppOfArity e `optParam 2 = true then some (Lean.Expr.appArg! e) else none

def Lean.Expr.getAutoParamTactic? (e : Lean.Expr) : Option Lean.Expr

Return some e' if e is of the form autoParam _ e'

Equations

• Lean.Expr.getAutoParamTactic? e = if Lean.Expr.isAppOfArity e `autoParam 2 = true then some (Lean.Expr.appArg! e) else none

@[export lean_is_out_param]

def Lean.Expr.isOutParam (e : Lean.Expr) : Bool

Return true if e is of the form outParam _

Equations

• Lean.Expr.isOutParam e = Lean.Expr.isAppOfArity e `outParam 1

def Lean.Expr.isSemiOutParam (e : Lean.Expr) : Bool

Return true if e is of the form semiOutParam _

Equations

• Lean.Expr.isSemiOutParam e = Lean.Expr.isAppOfArity e `semiOutParam 1

def Lean.Expr.isOptParam (e : Lean.Expr) : Bool

Return true if e is of the form optParam _ _

Equations

• Lean.Expr.isOptParam e = Lean.Expr.isAppOfArity e `optParam 2

def Lean.Expr.isAutoParam (e : Lean.Expr) : Bool

Return true if e is of the form autoParam _ _

Equations

• Lean.Expr.isAutoParam e = Lean.Expr.isAppOfArity e `autoParam 2

@[export lean_expr_consume_type_annotations]

partial def Lean.Expr.consumeTypeAnnotations (e : Lean.Expr) : Lean.Expr

Remove outParam, optParam, and autoParam applications/annotations from e. Note that it does not remove nested annotations. Examples:

• Given e of the form outParam (optParam Nat b), consumeTypeAnnotations e = b.
• Given e of the form Nat → outParam (optParam Nat b), consumeTypeAnnotations e = e.

partial def Lean.Expr.cleanupAnnotations (e : Lean.Expr) : Lean.Expr

Remove metadata annotations and outParam, optParam, and autoParam applications/annotations from e. Note that it does not remove nested annotations. Examples:

• Given e of the form outParam (optParam Nat b), cleanupAnnotations e = b.
• Given e of the form Nat → outParam (optParam Nat b), cleanupAnnotations e = e.

@[inline]

def Lean.Expr.hasAnyFVar (e : Lean.Expr) (p : Lean.FVarId → Bool) : Bool

Return true iff e contains a free variable which satisfies p.

Equations

• Lean.Expr.hasAnyFVar e p = Lean.Expr.hasAnyFVar.visit p e

@[specialize #[]]

def Lean.Expr.hasAnyFVar.visit (p : Lean.FVarId → Bool) (e : Lean.Expr) : Bool

def Lean.Expr.containsFVar (e : Lean.Expr) (fvarId : Lean.FVarId) : Bool

Return true if e contains the given free variable.

Equations

• Lean.Expr.containsFVar e fvarId = Lean.Expr.hasAnyFVar e fun (x : Lean.FVarId) => x == fvarId

The update functions try to avoid allocating new values using pointer equality. Note that if the update*! functions are used under a match-expression, the compiler will eliminate the double-match.

@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateApp!Impl]

def Lean.Expr.updateApp! (e : Lean.Expr) (newFn : Lean.Expr) (newArg : Lean.Expr) : Lean.Expr

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@[inline]

def Lean.Expr.updateFVar! (e : Lean.Expr) (fvarIdNew : Lean.FVarId) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateConst!Impl]

def Lean.Expr.updateConst! (e : Lean.Expr) (newLevels : List Lean.Level) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateSort!Impl]

def Lean.Expr.updateSort! (e : Lean.Expr) (newLevel : Lean.Level) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateMData!Impl]

def Lean.Expr.updateMData! (e : Lean.Expr) (newExpr : Lean.Expr) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateProj!Impl]

def Lean.Expr.updateProj! (e : Lean.Expr) (newExpr : Lean.Expr) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateForall!Impl]

def Lean.Expr.updateForall! (e : Lean.Expr) (newBinfo : Lean.BinderInfo) (newDomain : Lean.Expr) (newBody : Lean.Expr) : Lean.Expr

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@[inline]

def Lean.Expr.updateForallE! (e : Lean.Expr) (newDomain : Lean.Expr) (newBody : Lean.Expr) : Lean.Expr

Equations

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateLambda!Impl]

def Lean.Expr.updateLambda! (e : Lean.Expr) (newBinfo : Lean.BinderInfo) (newDomain : Lean.Expr) (newBody : Lean.Expr) : Lean.Expr

Equations

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@[inline]

def Lean.Expr.updateLambdaE! (e : Lean.Expr) (newDomain : Lean.Expr) (newBody : Lean.Expr) : Lean.Expr

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@[implemented_by _private.Lean.Expr.0.Lean.Expr.updateLet!Impl]

def Lean.Expr.updateLet! (e : Lean.Expr) (newType : Lean.Expr) (newVal : Lean.Expr) (newBody : Lean.Expr) : Lean.Expr

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def Lean.Expr.updateFn : Lean.Expr → Lean.Expr → Lean.Expr

Equations

• Lean.Expr.updateFn (Lean.Expr.app f a) x = Lean.Expr.updateApp! (Lean.Expr.app f a) (Lean.Expr.updateFn f x) a
• Lean.Expr.updateFn x✝ x = x

partial def Lean.Expr.eta (e : Lean.Expr) : Lean.Expr

Eta reduction. If e is of the form (fun x => f x), then return f.

def Lean.Expr.setOption {α : Type} (e : Lean.Expr) (optionName : Lake.Name) [Lean.KVMap.Value α] (val : α) : Lean.Expr

Annotate e with the given option. The information is stored using metadata around e.

Equations

• Lean.Expr.setOption e optionName val = Lean.mkMData (Lean.KVMap.set Lean.MData.empty optionName val) e

def Lean.Expr.setPPExplicit (e : Lean.Expr) (flag : Bool) : Lean.Expr

Annotate e with pp.explicit := flag The delaborator uses pp options.

Equations

• Lean.Expr.setPPExplicit e flag = Lean.Expr.setOption e `pp.explicit flag

def Lean.Expr.setPPUniverses (e : Lean.Expr) (flag : Bool) : Lean.Expr

Annotate e with pp.universes := flag The delaborator uses pp options.

Equations

• Lean.Expr.setPPUniverses e flag = Lean.Expr.setOption e `pp.universes flag

def Lean.Expr.setAppPPExplicit (e : Lean.Expr) : Lean.Expr

If e is an application f a_1 ... a_n annotate f, a_1 ... a_n with pp.explicit := false, and annotate e with pp.explicit := true.

Equations

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def Lean.Expr.setAppPPExplicitForExposingMVars (e : Lean.Expr) : Lean.Expr

Similar for setAppPPExplicit, but only annotate children with pp.explicit := false if e does not contain metavariables.

Equations

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def Lean.Expr.isLetFun (e : Lean.Expr) : Bool

Returns true if e is a let_fun expression, which is an expression of the form letFun v f. Ideally f is a lambda, but we do not require that here. Warning: if the let_fun is applied to additional arguments (such as in (let_fun f := id; id) 1), this function returns false.

Equations

• Lean.Expr.isLetFun e = Lean.Expr.isAppOfArity e `letFun 4

def Lean.Expr.letFun? (e : Lean.Expr) : Option (Lake.Name × Lean.Expr × Lean.Expr × Lean.Expr)

Recognizes a let_fun expression. For let_fun n : t := v; b, returns some (n, t, v, b), which are the first four arguments to Lean.Expr.letE. Warning: if the let_fun is applied to additional arguments (such as in (let_fun f := id; id) 1), this function returns none.

let_fun expressions are encoded as letFun v (fun (n : t) => b). They can be created using Lean.Meta.mkLetFun.

If in the encoding of let_fun the last argument to letFun is eta reduced, this returns Name.anonymous for the binder name.

Equations

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def Lean.Expr.letFunAppArgs? (e : Lean.Expr) : Option (Array Lean.Expr × Lake.Name × Lean.Expr × Lean.Expr × Lean.Expr)

Like Lean.Expr.letFun?, but handles the case when the let_fun expression is possibly applied to additional arguments. Returns those arguments in addition to the values returned by letFun?.

Equations

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def Lean.mkAnnotation (kind : Lake.Name) (e : Lean.Expr) : Lean.Expr

Annotate e with the given annotation name kind. It uses metadata to store the annotation.

Equations

• Lean.mkAnnotation kind e = Lean.mkMData (Lean.KVMap.insert Lean.KVMap.empty kind (Lean.DataValue.ofBool true)) e

def Lean.annotation? (kind : Lake.Name) (e : Lean.Expr) : Option Lean.Expr

Return some e' if e = mkAnnotation kind e'

Equations

• Lean.annotation? kind e = match e with | Lean.Expr.mdata d b => if (Lean.KVMap.size d == 1 && Lean.KVMap.getBool d kind) = true then some b else none | x => none

def Lean.mkInaccessible (e : Lean.Expr) : Lean.Expr

Auxiliary annotation used to mark terms marked with the "inaccessible" annotation .(t) and _ in patterns.

Equations

• Lean.mkInaccessible e = Lean.mkAnnotation `_inaccessible e

def Lean.inaccessible? (e : Lean.Expr) : Option Lean.Expr

Return some e' if e = mkInaccessible e'.

Equations

• Lean.inaccessible? e = Lean.annotation? `_inaccessible e

def Lean.patternWithRef? (p : Lean.Expr) : Option (Lean.Syntax × Lean.Expr)

During elaboration expressions corresponding to pattern matching terms are annotated with Syntax objects. This function returns some (stx, p') if p is the pattern p' annotated with stx

Equations

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def Lean.isPatternWithRef (p : Lean.Expr) : Bool

Equations

• Lean.isPatternWithRef p = Option.isSome (Lean.patternWithRef? p)

def Lean.mkPatternWithRef (p : Lean.Expr) (stx : Lean.Syntax) : Lean.Expr

Annotate the pattern p with stx. This is an auxiliary annotation for producing better hover information.

Equations

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def Lean.patternAnnotation? (e : Lean.Expr) : Option Lean.Expr

Return some p if e is an annotated pattern (inaccessible? or patternWithRef?)

Equations

• Lean.patternAnnotation? e = match Lean.inaccessible? e with | some e => some e | x => match Lean.patternWithRef? e with | some (fst, e) => some e | x => none

def Lean.mkLHSGoalRaw (e : Lean.Expr) : Lean.Expr

Annotate e with the LHS annotation. The delaborator displays expressions of the form lhs = rhs as lhs when they have this annotation. This is used to implement the infoview for the conv mode.

This version of mkLHSGoal does not check that the argument is an equality.

Equations

• Lean.mkLHSGoalRaw e = Lean.mkAnnotation `_lhsGoal e

def Lean.isLHSGoal? (e : Lean.Expr) : Option Lean.Expr

Return some lhs if e = mkLHSGoal e', where e' is of the form lhs = rhs.

Equations

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

def Lean.mkFreshFVarId {m : Type → Type} [Monad m] [Lean.MonadNameGenerator m] : m Lean.FVarId

Polymorphic operation for generating unique/fresh free variable identifiers. It is available in any monad m that implements the interface MonadNameGenerator.

Equations

• Lean.mkFreshFVarId = do let __do_lift ← Lean.mkFreshId pure { name := __do_lift }

def Lean.mkFreshMVarId {m : Type → Type} [Monad m] [Lean.MonadNameGenerator m] : m Lean.MVarId

Polymorphic operation for generating unique/fresh metavariable identifiers. It is available in any monad m that implements the interface MonadNameGenerator.

Equations

• Lean.mkFreshMVarId = do let __do_lift ← Lean.mkFreshId pure { name := __do_lift }

def Lean.mkFreshLMVarId {m : Type → Type} [Monad m] [Lean.MonadNameGenerator m] : m Lean.LMVarId

Polymorphic operation for generating unique/fresh universe metavariable identifiers. It is available in any monad m that implements the interface MonadNameGenerator.

Equations

• Lean.mkFreshLMVarId = do let __do_lift ← Lean.mkFreshId pure { name := __do_lift }

def Lean.mkNot (p : Lean.Expr) : Lean.Expr

Return Not p

Equations

• Lean.mkNot p = Lean.mkApp (Lean.mkConst `Not) p

def Lean.mkOr (p : Lean.Expr) (q : Lean.Expr) : Lean.Expr

Return p ∨ q

Equations

• Lean.mkOr p q = Lean.mkApp2 (Lean.mkConst `Or) p q

def Lean.mkAnd (p : Lean.Expr) (q : Lean.Expr) : Lean.Expr

Return p ∧ q

Equations

• Lean.mkAnd p q = Lean.mkApp2 (Lean.mkConst `And) p q

def Lean.mkEM (p : Lean.Expr) : Lean.Expr

Return Classical.em p

Equations

• Lean.mkEM p = Lean.mkApp (Lean.mkConst `Classical.em) p