MutualDef.lean

/-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Parser.Term
import Lean.Meta.Closure
import Lean.Meta.Check
import Lean.Elab.Command
import Lean.Elab.DefView
import Lean.Elab.PreDefinition
import Lean.Elab.DeclarationRange

namespace Lean.Elab
open Lean.Parser.Term

/- DefView after elaborating the header. -/
structure DefViewElabHeader where
  ref           : Syntax
  modifiers     : Modifiers
  kind          : DefKind
  shortDeclName : Name
  declName      : Name
  levelNames    : List Name
  numParams     : Nat
  type          : Expr -- including the parameters
  valueStx      : Syntax
  deriving Inhabited

namespace Term

open Meta

private def checkModifiers (m₁ m₂ : Modifiers) : TermElabM Unit := do
  unless m₁.isUnsafe == m₂.isUnsafe do
    throwError "cannot mix unsafe and safe definitions"
  unless m₁.isNoncomputable == m₂.isNoncomputable do
    throwError "cannot mix computable and non-computable definitions"
  unless m₁.isPartial == m₂.isPartial do
    throwError "cannot mix partial and non-partial definitions"

private def checkKinds (k₁ k₂ : DefKind) : TermElabM Unit := do
  unless k₁.isExample == k₂.isExample do
    throwError "cannot mix examples and definitions" -- Reason: we should discard examples
  unless k₁.isTheorem == k₂.isTheorem do
    throwError "cannot mix theorems and definitions" -- Reason: we will eventually elaborate theorems in `Task`s.

private def check (prevHeaders : Array DefViewElabHeader) (newHeader : DefViewElabHeader) : TermElabM Unit := do
  if newHeader.kind.isTheorem && newHeader.modifiers.isUnsafe then
    throwError "'unsafe' theorems are not allowed"
  if newHeader.kind.isTheorem && newHeader.modifiers.isPartial then
    throwError "'partial' theorems are not allowed, 'partial' is a code generation directive"
  if newHeader.kind.isTheorem && newHeader.modifiers.isNoncomputable then
    throwError "'theorem' subsumes 'noncomputable', code is not generated for theorems"
  if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isUnsafe then
    throwError "'noncomputable unsafe' is not allowed"
  if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isPartial then
    throwError "'noncomputable partial' is not allowed"
  if newHeader.modifiers.isPartial && newHeader.modifiers.isUnsafe then
    throwError "'unsafe' subsumes 'partial'"
  if h : 0 < prevHeaders.size then
    let firstHeader := prevHeaders.get ⟨0, h⟩
    try
      unless newHeader.levelNames == firstHeader.levelNames do
        throwError "universe parameters mismatch"
      checkModifiers newHeader.modifiers firstHeader.modifiers
      checkKinds newHeader.kind firstHeader.kind
    catch
       | Exception.error ref msg => throw (Exception.error ref m!"invalid mutually recursive definitions, {msg}")
       | ex => throw ex
  else
    pure ()

private def registerFailedToInferDefTypeInfo (type : Expr) (ref : Syntax) : TermElabM Unit :=
  registerCustomErrorIfMVar type ref "failed to infer definition type"

private def elabHeaders (views : Array DefView) : TermElabM (Array DefViewElabHeader) := do
  let mut headers := #[]
  for view in views do
    let newHeader ← withRef view.ref do
      let ⟨shortDeclName, declName, levelNames⟩ ← expandDeclId (← getCurrNamespace) (← getLevelNames) view.declId view.modifiers
      addDeclarationRanges declName view.ref
      applyAttributesAt declName view.modifiers.attrs AttributeApplicationTime.beforeElaboration
      withDeclName declName <| withAutoBoundImplicit <| withLevelNames levelNames <|
        elabBinders view.binders.getArgs fun xs => do
          let refForElabFunType := view.value
          let type ← match view.type? with
            | some typeStx =>
              let type ← elabType typeStx
              registerFailedToInferDefTypeInfo type typeStx
              pure type
            | none =>
              let hole := mkHole refForElabFunType
              let type ← elabType hole
              registerFailedToInferDefTypeInfo type refForElabFunType
              pure type
          Term.synthesizeSyntheticMVarsNoPostponing
          let type ← mkForallFVars xs type
          let type ← mkForallFVars (← read).autoBoundImplicits.toArray type
          let type ← instantiateMVars type
          let xs ← addAutoBoundImplicits xs
          let levelNames ← getLevelNames
          if view.type?.isSome then
            let pendingMVarIds ← getMVars type
            discard <| logUnassignedUsingErrorInfos pendingMVarIds <|
              m!"\nwhen the resulting type of a declaration is explicitly provided, all holes (e.g., `_`) in the header are resolved before the declaration body is processed"
          let newHeader := {
            ref           := view.ref,
            modifiers     := view.modifiers,
            kind          := view.kind,
            shortDeclName := shortDeclName,
            declName      := declName,
            levelNames    := levelNames,
            numParams     := xs.size,
            type          := type,
            valueStx      := view.value : DefViewElabHeader }
          check headers newHeader
          pure newHeader
    headers := headers.push newHeader
  pure headers

private partial def withFunLocalDecls {α} (headers : Array DefViewElabHeader) (k : Array Expr → TermElabM α) : TermElabM α :=
  let rec loop (i : Nat) (fvars : Array Expr) := do
    if h : i < headers.size then
      let header := headers.get ⟨i, h⟩
      withLocalDecl header.shortDeclName BinderInfo.auxDecl header.type fun fvar => loop (i+1) (fvars.push fvar)
    else
      k fvars
  loop 0 #[]

private def expandWhereDeclsAsStructInst : Macro
  | `(whereDecls|where $[$decls:letRecDecl$[;]?]*) => do
    let letIdDecls ← decls.mapM fun stx => match stx with
      | `(letRecDecl|$attrs:attributes $decl:letDecl) => Macro.throwErrorAt stx "attributes are 'where' elements are currently not supported here"
      | `(letRecDecl|$decl:letPatDecl)  => Macro.throwErrorAt stx "patterns are not allowed here"
      | `(letRecDecl|$decl:letEqnsDecl) => expandLetEqnsDecl decl
      | `(letRecDecl|$decl:letIdDecl)   => pure decl
      | _                               => Macro.throwUnsupported
    let structInstFields ← letIdDecls.mapM fun
      | stx@`(letIdDecl|$id:ident $[$binders]* $[: $ty?]? := $val) => withRef stx do
        let mut val := val
        if let some ty := ty? then
          val ← `(($val : $ty))
        val ← if binders.size > 0 then `(fun $[$binders]* => $val:term) else val
        `(structInstField|$id:ident := $val)
      | _ => Macro.throwUnsupported
    `({ $[$structInstFields,]* })
  | _ => Macro.throwUnsupported

/-
Recall that
```
def declValSimple    := leading_parser " :=\n" >> termParser >> optional Term.whereDecls
def declValEqns      := leading_parser Term.matchAltsWhereDecls
def declVal          := declValSimple <|> declValEqns <|> Term.whereDecls
```
-/
private def declValToTerm (declVal : Syntax) : MacroM Syntax := withRef declVal do
  if declVal.isOfKind `Lean.Parser.Command.declValSimple then
    expandWhereDeclsOpt declVal[2] declVal[1]
  else if declVal.isOfKind `Lean.Parser.Command.declValEqns then
    expandMatchAltsWhereDecls declVal[0]
  else if declVal.isOfKind `Lean.Parser.Term.whereDecls then
    expandWhereDeclsAsStructInst declVal
  else if declVal.isMissing then
    Macro.throwErrorAt declVal "declaration body is missing"
  else
    Macro.throwErrorAt declVal "unexpected declaration body"

private def elabFunValues (headers : Array DefViewElabHeader) : TermElabM (Array Expr) :=
  headers.mapM fun header => withDeclName header.declName $ withLevelNames header.levelNames do
    let valStx ← liftMacroM $ declValToTerm header.valueStx
    forallBoundedTelescope header.type header.numParams fun xs type => do
      let val ← elabTermEnsuringType valStx type
      mkLambdaFVars xs val

private def collectUsed (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)
    : StateRefT CollectFVars.State MetaM Unit := do
  headers.forM fun header => collectUsedFVars header.type
  values.forM collectUsedFVars
  toLift.forM fun letRecToLift => do
    collectUsedFVars letRecToLift.type
    collectUsedFVars letRecToLift.val

private def removeUnusedVars (vars : Array Expr) (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)
    : TermElabM (LocalContext × LocalInstances × Array Expr) := do
  let (_, used) ← (collectUsed headers values toLift).run {}
  removeUnused vars used

private def withUsed {α} (vars : Array Expr) (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)
    (k : Array Expr → TermElabM α) : TermElabM α := do
  let (lctx, localInsts, vars) ← removeUnusedVars vars headers values toLift
  withLCtx lctx localInsts $ k vars

private def isExample (views : Array DefView) : Bool :=
  views.any (·.kind.isExample)

private def isTheorem (views : Array DefView) : Bool :=
  views.any (·.kind.isTheorem)

private def instantiateMVarsAtHeader (header : DefViewElabHeader) : TermElabM DefViewElabHeader := do
  let type ← instantiateMVars header.type
  pure { header with type := type }

private def instantiateMVarsAtLetRecToLift (toLift : LetRecToLift) : TermElabM LetRecToLift := do
  let type ← instantiateMVars toLift.type
  let val ← instantiateMVars toLift.val
  pure { toLift with type := type, val := val }

private def typeHasRecFun (type : Expr) (funFVars : Array Expr) (letRecsToLift : List LetRecToLift) : Option FVarId :=
  let occ? := type.find? fun e => match e with
    | Expr.fvar fvarId _ => funFVars.contains e || letRecsToLift.any fun toLift => toLift.fvarId == fvarId
    | _ => false
  match occ? with
  | some (Expr.fvar fvarId _) => some fvarId
  | _ => none

private def getFunName (fvarId : FVarId) (letRecsToLift : List LetRecToLift) : TermElabM Name := do
  match (← findLocalDecl? fvarId) with
  | some decl => pure decl.userName
  | none =>
    /- Recall that the FVarId of nested let-recs are not in the current local context. -/
    match letRecsToLift.findSome? fun toLift => if toLift.fvarId == fvarId then some toLift.shortDeclName else none with
    | none   => throwError "unknown function"
    | some n => pure n

/-
Ensures that the of let-rec definition types do not contain functions being defined.
In principle, this test can be improved. We could perform it after we separate the set of functions is strongly connected components.
However, this extra complication doesn't seem worth it.
-/
private def checkLetRecsToLiftTypes (funVars : Array Expr) (letRecsToLift : List LetRecToLift) : TermElabM Unit :=
  letRecsToLift.forM fun toLift =>
    match typeHasRecFun toLift.type funVars letRecsToLift with
    | none        => pure ()
    | some fvarId => do
      let fnName ← getFunName fvarId letRecsToLift
      throwErrorAt toLift.ref "invalid type in 'let rec', it uses '{fnName}' which is being defined simultaneously"

namespace MutualClosure

/- A mapping from FVarId to Set of FVarIds. -/
abbrev UsedFVarsMap := NameMap NameSet

/-
Create the `UsedFVarsMap` mapping that takes the variable id for the mutually recursive functions being defined to the set of
free variables in its definition.

For `mainFVars`, this is just the set of section variables `sectionVars` used.
For nested let-rec functions, we collect their free variables.

Recall that a `let rec` expressions are encoded as follows in the elaborator.
```lean
let rec
  f : A := t,
  g : B := s;
body
```
is encoded as
```lean
let f : A := ?m₁;
let g : B := ?m₂;
body
```
where `?m₁` and `?m₂` are synthetic opaque metavariables. That are assigned by this module.
We may have nested `let rec`s.
```lean
let rec f : A :=
    let rec g : B := t;
    s;
body
```
is encoded as
```lean
let f : A := ?m₁;
body
```
and the body of `f` is stored the field `val` of a `LetRecToLift`. For the example above,
we would have a `LetRecToLift` containing:
```
{
  mvarId := m₁,
  val    := `(let g : B := ?m₂; body)
  ...
}
```
Note that `g` is not a free variable at `(let g : B := ?m₂; body)`. We recover the fact that
`f` depends on `g` because it contains `m₂`
-/
private def mkInitialUsedFVarsMap (mctx : MetavarContext) (sectionVars : Array Expr) (mainFVarIds : Array FVarId) (letRecsToLift : List LetRecToLift)
    : UsedFVarsMap := do
  let mut sectionVarSet := {}
  for var in sectionVars do
    sectionVarSet := sectionVarSet.insert var.fvarId!
  let mut usedFVarMap := {}
  for mainFVarId in mainFVarIds do
    usedFVarMap := usedFVarMap.insert mainFVarId sectionVarSet
  for toLift in letRecsToLift do
    let state := Lean.collectFVars {} toLift.val
    let state := Lean.collectFVars state toLift.type
    let mut set := state.fvarSet
    /- toLift.val may contain metavariables that are placeholders for nested let-recs. We should collect the fvarId
       for the associated let-rec because we need this information to compute the fixpoint later. -/
    let mvarIds := (toLift.val.collectMVars {}).result
    for mvarId in mvarIds do
      match letRecsToLift.findSome? fun (toLift : LetRecToLift) => if toLift.mvarId == mctx.getDelayedRoot mvarId then some toLift.fvarId else none with
      | some fvarId => set := set.insert fvarId
      | none        => pure ()
    usedFVarMap := usedFVarMap.insert toLift.fvarId set
  pure usedFVarMap

/-
The let-recs may invoke each other. Example:
```
let rec
  f (x : Nat) := g x + y
  g : Nat → Nat
    | 0   => 1
    | x+1 => f x + z
```
`y` is free variable in `f`, and `z` is a free variable in `g`.
To close `f` and `g`, `y` and `z` must be in the closure of both.
That is, we need to generate the top-level definitions.
```
def f (y z x : Nat) := g y z x + y
def g (y z : Nat) : Nat → Nat
  | 0 => 1
  | x+1 => f y z x + z
```
-/
namespace FixPoint

structure State where
  usedFVarsMap : UsedFVarsMap := {}
  modified     : Bool         := false

abbrev M := ReaderT (List FVarId) $ StateM State

private def isModified : M Bool := do pure (← get).modified
private def resetModified : M Unit := modify fun s => { s with modified := false }
private def markModified : M Unit := modify fun s => { s with modified := true }
private def getUsedFVarsMap : M UsedFVarsMap := do pure (← get).usedFVarsMap
private def modifyUsedFVars (f : UsedFVarsMap → UsedFVarsMap) : M Unit := modify fun s => { s with usedFVarsMap := f s.usedFVarsMap }

-- merge s₂ into s₁
private def merge (s₁ s₂ : NameSet) : M NameSet :=
  s₂.foldM (init := s₁) fun s₁ k => do
    if s₁.contains k then
      pure s₁
    else
      markModified
      pure $ s₁.insert k

private def updateUsedVarsOf (fvarId : FVarId) : M Unit := do
  let usedFVarsMap ← getUsedFVarsMap
  match usedFVarsMap.find? fvarId with
  | none         => pure ()
  | some fvarIds =>
    let fvarIdsNew ← fvarIds.foldM (init := fvarIds) fun fvarIdsNew fvarId' =>
      if fvarId == fvarId' then
        pure fvarIdsNew
      else
        match usedFVarsMap.find? fvarId' with
        | none => pure fvarIdsNew
          /- We are being sloppy here `otherFVarIds` may contain free variables that are
             not in the context of the let-rec associated with fvarId.
             We filter these out-of-context free variables later. -/
        | some otherFVarIds => merge fvarIdsNew otherFVarIds
    modifyUsedFVars fun usedFVars => usedFVars.insert fvarId fvarIdsNew

private partial def fixpoint : Unit → M Unit
  | _ => do
    resetModified
    let letRecFVarIds ← read
    letRecFVarIds.forM updateUsedVarsOf
    if (← isModified) then
      fixpoint ()

def run (letRecFVarIds : List FVarId) (usedFVarsMap : UsedFVarsMap) : UsedFVarsMap :=
  let (_, s) := ((fixpoint ()).run letRecFVarIds).run { usedFVarsMap := usedFVarsMap }
  s.usedFVarsMap

end FixPoint

abbrev FreeVarMap := NameMap (Array FVarId)

private def mkFreeVarMap
    (mctx : MetavarContext) (sectionVars : Array Expr) (mainFVarIds : Array FVarId)
    (recFVarIds : Array FVarId) (letRecsToLift : List LetRecToLift) : FreeVarMap := do
  let usedFVarsMap  := mkInitialUsedFVarsMap mctx sectionVars mainFVarIds letRecsToLift
  let letRecFVarIds := letRecsToLift.map fun toLift => toLift.fvarId
  let usedFVarsMap  := FixPoint.run letRecFVarIds usedFVarsMap
  let mut freeVarMap := {}
  for toLift in letRecsToLift do
    let lctx       := toLift.lctx
    let fvarIdsSet := (usedFVarsMap.find? toLift.fvarId).get!
    let fvarIds    := fvarIdsSet.fold (init := #[]) fun fvarIds fvarId =>
      if lctx.contains fvarId && !recFVarIds.contains fvarId then
        fvarIds.push fvarId
      else
        fvarIds
    freeVarMap := freeVarMap.insert toLift.fvarId fvarIds
  pure freeVarMap

structure ClosureState where
  newLocalDecls : Array LocalDecl := #[]
  localDecls    : Array LocalDecl := #[]
  newLetDecls   : Array LocalDecl := #[]
  exprArgs      : Array Expr      := #[]

private def pickMaxFVar? (lctx : LocalContext) (fvarIds : Array FVarId) : Option FVarId :=
  fvarIds.getMax? fun fvarId₁ fvarId₂ => (lctx.get! fvarId₁).index < (lctx.get! fvarId₂).index

private def preprocess (e : Expr) : TermElabM Expr := do
  let e ← instantiateMVars e
  -- which let-decls are dependent. We say a let-decl is dependent if its lambda abstraction is type incorrect.
  Meta.check e
  pure e

/- Push free variables in `s` to `toProcess` if they are not already there. -/
private def pushNewVars (toProcess : Array FVarId) (s : CollectFVars.State) : Array FVarId :=
  s.fvarSet.fold (init := toProcess) fun toProcess fvarId =>
    if toProcess.contains fvarId then toProcess else toProcess.push fvarId

private def pushLocalDecl (toProcess : Array FVarId) (fvarId : FVarId) (userName : Name) (type : Expr) (bi := BinderInfo.default)
    : StateRefT ClosureState TermElabM (Array FVarId) := do
  let type ← preprocess type
  modify fun s => { s with
    newLocalDecls := s.newLocalDecls.push $ LocalDecl.cdecl arbitrary fvarId userName type bi,
    exprArgs      := s.exprArgs.push (mkFVar fvarId)
  }
  pure $ pushNewVars toProcess (collectFVars {} type)

private partial def mkClosureForAux (toProcess : Array FVarId) : StateRefT ClosureState TermElabM Unit := do
  let lctx ← getLCtx
  match pickMaxFVar? lctx toProcess with
  | none        => pure ()
  | some fvarId =>
    trace[Elab.definition.mkClosure] "toProcess: {toProcess.map mkFVar}, maxVar: {mkFVar fvarId}"
    let toProcess := toProcess.erase fvarId
    let localDecl ← getLocalDecl fvarId
    match localDecl with
    | LocalDecl.cdecl _ _ userName type bi =>
      let toProcess ← pushLocalDecl toProcess fvarId userName type bi
      mkClosureForAux toProcess
    | LocalDecl.ldecl _ _ userName type val _ =>
      let zetaFVarIds ← getZetaFVarIds
      if !zetaFVarIds.contains fvarId then
        /- Non-dependent let-decl. See comment at src/Lean/Meta/Closure.lean -/
        let toProcess ← pushLocalDecl toProcess fvarId userName type
        mkClosureForAux toProcess
      else
        /- Dependent let-decl. -/
        let type ← preprocess type
        let val  ← preprocess val
        modify fun s => { s with
          newLetDecls   := s.newLetDecls.push $ LocalDecl.ldecl arbitrary fvarId userName type val false,
          /- We don't want to interleave let and lambda declarations in our closure. So, we expand any occurrences of fvarId
             at `newLocalDecls` and `localDecls` -/
          newLocalDecls := s.newLocalDecls.map (replaceFVarIdAtLocalDecl fvarId val),
          localDecls := s.localDecls.map (replaceFVarIdAtLocalDecl fvarId val)
        }
        mkClosureForAux (pushNewVars toProcess (collectFVars (collectFVars {} type) val))

private partial def mkClosureFor (freeVars : Array FVarId) (localDecls : Array LocalDecl) : TermElabM ClosureState := do
  let (_, s) ← (mkClosureForAux freeVars).run { localDecls := localDecls }
  pure { s with
    newLocalDecls := s.newLocalDecls.reverse,
    newLetDecls   := s.newLetDecls.reverse,
    exprArgs      := s.exprArgs.reverse
  }

structure LetRecClosure where
  ref        : Syntax
  localDecls : Array LocalDecl
  closed     : Expr -- expression used to replace occurrences of the let-rec FVarId
  toLift     : LetRecToLift

private def mkLetRecClosureFor (toLift : LetRecToLift) (freeVars : Array FVarId) : TermElabM LetRecClosure := do
  let lctx := toLift.lctx
  withLCtx lctx toLift.localInstances do
  lambdaTelescope toLift.val fun xs val => do
    let type ← instantiateForall toLift.type xs
    let lctx ← getLCtx
    let s ← mkClosureFor freeVars $ xs.map fun x => lctx.get! x.fvarId!
    let type := Closure.mkForall s.localDecls $ Closure.mkForall s.newLetDecls type
    let val  := Closure.mkLambda s.localDecls $ Closure.mkLambda s.newLetDecls val
    let c    := mkAppN (Lean.mkConst toLift.declName) s.exprArgs
    assignExprMVar toLift.mvarId c
    return {
      ref        := toLift.ref
      localDecls := s.newLocalDecls
      closed     := c
      toLift     := { toLift with val := val, type := type }
    }

private def mkLetRecClosures (letRecsToLift : List LetRecToLift) (freeVarMap : FreeVarMap) : TermElabM (List LetRecClosure) :=
  letRecsToLift.mapM fun toLift => mkLetRecClosureFor toLift (freeVarMap.find? toLift.fvarId).get!

/- Mapping from FVarId of mutually recursive functions being defined to "closure" expression. -/
abbrev Replacement := NameMap Expr

def insertReplacementForMainFns (r : Replacement) (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainFVars : Array Expr) : Replacement :=
  mainFVars.size.fold (init := r) fun i r =>
    r.insert mainFVars[i].fvarId! (mkAppN (Lean.mkConst mainHeaders[i].declName) sectionVars)


def insertReplacementForLetRecs (r : Replacement) (letRecClosures : List LetRecClosure) : Replacement :=
  letRecClosures.foldl (init := r) fun r c =>
    r.insert c.toLift.fvarId c.closed

def Replacement.apply (r : Replacement) (e : Expr) : Expr :=
  e.replace fun e => match e with
    | Expr.fvar fvarId _ => match r.find? fvarId with
      | some c => some c
      | _      => none
    | _ => none

def pushMain (preDefs : Array PreDefinition) (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainVals : Array Expr)
    : TermElabM (Array PreDefinition) :=
  mainHeaders.size.foldM (init := preDefs) fun i preDefs => do
    let header := mainHeaders[i]
    let val  ← mkLambdaFVars sectionVars mainVals[i]
    let type ← mkForallFVars sectionVars header.type
    return preDefs.push {
      ref         := getDeclarationSelectionRef header.ref
      kind        := header.kind
      declName    := header.declName
      levelParams := [], -- we set it later
      modifiers   := header.modifiers
      type        := type
      value       := val
    }

def pushLetRecs (preDefs : Array PreDefinition) (letRecClosures : List LetRecClosure) (kind : DefKind) (modifiers : Modifiers) : Array PreDefinition :=
  letRecClosures.foldl (init := preDefs) fun preDefs c =>
    let type := Closure.mkForall c.localDecls c.toLift.type
    let val  := Closure.mkLambda c.localDecls c.toLift.val
    preDefs.push {
      ref         := c.ref
      kind        := kind
      declName    := c.toLift.declName
      levelParams := [] -- we set it later
      modifiers   := { modifiers with attrs := c.toLift.attrs }
      type        := type
      value       := val
    }

def getKindForLetRecs (mainHeaders : Array DefViewElabHeader) : DefKind :=
  if mainHeaders.any fun h => h.kind.isTheorem then DefKind.«theorem»
  else DefKind.«def»

def getModifiersForLetRecs (mainHeaders : Array DefViewElabHeader) : Modifiers := {
  isNoncomputable := mainHeaders.any fun h => h.modifiers.isNoncomputable,
  isPartial       := mainHeaders.any fun h => h.modifiers.isPartial,
  isUnsafe        := mainHeaders.any fun h => h.modifiers.isUnsafe
}

/-
- `sectionVars`:   The section variables used in the `mutual` block.
- `mainHeaders`:   The elaborated header of the top-level definitions being defined by the mutual block.
- `mainFVars`:     The auxiliary variables used to represent the top-level definitions being defined by the mutual block.
- `mainVals`:      The elaborated value for the top-level definitions
- `letRecsToLift`: The let-rec's definitions that need to be lifted
-/
def main (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainFVars : Array Expr) (mainVals : Array Expr) (letRecsToLift : List LetRecToLift)
    : TermElabM (Array PreDefinition) := do
  -- Store in recFVarIds the fvarId of every function being defined by the mutual block.
  let mainFVarIds := mainFVars.map Expr.fvarId!
  let recFVarIds  := (letRecsToLift.toArray.map fun toLift => toLift.fvarId) ++ mainFVarIds
  -- Compute the set of free variables (excluding `recFVarIds`) for each let-rec.
  let mctx ← getMCtx
  let freeVarMap := mkFreeVarMap mctx sectionVars mainFVarIds recFVarIds letRecsToLift
  resetZetaFVarIds
  withTrackingZeta do
    -- By checking `toLift.type` and `toLift.val` we populate `zetaFVarIds`. See comments at `src/Lean/Meta/Closure.lean`.
    letRecsToLift.forM fun toLift => withLCtx toLift.lctx toLift.localInstances do Meta.check toLift.type; Meta.check toLift.val
    let letRecClosures ← mkLetRecClosures letRecsToLift freeVarMap
    -- mkLetRecClosures assign metavariables that were placeholders for the lifted declarations.
    let mainVals    ← mainVals.mapM (instantiateMVars ·)
    let mainHeaders ← mainHeaders.mapM instantiateMVarsAtHeader
    let letRecClosures ← letRecClosures.mapM fun closure => do pure { closure with toLift := (← instantiateMVarsAtLetRecToLift closure.toLift) }
    -- Replace fvarIds for functions being defined with closed terms
    let r              := insertReplacementForMainFns {} sectionVars mainHeaders mainFVars
    let r              := insertReplacementForLetRecs r letRecClosures
    let mainVals       := mainVals.map r.apply
    let mainHeaders    := mainHeaders.map fun h => { h with type := r.apply h.type }
    let letRecClosures := letRecClosures.map fun c => { c with toLift := { c.toLift with type := r.apply c.toLift.type, val := r.apply c.toLift.val } }
    let letRecKind     := getKindForLetRecs mainHeaders
    let letRecMods     := getModifiersForLetRecs mainHeaders
    pushMain (pushLetRecs #[] letRecClosures letRecKind letRecMods) sectionVars mainHeaders mainVals

end MutualClosure

private def getAllUserLevelNames (headers : Array DefViewElabHeader) : List Name :=
  if h : 0 < headers.size then
    -- Recall that all top-level functions must have the same levels. See `check` method above
    (headers.get ⟨0, h⟩).levelNames
  else
    []

/-- Eagerly convert universe metavariables occurring in theorem headers to universe parameters. -/
private def levelMVarToParamHeaders (views : Array DefView) (headers : Array DefViewElabHeader) : TermElabM (Array DefViewElabHeader) := do
  let rec process : StateRefT Nat TermElabM (Array DefViewElabHeader) := do
    let mut newHeaders := #[]
    for view in views, header in headers do
      if view.kind.isTheorem then
        newHeaders := newHeaders.push { header with type := (← levelMVarToParam' header.type) }
      else
        newHeaders := newHeaders.push header
    return newHeaders
  let newHeaders ← process.run' 1
  newHeaders.mapM fun header => return { header with type := (← instantiateMVars header.type) }

def elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM Unit :=
  if isExample views then
    withoutModifyingEnv go
  else
    go
where go := do
  let scopeLevelNames ← getLevelNames
  let headers ← elabHeaders views
  let headers ← levelMVarToParamHeaders views headers
  let allUserLevelNames := getAllUserLevelNames headers
  withFunLocalDecls headers fun funFVars => do
    let values ← elabFunValues headers
    Term.synthesizeSyntheticMVarsNoPostponing
    let values ← values.mapM (instantiateMVars ·)
    let headers ← headers.mapM instantiateMVarsAtHeader
    let letRecsToLift ← getLetRecsToLift
    let letRecsToLift ← letRecsToLift.mapM instantiateMVarsAtLetRecToLift
    checkLetRecsToLiftTypes funFVars letRecsToLift
    withUsed vars headers values letRecsToLift fun vars => do
      let preDefs ← MutualClosure.main vars headers funFVars values letRecsToLift
      let preDefs ← levelMVarToParamPreDecls preDefs
      let preDefs ← instantiateMVarsAtPreDecls preDefs
      let preDefs ← fixLevelParams preDefs scopeLevelNames allUserLevelNames
      addPreDefinitions preDefs

end Term
namespace Command

def elabMutualDef (ds : Array Syntax) : CommandElabM Unit := do
  let views ← ds.mapM fun d => do
    let modifiers ← elabModifiers d[0]
    mkDefView modifiers d[1]
  runTermElabM none fun vars => Term.elabMutualDef vars views

end Command
end Lean.Elab