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

@[export lean_mk_reducibility_hints_regular]

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

@[export lean_reducibility_hints_get_height]

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

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

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

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

structure Lean.ConstantVal : Type

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

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

instance Lean.instInhabitedConstantVal : Inhabited Lean.ConstantVal

structure Lean.AxiomValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• isUnsafe : Bool

instance Lean.instInhabitedAxiomVal : Inhabited Lean.AxiomVal

@[export lean_mk_axiom_val]

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

@[export lean_axiom_val_is_unsafe]

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

inductive Lean.DefinitionSafety : Type

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

instance Lean.instInhabitedDefinitionSafety : Inhabited Lean.DefinitionSafety

instance Lean.instBEqDefinitionSafety : BEq Lean.DefinitionSafety

instance Lean.instReprDefinitionSafety : Repr Lean.DefinitionSafety

structure Lean.DefinitionValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• value : Lean.Expr
• hints : Lean.ReducibilityHints
• safety : Lean.DefinitionSafety
• all : List Lake.Name

  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.

instance Lean.instInhabitedDefinitionVal : Inhabited Lean.DefinitionVal

@[export lean_mk_definition_val]

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

@[export lean_definition_val_get_safety]

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

structure Lean.TheoremValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• value : Lean.Expr
• all : List Lake.Name

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

instance Lean.instInhabitedTheoremVal : Inhabited Lean.TheoremVal

structure Lean.OpaqueValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• value : Lean.Expr
• isUnsafe : Bool
• all : List Lake.Name

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

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

instance Lean.instInhabitedOpaqueVal : Inhabited Lean.OpaqueVal

@[export lean_mk_opaque_val]

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

@[export lean_opaque_val_is_unsafe]

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

structure Lean.Constructor : Type

• name : Lake.Name
• type : Lean.Expr

instance Lean.instInhabitedConstructor : Inhabited Lean.Constructor

structure Lean.InductiveType : Type

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

instance Lean.instInhabitedInductiveType : Inhabited Lean.InductiveType

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 Lake.Name → Nat → List Lean.InductiveType → Bool → Lean.Declaration

Declaration object that can be sent to the kernel.

instance Lean.instInhabitedDeclaration : Inhabited Lean.Declaration

@[export lean_mk_inductive_decl]

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

@[export lean_is_unsafe_inductive_decl]

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

@[specialize #[]]

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

@[inline]

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

structure Lean.InductiveValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• numParams : Nat

  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.

• numIndices : 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).

• all : List Lake.Name

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

• ctors : List Lake.Name

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

• isRec : Bool

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

• isUnsafe : Bool

  Whether the definition is flagged as unsafe.

• isReflexive : 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

• isNested : 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, ...).

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

@[export lean_mk_inductive_val]

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

@[export lean_inductive_val_is_rec]

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

@[export lean_inductive_val_is_unsafe]

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

@[export lean_inductive_val_is_reflexive]

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

@[export lean_inductive_val_is_nested]

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

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

structure Lean.ConstructorValextends Lean.ConstantVal : Type

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

  Inductive type this constructor is a member of

• cidx : Nat

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

• numParams : Nat

  Number of parameters in inductive datatype.

• numFields : Nat

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

• isUnsafe : Bool

instance Lean.instInhabitedConstructorVal : Inhabited Lean.ConstructorVal

@[export lean_mk_constructor_val]

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

@[export lean_constructor_val_is_unsafe]

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

structure Lean.RecursorRule : Type

• ctor : Lake.Name

  Reduction rule for this Constructor

• nfields : Nat

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

• rhs : Lean.Expr

  Right hand side of the reduction rule

Information for reducing a recursor

instance Lean.instInhabitedRecursorRule : Inhabited Lean.RecursorRule

structure Lean.RecursorValextends Lean.ConstantVal : Type

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

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

• numParams : Nat

  Number of parameters

• numIndices : Nat

  Number of indices

• numMotives : Nat

  Number of motives

• numMinors : Nat

  Number of minor premises

• rules : List Lean.RecursorRule

  A reduction for each Constructor

• k : Bool

  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).

• isUnsafe : Bool

instance Lean.instInhabitedRecursorVal : Inhabited Lean.RecursorVal

@[export lean_mk_recursor_val]

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

@[export lean_recursor_k]

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

@[export lean_recursor_is_unsafe]

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

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

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

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

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

inductive Lean.QuotKind : Type

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

instance Lean.instInhabitedQuotKind : Inhabited Lean.QuotKind

structure Lean.QuotValextends Lean.ConstantVal : Type

• name : Lake.Name
• levelParams : List Lake.Name
• type : Lean.Expr
• kind : Lean.QuotKind

instance Lean.instInhabitedQuotVal : Inhabited Lean.QuotVal

@[export lean_mk_quot_val]

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

@[export lean_quot_val_kind]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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