inductive Lean.ReducibilityHints : Type

• opaque: Lean.ReducibilityHints
• abbrev: Lean.ReducibilityHints
• regular: UInt32 → Lean.ReducibilityHints

Reducibility hints are used in the convertibility checker. When trying to solve a constraint such a

       (f ...) =?= (g ...)

where f and g are definitions, the checker has to decide which one will be unfolded. If f (g) is opaque, then g (f) is unfolded if it is also not marked as opaque, Else if f (g) is abbrev, then f (g) is unfolded if g (f) is also not marked as abbrev, Else if f and g are regular, then we unfold the one with the biggest definitional height. Otherwise both are unfolded.

The arguments of the regular Constructor are: the definitional height and the flag selfOpt.

The definitional height is by default computed by the kernel. It only takes into account other regular definitions used in a definition. When creating declarations using meta-programming, we can specify the definitional depth manually.

Remark: the hint only affects performance. None of the hints prevent the kernel from unfolding a declaration during Type checking.

Remark: the ReducibilityHints are not related to the attributes: reducible/irrelevance/semireducible. These attributes are used by the Elaborator. The ReducibilityHints are used by the kernel (and Elaborator). Moreover, the ReducibilityHints cannot be changed after a declaration is added to the kernel.

instance Lean.instInhabitedReducibilityHints : Inhabited Lean.ReducibilityHints

Equations

• Lean.instInhabitedReducibilityHints = { default := Lean.ReducibilityHints.opaque }

@[export lean_mk_reducibility_hints_regular]

def Lean.mkReducibilityHintsRegularEx (h : UInt32) : Lean.ReducibilityHints

Equations

• Lean.mkReducibilityHintsRegularEx h = Lean.ReducibilityHints.regular h

@[export lean_reducibility_hints_get_height]

def Lean.ReducibilityHints.getHeightEx (h : Lean.ReducibilityHints) : UInt32

Equations

• Lean.ReducibilityHints.getHeightEx h = match h with | Lean.ReducibilityHints.regular h => h | x => 0

def Lean.ReducibilityHints.lt : Lean.ReducibilityHints → Lean.ReducibilityHints → Bool

Equations

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

def Lean.ReducibilityHints.isAbbrev : Lean.ReducibilityHints → Bool

Equations

• Lean.ReducibilityHints.isAbbrev x = match x with | Lean.ReducibilityHints.abbrev => true | x => false

def Lean.ReducibilityHints.isRegular : Lean.ReducibilityHints → Bool

Equations

• Lean.ReducibilityHints.isRegular x = match x with | Lean.ReducibilityHints.regular a => true | x => false

structure Lean.ConstantVal : Type

• name : Lean.Name
• levelParams : List Lean.Name
• type : Lean.Expr

Base structure for AxiomVal, DefinitionVal, TheoremVal, InductiveVal, ConstructorVal, RecursorVal and QuotVal.

instance Lean.instInhabitedConstantVal : Inhabited Lean.ConstantVal

Equations

• Lean.instInhabitedConstantVal = { default := { name := default, levelParams := default, type := default } }

structure Lean.AxiomValextends Lean.ConstantVal : Type

• isUnsafe : Bool

instance Lean.instInhabitedAxiomVal : Inhabited Lean.AxiomVal

Equations

• Lean.instInhabitedAxiomVal = { default := { toConstantVal := default, isUnsafe := default } }

@[export lean_mk_axiom_val]

def Lean.mkAxiomValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (isUnsafe : Bool) : Lean.AxiomVal

Equations

• Lean.mkAxiomValEx name levelParams type isUnsafe = { toConstantVal := { name := name, levelParams := levelParams, type := type }, isUnsafe := isUnsafe }

@[export lean_axiom_val_is_unsafe]

def Lean.AxiomVal.isUnsafeEx (v : Lean.AxiomVal) : Bool

Equations

• Lean.AxiomVal.isUnsafeEx v = v.isUnsafe

inductive Lean.DefinitionSafety : Type

• unsafe: Lean.DefinitionSafety
• safe: Lean.DefinitionSafety
• partial: Lean.DefinitionSafety

instance Lean.instInhabitedDefinitionSafety : Inhabited Lean.DefinitionSafety

Equations

• Lean.instInhabitedDefinitionSafety = { default := Lean.DefinitionSafety.unsafe }

instance Lean.instBEqDefinitionSafety : BEq Lean.DefinitionSafety

Equations

• Lean.instBEqDefinitionSafety = { beq := Lean.beqDefinitionSafety✝ }

instance Lean.instReprDefinitionSafety : Repr Lean.DefinitionSafety

Equations

• Lean.instReprDefinitionSafety = { reprPrec := Lean.reprDefinitionSafety✝ }

structure Lean.DefinitionValextends Lean.ConstantVal : Type

• value : Lean.Expr
• hints : Lean.ReducibilityHints
• safety : Lean.DefinitionSafety

• List of all (including this one) declarations in the same mutual block. Note that this information is not used by the kernel, and is only used to save the information provided by the user when using mutual blocks. Recall that the Lean kernel does not support recursive definitions and they are compiled using recursors and WellFounded.fix.

  all : List Lean.Name

instance Lean.instInhabitedDefinitionVal : Inhabited Lean.DefinitionVal

Equations

• Lean.instInhabitedDefinitionVal = { default := { toConstantVal := default, value := default, hints := default, safety := default, all := default } }

@[export lean_mk_definition_val]

def Lean.mkDefinitionValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (value : Lean.Expr) (hints : Lean.ReducibilityHints) (safety : Lean.DefinitionSafety) (all : List Lean.Name) : Lean.DefinitionVal

Equations

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

@[export lean_definition_val_get_safety]

def Lean.DefinitionVal.getSafetyEx (v : Lean.DefinitionVal) : Lean.DefinitionSafety

Equations

• Lean.DefinitionVal.getSafetyEx v = v.safety

structure Lean.TheoremValextends Lean.ConstantVal : Type

• value : Lean.Expr

• List of all (including this one) declarations in the same mutual block. See comment at DefinitionVal.all.

  all : List Lean.Name

instance Lean.instInhabitedTheoremVal : Inhabited Lean.TheoremVal

Equations

• Lean.instInhabitedTheoremVal = { default := { toConstantVal := default, value := default, all := default } }

structure Lean.OpaqueValextends Lean.ConstantVal : Type

• value : Lean.Expr
• isUnsafe : Bool

• List of all (including this one) declarations in the same mutual block. See comment at DefinitionVal.all.

  all : List Lean.Name

Value for an opaque constant declaration opaque x : t := e

instance Lean.instInhabitedOpaqueVal : Inhabited Lean.OpaqueVal

Equations

• Lean.instInhabitedOpaqueVal = { default := { toConstantVal := default, value := default, isUnsafe := default, all := default } }

@[export lean_mk_opaque_val]

def Lean.mkOpaqueValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (value : Lean.Expr) (isUnsafe : Bool) (all : List Lean.Name) : Lean.OpaqueVal

Equations

• Lean.mkOpaqueValEx name levelParams type value isUnsafe all = { toConstantVal := { name := name, levelParams := levelParams, type := type }, value := value, isUnsafe := isUnsafe, all := all }

@[export lean_opaque_val_is_unsafe]

def Lean.OpaqueVal.isUnsafeEx (v : Lean.OpaqueVal) : Bool

Equations

• Lean.OpaqueVal.isUnsafeEx v = v.isUnsafe

structure Lean.Constructor : Type

• name : Lean.Name
• type : Lean.Expr

instance Lean.instInhabitedConstructor : Inhabited Lean.Constructor

Equations

• Lean.instInhabitedConstructor = { default := { name := default, type := default } }

structure Lean.InductiveType : Type

• name : Lean.Name
• type : Lean.Expr
• ctors : List Lean.Constructor

instance Lean.instInhabitedInductiveType : Inhabited Lean.InductiveType

Equations

• Lean.instInhabitedInductiveType = { default := { name := default, type := default, ctors := default } }

inductive Lean.Declaration : Type

• axiomDecl: Lean.AxiomVal → Lean.Declaration
• defnDecl: Lean.DefinitionVal → Lean.Declaration
• thmDecl: Lean.TheoremVal → Lean.Declaration
• opaqueDecl: Lean.OpaqueVal → Lean.Declaration
• quotDecl: Lean.Declaration
• mutualDefnDecl: List Lean.DefinitionVal → Lean.Declaration
• inductDecl: List Lean.Name → Nat → List Lean.InductiveType → Bool → Lean.Declaration

Declaration object that can be sent to the kernel.

instance Lean.instInhabitedDeclaration : Inhabited Lean.Declaration

Equations

• Lean.instInhabitedDeclaration = { default := Lean.Declaration.axiomDecl default }

@[export lean_mk_inductive_decl]

def Lean.mkInductiveDeclEs (lparams : List Lean.Name) (nparams : Nat) (types : List Lean.InductiveType) (isUnsafe : Bool) : Lean.Declaration

Equations

• Lean.mkInductiveDeclEs lparams nparams types isUnsafe = Lean.Declaration.inductDecl lparams nparams types isUnsafe

@[export lean_is_unsafe_inductive_decl]

def Lean.Declaration.isUnsafeInductiveDeclEx : Lean.Declaration → Bool

Equations

• Lean.Declaration.isUnsafeInductiveDeclEx x = match x with | Lean.Declaration.inductDecl lparams nparams types isUnsafe => isUnsafe | x => false

@[specialize #[]]

def Lean.Declaration.foldExprM {α : Type} {m : Type → Type} [inst : Monad m] (d : Lean.Declaration) (f : α → Lean.Expr → m α) (a : α) : m α

Equations

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

@[inline]

def Lean.Declaration.forExprM {m : Type → Type} [inst : Monad m] (d : Lean.Declaration) (f : Lean.Expr → m Unit) : m Unit

Equations

• Lean.Declaration.forExprM d f = Lean.Declaration.foldExprM d (fun x a => f a) ()

structure Lean.InductiveValextends Lean.ConstantVal : Type

• Number of parameters. A parameter is an argument to the defined type that is fixed over constructors. An example of this is the α : Type argument in the vector constructors nil : Vector α 0 and cons : α → Vector α n → Vector α (n+1).

  The intuition is that the inductive type must exhibit parametric polymorphism over the inductive parameter, as opposed to ad-hoc polymorphism.

  numParams : Nat

• Number of indices. An index is an argument that varies over constructors.

  An example of this is the n : Nat argument in the vector constructor cons : α → Vector α n → Vector α (n+1).

  numIndices : Nat

• List of all (including this one) inductive datatypes in the mutual declaration containing this one

  all : List Lean.Name

• List of the names of the constructors for this inductive datatype.

  ctors : List Lean.Name

• true when recursive (that is, the inductive type appears as an argument in a constructor).

  isRec : Bool

• Whether the definition is flagged as unsafe.

  isUnsafe : Bool

• An inductive type is called reflexive if it has at least one constructor that takes as an argument a function returning the same type we are defining. Consider the type:

  inductive WideTree where
  | branch: (Nat -> WideTree) -> WideTree
  | leaf: WideTree
  

  this is reflexive due to the presence of the branch : (Nat -> WideTree) -> WideTree constructor.

  See also: 'Inductive Definitions in the system Coq Rules and Properties' by Christine Paulin-Mohring Section 2.2, Definition 3

  isReflexive : Bool

• An inductive definition T is nested when there is a constructor with an argument x : F T, where F : Type → Type is some suitably behaved (ie strictly positive) function (Eg Array T, List T, T × T, ...).

  isNested : Bool

The kernel compiles (mutual) inductive declarations (see inductiveDecls) into a set of

• Declaration.inductDecl (for each inductive datatype in the mutual Declaration),
• Declaration.ctorDecl (for each Constructor in the mutual Declaration),
• Declaration.recDecl (automatically generated recursors).

This data is used to implement iota-reduction efficiently and compile nested inductive declarations.

A series of checks are performed by the kernel to check whether a inductiveDecls is valid or not.

instance Lean.instInhabitedInductiveVal : Inhabited Lean.InductiveVal

Equations

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

@[export lean_mk_inductive_val]

def Lean.mkInductiveValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (numParams : Nat) (numIndices : Nat) (all : List Lean.Name) (ctors : List Lean.Name) (isRec : Bool) (isUnsafe : Bool) (isReflexive : Bool) (isNested : Bool) : Lean.InductiveVal

Equations

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

@[export lean_inductive_val_is_rec]

def Lean.InductiveVal.isRecEx (v : Lean.InductiveVal) : Bool

Equations

• Lean.InductiveVal.isRecEx v = v.isRec

@[export lean_inductive_val_is_unsafe]

def Lean.InductiveVal.isUnsafeEx (v : Lean.InductiveVal) : Bool

Equations

• Lean.InductiveVal.isUnsafeEx v = v.isUnsafe

@[export lean_inductive_val_is_reflexive]

def Lean.InductiveVal.isReflexiveEx (v : Lean.InductiveVal) : Bool

Equations

• Lean.InductiveVal.isReflexiveEx v = v.isReflexive

@[export lean_inductive_val_is_nested]

def Lean.InductiveVal.isNestedEx (v : Lean.InductiveVal) : Bool

Equations

• Lean.InductiveVal.isNestedEx v = v.isNested

def Lean.InductiveVal.numCtors (v : Lean.InductiveVal) : Nat

Equations

• Lean.InductiveVal.numCtors v = List.length v.ctors

structure Lean.ConstructorValextends Lean.ConstantVal : Type

• Inductive type this constructor is a member of

  induct : Lean.Name

• Constructor index (i.e., Position in the inductive declaration)

  cidx : Nat

• Number of parameters in inductive datatype.

  numParams : Nat

• Number of fields (i.e., arity - nparams)

  numFields : Nat
• isUnsafe : Bool

instance Lean.instInhabitedConstructorVal : Inhabited Lean.ConstructorVal

Equations

• Lean.instInhabitedConstructorVal = { default := { toConstantVal := default, induct := default, cidx := default, numParams := default, numFields := default, isUnsafe := default } }

@[export lean_mk_constructor_val]

def Lean.mkConstructorValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (induct : Lean.Name) (cidx : Nat) (numParams : Nat) (numFields : Nat) (isUnsafe : Bool) : Lean.ConstructorVal

Equations

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

@[export lean_constructor_val_is_unsafe]

def Lean.ConstructorVal.isUnsafeEx (v : Lean.ConstructorVal) : Bool

Equations

• Lean.ConstructorVal.isUnsafeEx v = v.isUnsafe

structure Lean.RecursorRule : Type

• Reduction rule for this Constructor

  ctor : Lean.Name

• Number of fields (i.e., without counting inductive datatype parameters)

  nfields : Nat

• Right hand side of the reduction rule

  rhs : Lean.Expr

Information for reducing a recursor

instance Lean.instInhabitedRecursorRule : Inhabited Lean.RecursorRule

Equations

• Lean.instInhabitedRecursorRule = { default := { ctor := default, nfields := default, rhs := default } }

structure Lean.RecursorValextends Lean.ConstantVal : Type

• List of all inductive datatypes in the mutual declaration that generated this recursor

  all : List Lean.Name

• Number of parameters

  numParams : Nat

• Number of indices

  numIndices : Nat

• Number of motives

  numMotives : Nat

• Number of minor premises

  numMinors : Nat

• A reduction for each Constructor

  rules : List Lean.RecursorRule

• It supports K-like reduction. A recursor is said to support K-like reduction if one can assume it behaves like Eq under axiom K --- that is, it has one constructor, the constructor has 0 arguments, and it is an inductive predicate (ie, it lives in Prop).

  Examples of inductives with K-like reduction is Eq, Acc, and And.intro. Non-examples are exists (where the constructor has arguments) and Or.intro (which has multiple constructors).

  k : Bool
• isUnsafe : Bool

instance Lean.instInhabitedRecursorVal : Inhabited Lean.RecursorVal

Equations

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

@[export lean_mk_recursor_val]

def Lean.mkRecursorValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (all : List Lean.Name) (numParams : Nat) (numIndices : Nat) (numMotives : Nat) (numMinors : Nat) (rules : List Lean.RecursorRule) (k : Bool) (isUnsafe : Bool) : Lean.RecursorVal

Equations

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

@[export lean_recursor_k]

def Lean.RecursorVal.kEx (v : Lean.RecursorVal) : Bool

Equations

• Lean.RecursorVal.kEx v = v.k

@[export lean_recursor_is_unsafe]

def Lean.RecursorVal.isUnsafeEx (v : Lean.RecursorVal) : Bool

Equations

• Lean.RecursorVal.isUnsafeEx v = v.isUnsafe

def Lean.RecursorVal.getMajorIdx (v : Lean.RecursorVal) : Nat

Equations

• Lean.RecursorVal.getMajorIdx v = v.numParams + v.numMotives + v.numMinors + v.numIndices

def Lean.RecursorVal.getFirstIndexIdx (v : Lean.RecursorVal) : Nat

Equations

• Lean.RecursorVal.getFirstIndexIdx v = v.numParams + v.numMotives + v.numMinors

def Lean.RecursorVal.getFirstMinorIdx (v : Lean.RecursorVal) : Nat

Equations

• Lean.RecursorVal.getFirstMinorIdx v = v.numParams + v.numMotives

def Lean.RecursorVal.getInduct (v : Lean.RecursorVal) : Lean.Name

Equations

• Lean.RecursorVal.getInduct v = Lean.Name.getPrefix v.toConstantVal.name

inductive Lean.QuotKind : Type

• type: Lean.QuotKind
• ctor: Lean.QuotKind
• lift: Lean.QuotKind
• ind: Lean.QuotKind

instance Lean.instInhabitedQuotKind : Inhabited Lean.QuotKind

Equations

• Lean.instInhabitedQuotKind = { default := Lean.QuotKind.type }

structure Lean.QuotValextends Lean.ConstantVal : Type

• kind : Lean.QuotKind

instance Lean.instInhabitedQuotVal : Inhabited Lean.QuotVal

Equations

• Lean.instInhabitedQuotVal = { default := { toConstantVal := default, kind := default } }

@[export lean_mk_quot_val]

def Lean.mkQuotValEx (name : Lean.Name) (levelParams : List Lean.Name) (type : Lean.Expr) (kind : Lean.QuotKind) : Lean.QuotVal

Equations

• Lean.mkQuotValEx name levelParams type kind = { toConstantVal := { name := name, levelParams := levelParams, type := type }, kind := kind }

@[export lean_quot_val_kind]

def Lean.QuotVal.kindEx (v : Lean.QuotVal) : Lean.QuotKind

Equations

• Lean.QuotVal.kindEx v = v.kind

inductive Lean.ConstantInfo : Type

• axiomInfo: Lean.AxiomVal → Lean.ConstantInfo
• defnInfo: Lean.DefinitionVal → Lean.ConstantInfo
• thmInfo: Lean.TheoremVal → Lean.ConstantInfo
• opaqueInfo: Lean.OpaqueVal → Lean.ConstantInfo
• quotInfo: Lean.QuotVal → Lean.ConstantInfo
• inductInfo: Lean.InductiveVal → Lean.ConstantInfo
• ctorInfo: Lean.ConstructorVal → Lean.ConstantInfo
• recInfo: Lean.RecursorVal → Lean.ConstantInfo

Information associated with constant declarations.

instance Lean.instInhabitedConstantInfo : Inhabited Lean.ConstantInfo

Equations

• Lean.instInhabitedConstantInfo = { default := Lean.ConstantInfo.axiomInfo default }

def Lean.ConstantInfo.toConstantVal : Lean.ConstantInfo → Lean.ConstantVal

Equations

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

def Lean.ConstantInfo.isUnsafe : Lean.ConstantInfo → Bool

Equations

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

def Lean.ConstantInfo.isPartial : Lean.ConstantInfo → Bool

Equations

• Lean.ConstantInfo.isPartial x = match x with | Lean.ConstantInfo.defnInfo v => v.safety == Lean.DefinitionSafety.partial | x => false

def Lean.ConstantInfo.name (d : Lean.ConstantInfo) : Lean.Name

Equations

• Lean.ConstantInfo.name d = (Lean.ConstantInfo.toConstantVal d).name

def Lean.ConstantInfo.levelParams (d : Lean.ConstantInfo) : List Lean.Name

Equations

• Lean.ConstantInfo.levelParams d = (Lean.ConstantInfo.toConstantVal d).levelParams

def Lean.ConstantInfo.numLevelParams (d : Lean.ConstantInfo) : Nat

Equations

• Lean.ConstantInfo.numLevelParams d = List.length (Lean.ConstantInfo.levelParams d)

def Lean.ConstantInfo.type (d : Lean.ConstantInfo) : Lean.Expr

Equations

• Lean.ConstantInfo.type d = (Lean.ConstantInfo.toConstantVal d).type

def Lean.ConstantInfo.value? : Lean.ConstantInfo → Option Lean.Expr

Equations

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

def Lean.ConstantInfo.hasValue : Lean.ConstantInfo → Bool

Equations

• Lean.ConstantInfo.hasValue x = match x with | Lean.ConstantInfo.defnInfo val => true | Lean.ConstantInfo.thmInfo val => true | x => false

def Lean.ConstantInfo.value! : Lean.ConstantInfo → Lean.Expr

Equations

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

def Lean.ConstantInfo.hints : Lean.ConstantInfo → Lean.ReducibilityHints

Equations

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

def Lean.ConstantInfo.isCtor : Lean.ConstantInfo → Bool

Equations

• Lean.ConstantInfo.isCtor x = match x with | Lean.ConstantInfo.ctorInfo val => true | x => false

def Lean.ConstantInfo.isInductive : Lean.ConstantInfo → Bool

Equations

• Lean.ConstantInfo.isInductive x = match x with | Lean.ConstantInfo.inductInfo val => true | x => false

def Lean.ConstantInfo.all : Lean.ConstantInfo → List Lean.Name

List of all (including this one) declarations in the same mutual block.

Equations

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

def Lean.mkRecName (declName : Lean.Name) : Lean.Name

Equations

• Lean.mkRecName declName = Lean.Name.mkStr declName "rec"