/-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude

universe u v w

@[inline] def id {α : Sort u} (a : α) : α := a

@[inline] def Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) : α → δ :=
  fun x => f (g x)

@[inline] def Function.const {α : Sort u} (β : Sort v) (a : α) : β → α :=
  fun x => a

set_option checkBinderAnnotations false in
@[reducible] def inferInstance {α : Sort u} [i : α] : α := i
set_option checkBinderAnnotations false in
@[reducible] def inferInstanceAs (α : Sort u) [i : α] : α := i

set_option bootstrap.inductiveCheckResultingUniverse false in
inductive PUnit : Sort u where
  | unit : PUnit

/-- An abbreviation for `PUnit.{0}`, its most common instantiation.
    This Type should be preferred over `PUnit` where possible to avoid
    unnecessary universe parameters. -/
abbrev Unit : Type := PUnit

@[matchPattern] abbrev Unit.unit : Unit := PUnit.unit

/-- Auxiliary unsafe constant used by the Compiler when erasing proofs from code. -/
unsafe axiom lcProof {α : Prop} : α

/-- Auxiliary unsafe constant used by the Compiler to mark unreachable code. -/
unsafe axiom lcUnreachable {α : Sort u} : α

inductive True : Prop where
  | intro : True

inductive False : Prop

inductive Empty : Type

set_option bootstrap.inductiveCheckResultingUniverse false in
inductive PEmpty : Sort u where

def Not (a : Prop) : Prop := a → False

@[macroInline] def False.elim {C : Sort u} (h : False) : C :=
  False.rec (fun _ => C) h

@[macroInline] def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : Not a) : b :=
  False.elim (h₂ h₁)

inductive Eq {α : Sort u} (a : α) : α → Prop where
  | refl {} : Eq a a

@[simp] abbrev Eq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : α → Sort u1} (m : motive a) {b : α} (h : Eq a b) : motive b :=
  Eq.rec (motive := fun α _ => motive α) m h

@[matchPattern] def rfl {α : Sort u} {a : α} : Eq a a := Eq.refl a

@[simp] theorem id_eq (a : α) : Eq (id a) a := rfl

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

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

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

@[macroInline] def cast {α β : Sort u} (h : Eq α β) (a : α) : β :=
  Eq.rec (motive := fun α _ => α) a h

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

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

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

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

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

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

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

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

inductive HEq {α : Sort u} (a : α) : {β : Sort u} → β → Prop where
  | refl {} : HEq a a

@[matchPattern] protected def HEq.rfl {α : Sort u} {a : α} : HEq a a :=
  HEq.refl a

theorem eq_of_heq {α : Sort u} {a a' : α} (h : HEq a a') : Eq a a' :=
  have : (α β : Sort u) → (a : α) → (b : β) → HEq a b → (h : Eq α β) → Eq (cast h a) b :=
    fun α β a b h₁ =>
      HEq.rec (motive := fun {β} (b : β) (h : HEq a b) => (h₂ : Eq α β) → Eq (cast h₂ a) b)
        (fun (h₂ : Eq α α) => rfl)
        h₁
  this α α a a' h rfl

structure Prod (α : Type u) (β : Type v) where
  fst : α
  snd : β

attribute [unbox] Prod

/-- Similar to `Prod`, but `α` and `β` can be propositions.
   We use this Type internally to automatically generate the brecOn recursor. -/
structure PProd (α : Sort u) (β : Sort v) where
  fst : α
  snd : β

/-- Similar to `Prod`, but `α` and `β` are in the same universe. -/
structure MProd (α β : Type u) where
  fst : α
  snd : β

structure And (a b : Prop) : Prop where
  intro :: (left : a) (right : b)

inductive Or (a b : Prop) : Prop where
  | inl (h : a) : Or a b
  | inr (h : b) : Or a b

theorem Or.intro_left (b : Prop) (h : a) : Or a b :=
  Or.inl h

theorem Or.intro_right (a : Prop) (h : b) : Or a b :=
  Or.inr h

theorem Or.elim {c : Prop} (h : Or a b) (left : a → c) (right : b → c) : c :=
  match h with
  | Or.inl h => left h
  | Or.inr h => right h

inductive Bool : Type where
  | false : Bool
  | true : Bool

export Bool (false true)

/- Remark: Subtype must take a Sort instead of Type because of the axiom strongIndefiniteDescription. -/
structure Subtype {α : Sort u} (p : α → Prop) where
  val : α
  property : p val

/-- Gadget for optional parameter support. -/
@[reducible] def optParam (α : Sort u) (default : α) : Sort u := α

/-- Gadget for marking output parameters in type classes. -/
@[reducible] def outParam (α : Sort u) : Sort u := α

/-- Auxiliary Declaration used to implement the notation (a : α) -/
@[reducible] def typedExpr (α : Sort u) (a : α) : α := a

/-- Auxiliary Declaration used to implement the named patterns `x@h:p` -/
@[reducible] def namedPattern {α : Sort u} (x a : α) (h : Eq x a) : α := a

/- Auxiliary axiom used to implement `sorry`. -/
@[extern "lean_sorry", neverExtract]
axiom sorryAx (α : Sort u) (synthetic := true) : α

theorem eq_false_of_ne_true : {b : Bool} → Not (Eq b true) → Eq b false
  | true, h => False.elim (h rfl)
  | false, h => rfl

theorem eq_true_of_ne_false : {b : Bool} → Not (Eq b false) → Eq b true
  | true, h => rfl
  | false, h => False.elim (h rfl)

theorem ne_false_of_eq_true : {b : Bool} → Eq b true → Not (Eq b false)
  | true, _  => fun h => Bool.noConfusion h
  | false, h => Bool.noConfusion h

theorem ne_true_of_eq_false : {b : Bool} → Eq b false → Not (Eq b true)
  | true, h  => Bool.noConfusion h
  | false, _ => fun h => Bool.noConfusion h

class Inhabited (α : Sort u) where
  mk {} :: (default : α)

export Inhabited (default)

class inductive Nonempty (α : Sort u) : Prop where
  | intro (val : α) : Nonempty α

axiom Classical.choice {α : Sort u} : Nonempty α → α

protected def Nonempty.elim {α : Sort u} {p : Prop} (h₁ : Nonempty α) (h₂ : α → p) : p :=
  h₂ h₁.1

instance {α : Sort u} [Inhabited α] : Nonempty α :=
  ⟨default⟩

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

instance (α : Sort u) {β : Sort v} [Nonempty β] : Nonempty (α → β) :=
  Nonempty.intro fun _ => Classical.ofNonempty

instance (α : Sort u) {β : α → Sort v} [(a : α) → Nonempty (β a)] : Nonempty ((a : α) → β a) :=
  Nonempty.intro fun _ => Classical.ofNonempty

instance : Inhabited (Sort u) where
  default := PUnit

instance (α : Sort u) {β : Sort v} [Inhabited β] : Inhabited (α → β) where
  default := fun _ => default

instance (α : Sort u) {β : α → Sort v} [(a : α) → Inhabited (β a)] : Inhabited ((a : α) → β a) where
  default := fun _ => default

deriving instance Inhabited for Bool

/-- Universe lifting operation from Sort to Type -/
structure PLift (α : Sort u) : Type u where
  up :: (down : α)

/- Bijection between α and PLift α -/
theorem PLift.up_down {α : Sort u} : ∀ (b : PLift α), Eq (up (down b)) b
  | up a => rfl

theorem PLift.down_up {α : Sort u} (a : α) : Eq (down (up a)) a :=
  rfl

/- Pointed types -/
def NonemptyType := Subtype fun α : Type u => Nonempty α

abbrev NonemptyType.type (type : NonemptyType.{u}) : Type u :=
  type.val

instance : Inhabited NonemptyType.{u} where
  default := ⟨PUnit.{u+1}, Nonempty.intro ⟨⟩⟩

/-- Universe lifting operation -/
structure ULift.{r, s} (α : Type s) : Type (max s r) where
  up :: (down : α)

/- Bijection between α and ULift.{v} α -/
theorem ULift.up_down {α : Type u} : ∀ (b : ULift.{v} α), Eq (up (down b)) b
  | up a => rfl

theorem ULift.down_up {α : Type u} (a : α) : Eq (down (up.{v} a)) a :=
  rfl

class inductive Decidable (p : Prop) where
  | isFalse (h : Not p) : Decidable p
  | isTrue  (h : p) : Decidable p

@[inlineIfReduce, nospecialize] def Decidable.decide (p : Prop) [h : Decidable p] : Bool :=
  Decidable.casesOn (motive := fun _ => Bool) h (fun _ => false) (fun _ => true)

export Decidable (isTrue isFalse decide)

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

abbrev DecidableRel {α : Sort u} (r : α → α → Prop) :=
  (a b : α) → Decidable (r a b)

abbrev DecidableEq (α : Sort u) :=
  (a b : α) → Decidable (Eq a b)

def decEq {α : Sort u} [s : DecidableEq α] (a b : α) : Decidable (Eq a b) :=
  s a b

theorem decide_eq_true : [s : Decidable p] → p → Eq (decide p) true
  | isTrue  _, _   => rfl
  | isFalse h₁, h₂ => absurd h₂ h₁

theorem decide_eq_false : [s : Decidable p] → Not p → Eq (decide p) false
  | isTrue  h₁, h₂ => absurd h₁ h₂
  | isFalse h, _   => rfl

theorem of_decide_eq_true [s : Decidable p] : Eq (decide p) true → p := fun h =>
  match (generalizing := false) s with
  | isTrue  h₁ => h₁
  | isFalse h₁ => absurd h (ne_true_of_eq_false (decide_eq_false h₁))

theorem of_decide_eq_false [s : Decidable p] : Eq (decide p) false → Not p := fun h =>
  match (generalizing := false) s with
  | isTrue  h₁ => absurd h (ne_false_of_eq_true (decide_eq_true h₁))
  | isFalse h₁ => h₁

@[inline] instance : DecidableEq Bool :=
  fun a b => match a, b with
   | false, false => isTrue rfl
   | false, true  => isFalse (fun h => Bool.noConfusion h)
   | true, false  => isFalse (fun h => Bool.noConfusion h)
   | true, true   => isTrue rfl

class BEq (α : Type u) where
  beq : α → α → Bool

open BEq (beq)

instance [DecidableEq α] : BEq α where
  beq a b := decide (Eq a b)

-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
@[macroInline] def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : Not c → α) : α :=
  Decidable.casesOn (motive := fun _ => α) h e t

/- if-then-else -/

@[macroInline] def ite {α : Sort u} (c : Prop) [h : Decidable c] (t e : α) : α :=
  Decidable.casesOn (motive := fun _ => α) h (fun _ => e) (fun _ => t)

@[macroInline] instance {p q} [dp : Decidable p] [dq : Decidable q] : Decidable (And p q) :=
  match dp with
  | isTrue  hp =>
    match dq with
    | isTrue hq  => isTrue ⟨hp, hq⟩
    | isFalse hq => isFalse (fun h => hq (And.right h))
  | isFalse hp =>
    isFalse (fun h => hp (And.left h))

@[macroInline] instance [dp : Decidable p] [dq : Decidable q] : Decidable (Or p q) :=
  match dp with
  | isTrue  hp => isTrue (Or.inl hp)
  | isFalse hp =>
    match dq with
    | isTrue hq  => isTrue (Or.inr hq)
    | isFalse hq =>
      isFalse fun h => match h with
        | Or.inl h => hp h
        | Or.inr h => hq h

instance [dp : Decidable p] : Decidable (Not p) :=
  match dp with
  | isTrue hp  => isFalse (absurd hp)
  | isFalse hp => isTrue hp

/- Boolean operators -/

@[macroInline] def cond {α : Type u} (c : Bool) (x y : α) : α :=
  match c with
  | true  => x
  | false => y

@[macroInline] def or (x y : Bool) : Bool :=
  match x with
  | true  => true
  | false => y

@[macroInline] def and (x y : Bool) : Bool :=
  match x with
  | false => false
  | true  => y

@[inline] def not : Bool → Bool
  | true  => false
  | false => true

inductive Nat where
  | zero : Nat
  | succ (n : Nat) : Nat

instance : Inhabited Nat where
  default := Nat.zero

/- For numeric literals notation -/
class OfNat (α : Type u) (n : Nat) where
  ofNat : α

@[defaultInstance 100] /- low prio -/
instance (n : Nat) : OfNat Nat n where
  ofNat := n

class LE (α : Type u) where le : α → α → Prop
class LT (α : Type u) where lt : α → α → Prop

@[reducible] def GE.ge {α : Type u} [LE α] (a b : α) : Prop := LE.le b a
@[reducible] def GT.gt {α : Type u} [LT α] (a b : α) : Prop := LT.lt b a

@[inline] def max [LT α] [DecidableRel (@LT.lt α _)] (a b : α) : α :=
  ite (LT.lt b a) a b

@[inline] def min [LE α] [DecidableRel (@LE.le α _)] (a b : α) : α :=
  ite (LE.le a b) a b

/-- Transitive chaining of proofs, used e.g. by `calc`. -/
class Trans (r : α → β → Prop) (s : β → γ → Prop) (t : outParam (α → γ → Prop)) where
  trans : r a b → s b c → t a c

export Trans (trans)

instance (r : α → γ → Prop) : Trans Eq r r where
  trans heq h' := heq ▸ h'

instance (r : α → β → Prop) : Trans r Eq r where
  trans h' heq := heq ▸ h'

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

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

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

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

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

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

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

class HOrElse (α : Type u) (β : Type v) (γ : outParam (Type w)) where
  hOrElse : α → (Unit → β) → γ

class HAndThen (α : Type u) (β : Type v) (γ : outParam (Type w)) where
  hAndThen : α → (Unit → β) → γ

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

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

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

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

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

class Add (α : Type u) where
  add : α → α → α

class Sub (α : Type u) where
  sub : α → α → α

class Mul (α : Type u) where
  mul : α → α → α

class Neg (α : Type u) where
  neg : α → α

class Div (α : Type u) where
  div : α → α → α

class Mod (α : Type u) where
  mod : α → α → α

class Pow (α : Type u) (β : Type v) where
  pow : α → β → α

class Append (α : Type u) where
  append : α → α → α

class OrElse (α : Type u) where
  orElse  : α → (Unit → α) → α

class AndThen (α : Type u) where
  andThen : α → (Unit → α) → α

class AndOp (α : Type u) where
  and : α → α → α

class Xor (α : Type u) where
  xor : α → α → α

class OrOp (α : Type u) where
  or : α → α → α

class Complement (α : Type u) where
  complement : α → α

class ShiftLeft (α : Type u) where
  shiftLeft : α → α → α

class ShiftRight (α : Type u) where
  shiftRight : α → α → α

@[defaultInstance]
instance [Add α] : HAdd α α α where
  hAdd a b := Add.add a b

@[defaultInstance]
instance [Sub α] : HSub α α α where
  hSub a b := Sub.sub a b

@[defaultInstance]
instance [Mul α] : HMul α α α where
  hMul a b := Mul.mul a b

@[defaultInstance]
instance [Div α] : HDiv α α α where
  hDiv a b := Div.div a b

@[defaultInstance]
instance [Mod α] : HMod α α α where
  hMod a b := Mod.mod a b

@[defaultInstance]
instance [Pow α β] : HPow α β α where
  hPow a b := Pow.pow a b

@[defaultInstance]
instance [Append α] : HAppend α α α where
  hAppend a b := Append.append a b

@[defaultInstance]
instance [OrElse α] : HOrElse α α α where
  hOrElse a b := OrElse.orElse a b

@[defaultInstance]
instance [AndThen α] : HAndThen α α α where
  hAndThen a b := AndThen.andThen a b

@[defaultInstance]
instance [AndOp α] : HAnd α α α where
  hAnd a b := AndOp.and a b

@[defaultInstance]
instance [Xor α] : HXor α α α where
  hXor a b := Xor.xor a b

@[defaultInstance]
instance [OrOp α] : HOr α α α where
  hOr a b := OrOp.or a b

@[defaultInstance]
instance [ShiftLeft α] : HShiftLeft α α α where
  hShiftLeft a b := ShiftLeft.shiftLeft a b

@[defaultInstance]
instance [ShiftRight α] : HShiftRight α α α where
  hShiftRight a b := ShiftRight.shiftRight a b

open HAdd (hAdd)
open HMul (hMul)
open HPow (hPow)
open HAppend (hAppend)

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_add"]
protected def Nat.add : (@& Nat) → (@& Nat) → Nat
  | a, Nat.zero   => a
  | a, Nat.succ b => Nat.succ (Nat.add a b)

instance : Add Nat where
  add := Nat.add

/- We mark the following definitions as pattern to make sure they can be used in recursive equations,
   and reduced by the equation Compiler. -/
attribute [matchPattern] Nat.add Add.add HAdd.hAdd Neg.neg

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_mul"]
protected def Nat.mul : (@& Nat) → (@& Nat) → Nat
  | a, 0          => 0
  | a, Nat.succ b => Nat.add (Nat.mul a b) a

instance : Mul Nat where
  mul := Nat.mul

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_pow"]
protected def Nat.pow (m : @& Nat) : (@& Nat) → Nat
  | 0      => 1
  | succ n => Nat.mul (Nat.pow m n) m

instance : Pow Nat Nat where
  pow := Nat.pow

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_dec_eq"]
def Nat.beq : (@& Nat) → (@& Nat) → Bool
  | zero,   zero   => true
  | zero,   succ m => false
  | succ n, zero   => false
  | succ n, succ m => beq n m

theorem Nat.eq_of_beq_eq_true : {n m : Nat} → Eq (beq n m) true → Eq n m
  | zero,   zero,   h => rfl
  | zero,   succ m, h => Bool.noConfusion h
  | succ n, zero,   h => Bool.noConfusion h
  | succ n, succ m, h =>
    have : Eq (beq n m) true := h
    have : Eq n m := eq_of_beq_eq_true this
    this ▸ rfl

theorem Nat.ne_of_beq_eq_false : {n m : Nat} → Eq (beq n m) false → Not (Eq n m)
  | zero,   zero,   h₁, h₂ => Bool.noConfusion h₁
  | zero,   succ m, h₁, h₂ => Nat.noConfusion h₂
  | succ n, zero,   h₁, h₂ => Nat.noConfusion h₂
  | succ n, succ m, h₁, h₂ =>
    have : Eq (beq n m) false := h₁
    Nat.noConfusion h₂ (fun h₂ => absurd h₂ (ne_of_beq_eq_false this))

@[extern "lean_nat_dec_eq"]
protected def Nat.decEq (n m : @& Nat) : Decidable (Eq n m) :=
  match h:beq n m with
  | true  => isTrue (eq_of_beq_eq_true h)
  | false => isFalse (ne_of_beq_eq_false h)

@[inline] instance : DecidableEq Nat := Nat.decEq

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_dec_le"]
def Nat.ble : @& Nat → @& Nat → Bool
  | zero,   zero   => true
  | zero,   succ m => true
  | succ n, zero   => false
  | succ n, succ m => ble n m

protected inductive Nat.le (n : Nat) : Nat → Prop
  | refl     : Nat.le n n
  | step {m} : Nat.le n m → Nat.le n (succ m)

instance : LE Nat where
  le := Nat.le

protected def Nat.lt (n m : Nat) : Prop :=
  Nat.le (succ n) m

instance : LT Nat where
  lt := Nat.lt

theorem Nat.not_succ_le_zero : ∀ (n : Nat), LE.le (succ n) 0 → False
  | 0,      h => nomatch h
  | succ n, h => nomatch h

theorem Nat.not_lt_zero (n : Nat) : Not (LT.lt n 0) :=
  not_succ_le_zero n

theorem Nat.zero_le : (n : Nat) → LE.le 0 n
  | zero   => Nat.le.refl
  | succ n => Nat.le.step (zero_le n)

theorem Nat.succ_le_succ : LE.le n m → LE.le (succ n) (succ m)
  | Nat.le.refl   => Nat.le.refl
  | Nat.le.step h => Nat.le.step (succ_le_succ h)

theorem Nat.zero_lt_succ (n : Nat) : LT.lt 0 (succ n) :=
  succ_le_succ (zero_le n)

theorem Nat.le_step (h : LE.le n m) : LE.le n (succ m) :=
  Nat.le.step h

protected theorem Nat.le_trans {n m k : Nat} : LE.le n m → LE.le m k → LE.le n k
  | h,  Nat.le.refl    => h
  | h₁, Nat.le.step h₂ => Nat.le.step (Nat.le_trans h₁ h₂)

protected theorem Nat.lt_trans {n m k : Nat} (h₁ : LT.lt n m) : LT.lt m k → LT.lt n k :=
  Nat.le_trans (le_step h₁)

theorem Nat.le_succ (n : Nat) : LE.le n (succ n) :=
  Nat.le.step Nat.le.refl

theorem Nat.le_succ_of_le {n m : Nat} (h : LE.le n m) : LE.le n (succ m) :=
  Nat.le_trans h (le_succ m)

protected theorem Nat.le_refl (n : Nat) : LE.le n n :=
  Nat.le.refl

theorem Nat.succ_pos (n : Nat) : LT.lt 0 (succ n) :=
  zero_lt_succ n

set_option bootstrap.genMatcherCode false in
@[extern c inline "lean_nat_sub(#1, lean_box(1))"]
def Nat.pred : (@& Nat) → Nat
  | 0      => 0
  | succ a => a

theorem Nat.pred_le_pred : {n m : Nat} → LE.le n m → LE.le (pred n) (pred m)
  | _,           _, Nat.le.refl   => Nat.le.refl
  | 0,      succ m, Nat.le.step h => h
  | succ n, succ m, Nat.le.step h => Nat.le_trans (le_succ _) h

theorem Nat.le_of_succ_le_succ {n m : Nat} : LE.le (succ n) (succ m) → LE.le n m :=
  pred_le_pred

theorem Nat.le_of_lt_succ {m n : Nat} : LT.lt m (succ n) → LE.le m n :=
  le_of_succ_le_succ

protected theorem Nat.eq_or_lt_of_le : {n m: Nat} → LE.le n m → Or (Eq n m) (LT.lt n m)
  | zero,   zero,   h => Or.inl rfl
  | zero,   succ n, h => Or.inr (Nat.succ_le_succ (Nat.zero_le _))
  | succ n, zero,   h => absurd h (not_succ_le_zero _)
  | succ n, succ m, h =>
    have : LE.le n m := Nat.le_of_succ_le_succ h
    match Nat.eq_or_lt_of_le this with
    | Or.inl h => Or.inl (h ▸ rfl)
    | Or.inr h => Or.inr (succ_le_succ h)

protected theorem Nat.lt_or_ge (n m : Nat) : Or (LT.lt n m) (GE.ge n m) :=
  match m with
  | zero   => Or.inr (zero_le n)
  | succ m =>
    match Nat.lt_or_ge n m with
    | Or.inl h => Or.inl (le_succ_of_le h)
    | Or.inr h =>
      match Nat.eq_or_lt_of_le h with
      | Or.inl h1 => Or.inl (h1 ▸ Nat.le_refl _)
      | Or.inr h1 => Or.inr h1

theorem Nat.not_succ_le_self : (n : Nat) → Not (LE.le (succ n) n)
  | 0      => not_succ_le_zero _
  | succ n => fun h => absurd (le_of_succ_le_succ h) (not_succ_le_self n)

protected theorem Nat.lt_irrefl (n : Nat) : Not (LT.lt n n) :=
  Nat.not_succ_le_self n

protected theorem Nat.lt_of_le_of_lt {n m k : Nat} (h₁ : LE.le n m) (h₂ : LT.lt m k) : LT.lt n k :=
  Nat.le_trans (Nat.succ_le_succ h₁) h₂

protected theorem Nat.le_antisymm {n m : Nat} (h₁ : LE.le n m) (h₂ : LE.le m n) : Eq n m :=
  match h₁ with
  | Nat.le.refl   => rfl
  | Nat.le.step h => absurd (Nat.lt_of_le_of_lt h h₂) (Nat.lt_irrefl n)

protected theorem Nat.lt_of_le_of_ne {n m : Nat} (h₁ : LE.le n m) (h₂ : Not (Eq n m)) : LT.lt n m :=
  match Nat.lt_or_ge n m with
  | Or.inl h₃ => h₃
  | Or.inr h₃ => absurd (Nat.le_antisymm h₁ h₃) h₂

theorem Nat.le_of_ble_eq_true (h : Eq (Nat.ble n m) true) : LE.le n m :=
  match n, m with
  | 0,      _      => Nat.zero_le _
  | succ _, succ _ => Nat.succ_le_succ (le_of_ble_eq_true h)

theorem Nat.ble_self_eq_true : (n : Nat) → Eq (Nat.ble n n) true
  | 0      => rfl
  | succ n => ble_self_eq_true n

theorem Nat.ble_succ_eq_true : {n m : Nat} → Eq (Nat.ble n m) true → Eq (Nat.ble n (succ m)) true
  | 0,      _,      _ => rfl
  | succ n, succ m, h => ble_succ_eq_true (n := n) h

theorem Nat.ble_eq_true_of_le (h : LE.le n m) : Eq (Nat.ble n m) true :=
  match h with
  | Nat.le.refl   => Nat.ble_self_eq_true n
  | Nat.le.step h => Nat.ble_succ_eq_true (ble_eq_true_of_le h)

theorem Nat.not_le_of_not_ble_eq_true (h : Not (Eq (Nat.ble n m) true)) : Not (LE.le n m) :=
  fun h' => absurd (Nat.ble_eq_true_of_le h') h

@[extern "lean_nat_dec_le"]
instance Nat.decLe (n m : @& Nat) : Decidable (LE.le n m) :=
  dite (Eq (Nat.ble n m) true) (fun h => isTrue (Nat.le_of_ble_eq_true h)) (fun h => isFalse (Nat.not_le_of_not_ble_eq_true h))

@[extern "lean_nat_dec_lt"]
instance Nat.decLt (n m : @& Nat) : Decidable (LT.lt n m) :=
  decLe (succ n) m

set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_sub"]
protected def Nat.sub : (@& Nat) → (@& Nat) → Nat
  | a, 0      => a
  | a, succ b => pred (Nat.sub a b)

instance : Sub Nat where
  sub := Nat.sub

@[extern "lean_system_platform_nbits"] constant System.Platform.getNumBits : Unit → Subtype fun (n : Nat) => Or (Eq n 32) (Eq n 64) :=
  fun _ => ⟨64, Or.inr rfl⟩ -- inhabitant

def System.Platform.numBits : Nat :=
  (getNumBits ()).val

theorem System.Platform.numBits_eq : Or (Eq numBits 32) (Eq numBits 64) :=
  (getNumBits ()).property

structure Fin (n : Nat) where
  val  : Nat
  isLt : LT.lt val n

theorem Fin.eq_of_val_eq {n} : ∀ {i j : Fin n}, Eq i.val j.val → Eq i j
  | ⟨v, h⟩, ⟨_, _⟩, rfl => rfl

theorem Fin.val_eq_of_eq {n} {i j : Fin n} (h : Eq i j) : Eq i.val j.val :=
  h ▸ rfl

theorem Fin.ne_of_val_ne {n} {i j : Fin n} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
  fun h' => absurd (val_eq_of_eq h') h

instance (n : Nat) : DecidableEq (Fin n) :=
  fun i j =>
    match decEq i.val j.val with
    | isTrue h  => isTrue (Fin.eq_of_val_eq h)
    | isFalse h => isFalse (Fin.ne_of_val_ne h)

instance {n} : LT (Fin n) where
  lt a b := LT.lt a.val b.val

instance {n} : LE (Fin n) where
  le a b := LE.le a.val b.val

instance Fin.decLt {n} (a b : Fin n) : Decidable (LT.lt a b)  := Nat.decLt ..
instance Fin.decLe {n} (a b : Fin n) : Decidable (LE.le a b) := Nat.decLe ..

def UInt8.size : Nat := 256
structure UInt8 where
  val : Fin UInt8.size

attribute [extern "lean_uint8_of_nat_mk"] UInt8.mk
attribute [extern "lean_uint8_to_nat"] UInt8.val

@[extern "lean_uint8_of_nat"]
def UInt8.ofNatCore (n : @& Nat) (h : LT.lt n UInt8.size) : UInt8 := {
  val := { val := n, isLt := h }
}

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint8_dec_eq"]
def UInt8.decEq (a b : UInt8) : Decidable (Eq a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ =>
    dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt8.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq UInt8 := UInt8.decEq

instance : Inhabited UInt8 where
  default := UInt8.ofNatCore 0 (by decide)

def UInt16.size : Nat := 65536
structure UInt16 where
  val : Fin UInt16.size

attribute [extern "lean_uint16_of_nat_mk"] UInt16.mk
attribute [extern "lean_uint16_to_nat"] UInt16.val

@[extern "lean_uint16_of_nat"]
def UInt16.ofNatCore (n : @& Nat) (h : LT.lt n UInt16.size) : UInt16 := {
  val := { val := n, isLt := h }
}

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint16_dec_eq"]
def UInt16.decEq (a b : UInt16) : Decidable (Eq a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ =>
    dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt16.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq UInt16 := UInt16.decEq

instance : Inhabited UInt16 where
  default := UInt16.ofNatCore 0 (by decide)

def UInt32.size : Nat := 4294967296
structure UInt32 where
  val : Fin UInt32.size

attribute [extern "lean_uint32_of_nat_mk"] UInt32.mk
attribute [extern "lean_uint32_to_nat"] UInt32.val

@[extern "lean_uint32_of_nat"]
def UInt32.ofNatCore (n : @& Nat) (h : LT.lt n UInt32.size) : UInt32 := {
  val := { val := n, isLt := h }
}

@[extern "lean_uint32_to_nat"]
def UInt32.toNat (n : UInt32) : Nat := n.val.val

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint32_dec_eq"]
def UInt32.decEq (a b : UInt32) : Decidable (Eq a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ =>
    dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt32.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq UInt32 := UInt32.decEq

instance : Inhabited UInt32 where
  default := UInt32.ofNatCore 0 (by decide)

instance : LT UInt32 where
  lt a b := LT.lt a.val b.val

instance : LE UInt32 where
  le a b := LE.le a.val b.val

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint32_dec_lt"]
def UInt32.decLt (a b : UInt32) : Decidable (LT.lt a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LT.lt n m))

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint32_dec_le"]
def UInt32.decLe (a b : UInt32) : Decidable (LE.le a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LE.le n m))

instance (a b : UInt32) : Decidable (LT.lt a b) := UInt32.decLt a b
instance (a b : UInt32) : Decidable (LE.le a b) := UInt32.decLe a b

def UInt64.size : Nat := 18446744073709551616
structure UInt64 where
  val : Fin UInt64.size

attribute [extern "lean_uint64_of_nat_mk"] UInt64.mk
attribute [extern "lean_uint64_to_nat"] UInt64.val

@[extern "lean_uint64_of_nat"]
def UInt64.ofNatCore (n : @& Nat) (h : LT.lt n UInt64.size) : UInt64 := {
  val := { val := n, isLt := h }
}

set_option bootstrap.genMatcherCode false in
@[extern "lean_uint64_dec_eq"]
def UInt64.decEq (a b : UInt64) : Decidable (Eq a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ =>
    dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt64.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq UInt64 := UInt64.decEq

instance : Inhabited UInt64 where
  default := UInt64.ofNatCore 0 (by decide)

def USize.size : Nat := hPow 2 System.Platform.numBits

theorem usize_size_eq : Or (Eq USize.size 4294967296) (Eq USize.size 18446744073709551616) :=
  show Or (Eq (hPow 2 System.Platform.numBits) 4294967296) (Eq (hPow 2 System.Platform.numBits) 18446744073709551616) from
  match System.Platform.numBits, System.Platform.numBits_eq with
  | _, Or.inl rfl => Or.inl (by decide)
  | _, Or.inr rfl => Or.inr (by decide)

structure USize where
  val : Fin USize.size

attribute [extern "lean_usize_of_nat_mk"] USize.mk
attribute [extern "lean_usize_to_nat"] USize.val

@[extern "lean_usize_of_nat"]
def USize.ofNatCore (n : @& Nat) (h : LT.lt n USize.size) : USize := {
  val := { val := n, isLt := h }
}

set_option bootstrap.genMatcherCode false in
@[extern "lean_usize_dec_eq"]
def USize.decEq (a b : USize) : Decidable (Eq a b) :=
  match a, b with
  | ⟨n⟩, ⟨m⟩ =>
    dite (Eq n m) (fun h =>isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => USize.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq USize := USize.decEq

instance : Inhabited USize where
  default := USize.ofNatCore 0 (match USize.size, usize_size_eq with
    | _, Or.inl rfl => by decide
    | _, Or.inr rfl => by decide)

@[extern "lean_usize_of_nat"]
def USize.ofNat32 (n : @& Nat) (h : LT.lt n 4294967296) : USize := {
  val := {
    val  := n
    isLt := match USize.size, usize_size_eq with
      | _, Or.inl rfl => h
      | _, Or.inr rfl => Nat.lt_trans h (by decide)
  }
}

abbrev Nat.isValidChar (n : Nat) : Prop :=
  Or (LT.lt n 0xd800) (And (LT.lt 0xdfff n) (LT.lt n 0x110000))

abbrev UInt32.isValidChar (n : UInt32) : Prop :=
  n.toNat.isValidChar

/-- The `Char` Type represents an unicode scalar value.
    See http://www.unicode.org/glossary/#unicode_scalar_value). -/
structure Char where
  val   : UInt32
  valid : val.isValidChar

private theorem isValidChar_UInt32 {n : Nat} (h : n.isValidChar) : LT.lt n UInt32.size :=
  match h with
  | Or.inl h      => Nat.lt_trans h (by decide)
  | Or.inr ⟨_, h⟩ => Nat.lt_trans h (by decide)

@[extern "lean_uint32_of_nat"]
def Char.ofNatAux (n : @& Nat) (h : n.isValidChar) : Char :=
  { val := ⟨{ val := n, isLt := isValidChar_UInt32 h }⟩, valid := h }

@[noinline, matchPattern]
def Char.ofNat (n : Nat) : Char :=
  dite (n.isValidChar)
    (fun h => Char.ofNatAux n h)
    (fun _ => { val := ⟨{ val := 0, isLt := by decide }⟩, valid := Or.inl (by decide) })

theorem Char.eq_of_val_eq : ∀ {c d : Char}, Eq c.val d.val → Eq c d
  | ⟨v, h⟩, ⟨_, _⟩, rfl => rfl

theorem Char.val_eq_of_eq : ∀ {c d : Char}, Eq c d → Eq c.val d.val
  | _, _, rfl => rfl

theorem Char.ne_of_val_ne {c d : Char} (h : Not (Eq c.val d.val)) : Not (Eq c d) :=
  fun h' => absurd (val_eq_of_eq h') h

theorem Char.val_ne_of_ne {c d : Char} (h : Not (Eq c d)) : Not (Eq c.val d.val) :=
  fun h' => absurd (eq_of_val_eq h') h

instance : DecidableEq Char :=
  fun c d =>
    match decEq c.val d.val with
    | isTrue h  => isTrue (Char.eq_of_val_eq h)
    | isFalse h => isFalse (Char.ne_of_val_ne h)

def Char.utf8Size (c : Char) : UInt32 :=
  let v := c.val
  ite (LE.le v (UInt32.ofNatCore 0x7F (by decide)))
    (UInt32.ofNatCore 1 (by decide))
    (ite (LE.le v (UInt32.ofNatCore 0x7FF (by decide)))
      (UInt32.ofNatCore 2 (by decide))
      (ite (LE.le v (UInt32.ofNatCore 0xFFFF (by decide)))
        (UInt32.ofNatCore 3 (by decide))
        (UInt32.ofNatCore 4 (by decide))))

inductive Option (α : Type u) where
  | none : Option α
  | some (val : α) : Option α

attribute [unbox] Option

export Option (none some)

instance {α} : Inhabited (Option α) where
  default := none

@[macroInline] def Option.getD : Option α → α → α
  | some x, _ => x
  | none,   e => e

inductive List (α : Type u) where
  | nil : List α
  | cons (head : α) (tail : List α) : List α

instance {α} : Inhabited (List α) where
  default := List.nil

protected def List.hasDecEq {α: Type u} [DecidableEq α] : (a b : List α) → Decidable (Eq a b)
  | nil,       nil       => isTrue rfl
  | cons a as, nil       => isFalse (fun h => List.noConfusion h)
  | nil,       cons b bs => isFalse (fun h => List.noConfusion h)
  | cons a as, cons b bs =>
    match decEq a b with
    | isTrue hab  =>
      match List.hasDecEq as bs with
      | isTrue habs  => isTrue (hab ▸ habs ▸ rfl)
      | isFalse nabs => isFalse (fun h => List.noConfusion h (fun _ habs => absurd habs nabs))
    | isFalse nab => isFalse (fun h => List.noConfusion h (fun hab _ => absurd hab nab))

instance {α : Type u} [DecidableEq α] : DecidableEq (List α) := List.hasDecEq

@[specialize]
def List.foldl {α β} (f : α → β → α) : (init : α) → List β → α
  | a, nil      => a
  | a, cons b l => foldl f (f a b) l

def List.set : List α → Nat → α → List α
  | cons a as, 0,          b => cons b as
  | cons a as, Nat.succ n, b => cons a (set as n b)
  | nil,       _,          _ => nil

def List.length : List α → Nat
  | nil       => 0
  | cons a as => HAdd.hAdd (length as) 1

def List.lengthTRAux : List α → Nat → Nat
  | nil,       n => n
  | cons a as, n => lengthTRAux as (Nat.succ n)

def List.lengthTR (as : List α) : Nat :=
  lengthTRAux as 0

@[simp] theorem List.length_cons {α} (a : α) (as : List α) : Eq (cons a as).length as.length.succ :=
  rfl

def List.concat {α : Type u} : List α → α → List α
  | nil,       b => cons b nil
  | cons a as, b => cons a (concat as b)

def List.get {α : Type u} : (as : List α) → (i : Nat) → LT.lt i as.length → α
  | nil,       i,          h => absurd h (Nat.not_lt_zero _)
  | cons a as, 0,          h => a
  | cons a as, Nat.succ i, h =>
    have : LT.lt i.succ as.length.succ := length_cons .. ▸ h
    get as i (Nat.le_of_succ_le_succ this)

structure String where
  data : List Char

attribute [extern "lean_string_mk"] String.mk
attribute [extern "lean_string_data"] String.data

@[extern "lean_string_dec_eq"]
def String.decEq (s₁ s₂ : @& String) : Decidable (Eq s₁ s₂) :=
  match s₁, s₂ with
  | ⟨s₁⟩, ⟨s₂⟩ =>
    dite (Eq s₁ s₂) (fun h => isTrue (congrArg _ h)) (fun h => isFalse (fun h' => String.noConfusion h' (fun h' => absurd h' h)))

instance : DecidableEq String := String.decEq

/-- A byte position in a `String`. Internally, `String`s are UTF-8 encoded.
Codepoint positions (counting the Unicode codepoints rather than bytes)
are represented by plain `Nat`s instead.
Indexing a `String` by a byte position is constant-time, while codepoint
positions need to be translated internally to byte positions in linear-time. -/
abbrev String.Pos := Nat

structure Substring where
  str : String
  startPos : String.Pos
  stopPos : String.Pos

instance : Inhabited Substring where
  default := ⟨"", 0, 0⟩

@[inline] def Substring.bsize : Substring → Nat
  | ⟨_, b, e⟩ => e.sub b

def String.csize (c : Char) : Nat :=
  c.utf8Size.toNat

private def String.utf8ByteSizeAux : List Char → Nat → Nat
  | List.nil,       r => r
  | List.cons c cs, r => utf8ByteSizeAux cs (hAdd r (csize c))

@[extern "lean_string_utf8_byte_size"]
def String.utf8ByteSize : (@& String) → Nat
  | ⟨s⟩ => utf8ByteSizeAux s 0

@[inline] def String.bsize (s : String) : Nat :=
  utf8ByteSize s

@[inline] def String.toSubstring (s : String) : Substring := {
  str      := s
  startPos := 0
  stopPos  := s.bsize
}

unsafe def unsafeCast {α : Type u} {β : Type v} (a : α) : β :=
  ULift.down.{max u v} (cast lcProof (ULift.up.{max u v} a))

@[neverExtract, extern "lean_panic_fn"]
constant panicCore {α : Type u} [Inhabited α] (msg : String) : α

/-
  This is workaround for `panic` occurring in monadic code. See issue #695.
  The `panicCore` definition cannot be specialized since it is an extern.
  When `panic` occurs in monadic code, the `Inhabited α` parameter depends on a `[inst : Monad m]` instance.
  The `inst` parameter will not be eliminated during specialization if it occurs inside of a binder (to avoid work duplication), and
  will prevent the the actual monad from being "copied" to the code being specialized. When we reimplement the specializer, we
  may consider copying `inst` if it also occurs outside binders or if it is an instance.
-/
@[noinline, neverExtract]
def panic {α : Type u} [Inhabited α] (msg : String) : α :=
  panicCore msg

-- TODO: this be applied directly to `Inhabited`'s definition when we remove the above workaround
attribute [nospecialize] Inhabited

/-
The Compiler has special support for arrays.
They are implemented using dynamic arrays: https://en.wikipedia.org/wiki/Dynamic_array
-/
structure Array (α : Type u) where
  data : List α

attribute [extern "lean_array_data"] Array.data
attribute [extern "lean_array_mk"] Array.mk

/- The parameter `c` is the initial capacity -/
@[extern "lean_mk_empty_array_with_capacity"]
def Array.mkEmpty {α : Type u} (c : @& Nat) : Array α := {
  data := List.nil
}

def Array.empty {α : Type u} : Array α :=
  mkEmpty 0

@[reducible, extern "lean_array_get_size"]
def Array.size {α : Type u} (a : @& Array α) : Nat :=
 a.data.length

@[extern "lean_array_fget"]
def Array.get {α : Type u} (a : @& Array α) (i : @& Fin a.size) : α :=
  a.data.get i.val i.isLt

@[inline] def Array.getD (a : Array α) (i : Nat) (v₀ : α) : α :=
  dite (LT.lt i a.size) (fun h => a.get ⟨i, h⟩) (fun _ => v₀)

/- "Comfortable" version of `fget`. It performs a bound check at runtime. -/
@[extern "lean_array_get"]
def Array.get! {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
  Array.getD a i default

def Array.getOp {α : Type u} [Inhabited α] (self : Array α) (idx : Nat) : α :=
  self.get! idx

@[extern "lean_array_push"]
def Array.push {α : Type u} (a : Array α) (v : α) : Array α := {
  data := List.concat a.data v
}

@[extern "lean_array_fset"]
def Array.set (a : Array α) (i : @& Fin a.size) (v : α) : Array α := {
  data := a.data.set i.val v
}

@[inline] def Array.setD (a : Array α) (i : Nat) (v : α) : Array α :=
  dite (LT.lt i a.size) (fun h => a.set ⟨i, h⟩ v) (fun _ => a)

@[extern "lean_array_set"]
def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
  Array.setD a i v

-- Slower `Array.append` used in quotations.
protected def Array.appendCore {α : Type u}  (as : Array α) (bs : Array α) : Array α :=
  let rec loop (i : Nat) (j : Nat) (as : Array α) : Array α :=
    dite (LT.lt j bs.size)
      (fun hlt =>
        match i with
        | 0           => as
        | Nat.succ i' => loop i' (hAdd j 1) (as.push (bs.get ⟨j, hlt⟩)))
      (fun _ => as)
  loop bs.size 0 as

@[inlineIfReduce]
def List.toArrayAux : List α → Array α → Array α
  | nil,       r => r
  | cons a as, r => toArrayAux as (r.push a)

@[inlineIfReduce]
def List.redLength : List α → Nat
  | nil       => 0
  | cons _ as => as.redLength.succ

@[inline, matchPattern, export lean_list_to_array]
def List.toArray (as : List α) : Array α :=
  as.toArrayAux (Array.mkEmpty as.redLength)

class Bind (m : Type u → Type v) where
  bind : {α β : Type u} → m α → (α → m β) → m β

export Bind (bind)

class Pure (f : Type u → Type v) where
  pure {α : Type u} : α → f α

export Pure (pure)

class Functor (f : Type u → Type v) : Type (max (u+1) v) where
  map      : {α β : Type u} → (α → β) → f α → f β
  mapConst : {α β : Type u} → α → f β → f α := Function.comp map (Function.const _)

class Seq (f : Type u → Type v) : Type (max (u+1) v) where
  seq  : {α β : Type u} → f (α → β) → (Unit → f α) → f β

class SeqLeft (f : Type u → Type v) : Type (max (u+1) v) where
  seqLeft : {α β : Type u} → f α → (Unit → f β) → f α

class SeqRight (f : Type u → Type v) : Type (max (u+1) v) where
  seqRight : {α β : Type u} → f α → (Unit → f β) → f β

class Applicative (f : Type u → Type v) extends Functor f, Pure f, Seq f, SeqLeft f, SeqRight f where
  map      := fun x y => Seq.seq (pure x) fun _ => y
  seqLeft  := fun a b => Seq.seq (Functor.map (Function.const _) a) b
  seqRight := fun a b => Seq.seq (Functor.map (Function.const _ id) a) b

class Monad (m : Type u → Type v) extends Applicative m, Bind m : Type (max (u+1) v) where
  map      f x := bind x (Function.comp pure f)
  seq      f x := bind f fun y => Functor.map y (x ())
  seqLeft  x y := bind x fun a => bind (y ()) (fun _ => pure a)
  seqRight x y := bind x fun _ => y ()

instance {α : Type u} {m : Type u → Type v} [Monad m] : Inhabited (α → m α) where
  default := pure

instance {α : Type u} {m : Type u → Type v} [Monad m] [Inhabited α] : Inhabited (m α) where
  default := pure default

-- A fusion of Haskell's `sequence` and `map`
def Array.sequenceMap {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m β) : m (Array β) :=
  let rec loop (i : Nat) (j : Nat) (bs : Array β) : m (Array β) :=
    dite (LT.lt j as.size)
      (fun hlt =>
        match i with
        | 0           => pure bs
        | Nat.succ i' => Bind.bind (f (as.get ⟨j, hlt⟩)) fun b => loop i' (hAdd j 1) (bs.push b))
      (fun _ => pure bs)
  loop as.size 0 Array.empty

/-- A Function for lifting a computation from an inner Monad to an outer Monad.
    Like [MonadTrans](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Class.html),
    but `n` does not have to be a monad transformer.
    Alternatively, an implementation of [MonadLayer](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLayer) without `layerInvmap` (so far). -/
class MonadLift (m : Type u → Type v) (n : Type u → Type w) where
  monadLift : {α : Type u} → m α → n α

/-- The reflexive-transitive closure of `MonadLift`.
    `monadLift` is used to transitively lift monadic computations such as `StateT.get` or `StateT.put s`.
    Corresponds to [MonadLift](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLift). -/
class MonadLiftT (m : Type u → Type v) (n : Type u → Type w) where
  monadLift : {α : Type u} → m α → n α

export MonadLiftT (monadLift)

abbrev liftM := @monadLift

instance (m n o) [MonadLift n o] [MonadLiftT m n] : MonadLiftT m o where
  monadLift x := MonadLift.monadLift (m := n) (monadLift x)

instance (m) : MonadLiftT m m where
  monadLift x := x

/-- A functor in the category of monads. Can be used to lift monad-transforming functions.
    Based on pipes' [MFunctor](https://hackage.haskell.org/package/pipes-2.4.0/docs/Control-MFunctor.html),
    but not restricted to monad transformers.
    Alternatively, an implementation of [MonadTransFunctor](http://duairc.netsoc.ie/layers-docs/Control-Monad-Layer.html#t:MonadTransFunctor). -/
class MonadFunctor (m : Type u → Type v) (n : Type u → Type w) where
  monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α

/-- The reflexive-transitive closure of `MonadFunctor`.
    `monadMap` is used to transitively lift Monad morphisms -/
class MonadFunctorT (m : Type u → Type v) (n : Type u → Type w) where
  monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α

export MonadFunctorT (monadMap)

instance (m n o) [MonadFunctor n o] [MonadFunctorT m n] : MonadFunctorT m o where
  monadMap f := MonadFunctor.monadMap (m := n) (monadMap (m := m) f)

instance monadFunctorRefl (m) : MonadFunctorT m m where
  monadMap f := f

inductive Except (ε : Type u) (α : Type v) where
  | error : ε → Except ε α
  | ok    : α → Except ε α

attribute [unbox] Except

instance {ε : Type u} {α : Type v} [Inhabited ε] : Inhabited (Except ε α) where
  default := Except.error default

/-- An implementation of [MonadError](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Except.html#t:MonadError) -/
class MonadExceptOf (ε : Type u) (m : Type v → Type w) where
  throw {α : Type v} : ε → m α
  tryCatch {α : Type v} : m α → (ε → m α) → m α

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

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

/-- Similar to `MonadExceptOf`, but `ε` is an outParam for convenience -/
class MonadExcept (ε : outParam (Type u)) (m : Type v → Type w) where
  throw {α : Type v} : ε → m α
  tryCatch {α : Type v} : m α → (ε → m α) → m α

export MonadExcept (throw tryCatch)

instance (ε : outParam (Type u)) (m : Type v → Type w) [MonadExceptOf ε m] : MonadExcept ε m where
  throw    := throwThe ε
  tryCatch := tryCatchThe ε

namespace MonadExcept
variable {ε : Type u} {m : Type v → Type w}

@[inline] protected def orElse [MonadExcept ε m] {α : Type v} (t₁ : m α) (t₂ : Unit → m α) : m α :=
  tryCatch t₁ fun _ => t₂ ()

instance [MonadExcept ε m] {α : Type v} : OrElse (m α) where
  orElse := MonadExcept.orElse

end MonadExcept

/-- An implementation of [ReaderT](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Reader.html#t:ReaderT) -/
def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
  ρ → m α

instance (ρ : Type u) (m : Type u → Type v) (α : Type u) [Inhabited (m α)] : Inhabited (ReaderT ρ m α) where
  default := fun _ => default

@[inline] def ReaderT.run {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : ReaderT ρ m α) (r : ρ) : m α :=
  x r

namespace ReaderT

section
variable {ρ : Type u} {m : Type u → Type v} {α : Type u}

instance  : MonadLift m (ReaderT ρ m) where
  monadLift x := fun _ => x

instance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ReaderT ρ m) where
  throw e  := liftM (m := m) (throw e)
  tryCatch := fun x c r => tryCatchThe ε (x r) (fun e => (c e) r)

end

section
variable {ρ : Type u} {m : Type u → Type v} [Monad m] {α β : Type u}

@[inline] protected def read : ReaderT ρ m ρ :=
  pure

@[inline] protected def pure (a : α) : ReaderT ρ m α :=
  fun r => pure a

@[inline] protected def bind (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) : ReaderT ρ m β :=
  fun r => bind (x r) fun a => f a r

@[inline] protected def map (f : α → β) (x : ReaderT ρ m α) : ReaderT ρ m β :=
  fun r => Functor.map f (x r)

instance : Monad (ReaderT ρ m) where
  pure := ReaderT.pure
  bind := ReaderT.bind
  map  := ReaderT.map

instance (ρ m) [Monad m] : MonadFunctor m (ReaderT ρ m) where
  monadMap f x := fun ctx => f (x ctx)

@[inline] protected def adapt {ρ' : Type u} [Monad m] {α : Type u} (f : ρ' → ρ) : ReaderT ρ m α → ReaderT ρ' m α :=
  fun x r => x (f r)

end
end ReaderT

/-- An implementation of [MonadReader](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader).
    It does not contain `local` because this Function cannot be lifted using `monadLift`.
    Instead, the `MonadReaderAdapter` class provides the more general `adaptReader` Function.

    Note: This class can be seen as a simplification of the more "principled" definition
    ```
    class MonadReader (ρ : outParam (Type u)) (n : Type u → Type u) where
      lift {α : Type u} : ({m : Type u → Type u} → [Monad m] → ReaderT ρ m α) → n α
    ```
    -/
class MonadReaderOf (ρ : Type u) (m : Type u → Type v) where
  read : m ρ

@[inline] def readThe (ρ : Type u) {m : Type u → Type v} [MonadReaderOf ρ m] : m ρ :=
  MonadReaderOf.read

/-- Similar to `MonadReaderOf`, but `ρ` is an outParam for convenience -/
class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) where
  read : m ρ

export MonadReader (read)

instance (ρ : Type u) (m : Type u → Type v) [MonadReaderOf ρ m] : MonadReader ρ m where
  read := readThe ρ

instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadReaderOf ρ m] : MonadReaderOf ρ n where
  read := liftM (m := m) read

instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadReaderOf ρ (ReaderT ρ m) where
  read := ReaderT.read

class MonadWithReaderOf (ρ : Type u) (m : Type u → Type v) where
  withReader {α : Type u} : (ρ → ρ) → m α → m α

@[inline] def withTheReader (ρ : Type u) {m : Type u → Type v} [MonadWithReaderOf ρ m] {α : Type u} (f : ρ → ρ) (x : m α) : m α :=
  MonadWithReaderOf.withReader f x

class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) where
  withReader {α : Type u} : (ρ → ρ) → m α → m α

export MonadWithReader (withReader)

instance (ρ : Type u) (m : Type u → Type v) [MonadWithReaderOf ρ m] : MonadWithReader ρ m where
  withReader := withTheReader ρ

instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type v} [MonadFunctor m n] [MonadWithReaderOf ρ m] : MonadWithReaderOf ρ n where
  withReader f := monadMap (m := m) (withTheReader ρ f)

instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadWithReaderOf ρ (ReaderT ρ m) where
  withReader f x := fun ctx => x (f ctx)

/-- An implementation of [MonadState](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-State-Class.html).
    In contrast to the Haskell implementation, we use overlapping instances to derive instances
    automatically from `monadLift`. -/
class MonadStateOf (σ : Type u) (m : Type u → Type v) where
  /- Obtain the top-most State of a Monad stack. -/
  get : m σ
  /- Set the top-most State of a Monad stack. -/
  set : σ → m PUnit
  /- Map the top-most State of a Monad stack.

     Note: `modifyGet f` may be preferable to `do s <- get; let (a, s) := f s; put s; pure a`
     because the latter does not use the State linearly (without sufficient inlining). -/
  modifyGet {α : Type u} : (σ → Prod α σ) → m α

export MonadStateOf (set)

abbrev getThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] : m σ :=
  MonadStateOf.get

@[inline] abbrev modifyThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → σ) : m PUnit :=
  MonadStateOf.modifyGet fun s => (PUnit.unit, f s)

@[inline] abbrev modifyGetThe {α : Type u} (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → Prod α σ) : m α :=
  MonadStateOf.modifyGet f

/-- Similar to `MonadStateOf`, but `σ` is an outParam for convenience -/
class MonadState (σ : outParam (Type u)) (m : Type u → Type v) where
  get : m σ
  set : σ → m PUnit
  modifyGet {α : Type u} : (σ → Prod α σ) → m α

export MonadState (get modifyGet)

instance (σ : Type u) (m : Type u → Type v) [MonadStateOf σ m] : MonadState σ m where
  set         := MonadStateOf.set
  get         := getThe σ
  modifyGet f := MonadStateOf.modifyGet f

@[inline] def modify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) : m PUnit :=
  modifyGet fun s => (PUnit.unit, f s)

@[inline] def getModify {σ : Type u} {m : Type u → Type v} [MonadState σ m] [Monad m] (f : σ → σ) : m σ :=
  modifyGet fun s => (s, f s)

-- NOTE: The Ordering of the following two instances determines that the top-most `StateT` Monad layer
-- will be picked first
instance {σ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadStateOf σ m] : MonadStateOf σ n where
  get         := liftM (m := m) MonadStateOf.get
  set       s := liftM (m := m) (MonadStateOf.set s)
  modifyGet f := monadLift (m := m) (MonadState.modifyGet f)

namespace EStateM

inductive Result (ε σ α : Type u) where
  | ok    : α → σ → Result ε σ α
  | error : ε → σ → Result ε σ α

variable {ε σ α : Type u}

instance [Inhabited ε] [Inhabited σ] : Inhabited (Result ε σ α) where
  default := Result.error default default

end EStateM

open EStateM (Result) in
def EStateM (ε σ α : Type u) := σ → Result ε σ α

namespace EStateM

variable {ε σ α β : Type u}

instance [Inhabited ε] : Inhabited (EStateM ε σ α) where
  default := fun s => Result.error default s

@[inline] protected def pure (a : α) : EStateM ε σ α := fun s =>
  Result.ok a s

@[inline] protected def set (s : σ) : EStateM ε σ PUnit := fun _ =>
  Result.ok ⟨⟩ s

@[inline] protected def get : EStateM ε σ σ := fun s =>
  Result.ok s s

@[inline] protected def modifyGet (f : σ → Prod α σ) : EStateM ε σ α := fun s =>
  match f s with
  | (a, s) => Result.ok a s

@[inline] protected def throw (e : ε) : EStateM ε σ α := fun s =>
  Result.error e s

/-- Auxiliary instance for saving/restoring the "backtrackable" part of the state. -/
class Backtrackable (δ : outParam (Type u)) (σ : Type u) where
  save    : σ → δ
  restore : σ → δ → σ

@[inline] protected def tryCatch {δ} [Backtrackable δ σ] {α} (x : EStateM ε σ α) (handle : ε → EStateM ε σ α) : EStateM ε σ α := fun s =>
  let d := Backtrackable.save s
  match x s with
  | Result.error e s => handle e (Backtrackable.restore s d)
  | ok               => ok

@[inline] protected def orElse {δ} [Backtrackable δ σ] (x₁ : EStateM ε σ α) (x₂ : Unit → EStateM ε σ α) : EStateM ε σ α := fun s =>
  let d := Backtrackable.save s;
  match x₁ s with
  | Result.error _ s => x₂ () (Backtrackable.restore s d)
  | ok               => ok

@[inline] def adaptExcept {ε' : Type u} (f : ε → ε') (x : EStateM ε σ α) : EStateM ε' σ α := fun s =>
  match x s with
  | Result.error e s => Result.error (f e) s
  | Result.ok a s    => Result.ok a s

@[inline] protected def bind (x : EStateM ε σ α) (f : α → EStateM ε σ β) : EStateM ε σ β := fun s =>
  match x s with
  | Result.ok a s    => f a s
  | Result.error e s => Result.error e s

@[inline] protected def map (f : α → β) (x : EStateM ε σ α) : EStateM ε σ β := fun s =>
  match x s with
  | Result.ok a s    => Result.ok (f a) s
  | Result.error e s => Result.error e s

@[inline] protected def seqRight (x : EStateM ε σ α) (y : Unit → EStateM ε σ β) : EStateM ε σ β := fun s =>
  match x s with
  | Result.ok _ s    => y () s
  | Result.error e s => Result.error e s

instance : Monad (EStateM ε σ) where
  bind     := EStateM.bind
  pure     := EStateM.pure
  map      := EStateM.map
  seqRight := EStateM.seqRight

instance {δ} [Backtrackable δ σ] : OrElse (EStateM ε σ α) where
  orElse := EStateM.orElse

instance : MonadStateOf σ (EStateM ε σ) where
  set       := EStateM.set
  get       := EStateM.get
  modifyGet := EStateM.modifyGet

instance {δ} [Backtrackable δ σ] : MonadExceptOf ε (EStateM ε σ) where
  throw    := EStateM.throw
  tryCatch := EStateM.tryCatch

@[inline] def run (x : EStateM ε σ α) (s : σ) : Result ε σ α :=
  x s

@[inline] def run' (x : EStateM ε σ α) (s : σ) : Option α :=
  match run x s with
  | Result.ok v _   => some v
  | Result.error .. => none

@[inline] def dummySave : σ → PUnit := fun _ => ⟨⟩

@[inline] def dummyRestore : σ → PUnit → σ := fun s _ => s

/- Dummy default instance -/
instance nonBacktrackable : Backtrackable PUnit σ where
  save    := dummySave
  restore := dummyRestore

end EStateM

class Hashable (α : Sort u) where
  hash : α → UInt64

export Hashable (hash)

@[extern "lean_uint64_to_usize"]
constant UInt64.toUSize (u : UInt64) : USize

@[extern "lean_usize_to_uint64"]
constant USize.toUInt64 (u : USize) : UInt64

@[extern "lean_uint64_mix_hash"]
constant mixHash (u₁ u₂ : UInt64) : UInt64

@[extern "lean_string_hash"]
protected constant String.hash (s : @& String) : UInt64

instance : Hashable String where
  hash := String.hash

namespace Lean

/- Hierarchical names -/
inductive Name where
  | anonymous : Name
  | str : Name → String → UInt64 → Name
  | num : Name → Nat → UInt64 → Name

instance : Inhabited Name where
  default := Name.anonymous

protected def Name.hash : Name → UInt64
  | Name.anonymous => UInt64.ofNatCore 1723 (by decide)
  | Name.str p s h => h
  | Name.num p v h => h

instance : Hashable Name where
  hash := Name.hash

namespace Name

@[export lean_name_mk_string]
def mkStr (p : Name) (s : String) : Name :=
  Name.str p s (mixHash (hash p) (hash s))

@[export lean_name_mk_numeral]
def mkNum (p : Name) (v : Nat) : Name :=
  Name.num p v (mixHash (hash p) (dite (LT.lt v UInt64.size) (fun h => UInt64.ofNatCore v h) (fun _ => UInt64.ofNatCore 17 (by decide))))

def mkSimple (s : String) : Name :=
  mkStr Name.anonymous s

@[extern "lean_name_eq"]
protected def beq : (@& Name) → (@& Name) → Bool
  | anonymous,   anonymous   => true
  | str p₁ s₁ _, str p₂ s₂ _ => and (BEq.beq s₁ s₂) (Name.beq p₁ p₂)
  | num p₁ n₁ _, num p₂ n₂ _ => and (BEq.beq n₁ n₂) (Name.beq p₁ p₂)
  | _,           _           => false

instance : BEq Name where
  beq := Name.beq

protected def append : Name → Name → Name
  | n, anonymous => n
  | n, str p s _ => Name.mkStr (Name.append n p) s
  | n, num p d _ => Name.mkNum (Name.append n p) d

instance : Append Name where
  append := Name.append

end Name

/- Syntax -/

/-- Source information of tokens. -/
inductive SourceInfo where
  /-
    Token from original input with whitespace and position information.
    `leading` will be inferred after parsing by `Syntax.updateLeading`. During parsing,
    it is not at all clear what the preceding token was, especially with backtracking. -/
  | original (leading : Substring) (pos : String.Pos) (trailing : Substring) (endPos : String.Pos)
  /-
    Synthesized token (e.g. from a quotation) annotated with a span from the original source.
    In the delaborator, we "misuse" this constructor to store synthetic positions identifying
    subterms. -/
  | synthetic (pos : String.Pos) (endPos : String.Pos)
  /- Synthesized token without position information. -/
  | protected none

instance : Inhabited SourceInfo := ⟨SourceInfo.none⟩

namespace SourceInfo

def getPos? (info : SourceInfo) (originalOnly := false) : Option String.Pos :=
  match info, originalOnly with
  | original (pos := pos) ..,  _     => some pos
  | synthetic (pos := pos) .., false => some pos
  | _,                         _     => none

end SourceInfo

abbrev SyntaxNodeKind := Name

/- Syntax AST -/

/--
Syntax objects used by the parser, macro expander, delaborator, etc.
-/
inductive Syntax where
  | missing : Syntax
  | /--
  Node in the syntax tree.

  The `info` field is used by the delaborator
  to store the position of the subexpression
  corresponding to this node.
  The parser sets the `info` field to `none`.

  (Remark: the `node` constructor
  did not have an `info` field in previous versions.
  This caused a bug in the interactive widgets,
  where the popup for `a + b` was the same as for `a`.
  The delaborator used to associate subexpressions
  with pretty-printed syntax by setting
  the (string) position of the first atom/identifier
  to the (expression) position of the subexpression.
  For example, both `a` and `a + b`
  have the same first identifier,
  and so their infos got mixed up.)
  -/ node   (info : SourceInfo) (kind : SyntaxNodeKind) (args : Array Syntax) : Syntax
  | atom   (info : SourceInfo) (val : String) : Syntax
  | ident  (info : SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List (Prod Name (List String))) : Syntax

instance : Inhabited Syntax where
  default := Syntax.missing

/- Builtin kinds -/
def choiceKind : SyntaxNodeKind := `choice
def nullKind : SyntaxNodeKind := `null
def groupKind : SyntaxNodeKind := `group
def identKind : SyntaxNodeKind := `ident
def strLitKind : SyntaxNodeKind := `strLit
def charLitKind : SyntaxNodeKind := `charLit
def numLitKind : SyntaxNodeKind := `numLit
def scientificLitKind : SyntaxNodeKind := `scientificLit
def nameLitKind : SyntaxNodeKind := `nameLit
def fieldIdxKind : SyntaxNodeKind := `fieldIdx
def interpolatedStrLitKind : SyntaxNodeKind := `interpolatedStrLitKind
def interpolatedStrKind : SyntaxNodeKind := `interpolatedStrKind

namespace Syntax

def getKind (stx : Syntax) : SyntaxNodeKind :=
  match stx with
  | Syntax.node _ k args => k
  -- We use these "pseudo kinds" for antiquotation kinds.
  -- For example, an antiquotation `$id:ident` (using Lean.Parser.Term.ident)
  -- is compiled to ``if stx.isOfKind `ident ...``
  | Syntax.missing     => `missing
  | Syntax.atom _ v    => Name.mkSimple v
  | Syntax.ident ..    => identKind

def setKind (stx : Syntax) (k : SyntaxNodeKind) : Syntax :=
  match stx with
  | Syntax.node info _ args => Syntax.node info k args
  | _                       => stx

def isOfKind (stx : Syntax) (k : SyntaxNodeKind) : Bool :=
  beq stx.getKind k

def getArg (stx : Syntax) (i : Nat) : Syntax :=
  match stx with
  | Syntax.node _ _ args => args.getD i Syntax.missing
  | _                    => Syntax.missing

-- Add `stx[i]` as sugar for `stx.getArg i`
@[inline] def getOp (self : Syntax) (idx : Nat) : Syntax :=
  self.getArg idx

def getArgs (stx : Syntax) : Array Syntax :=
  match stx with
  | Syntax.node _ _ args => args
  | _                    => Array.empty

def getNumArgs (stx : Syntax) : Nat :=
  match stx with
  | Syntax.node _ _ args => args.size
  | _                    => 0

def isMissing : Syntax → Bool
  | Syntax.missing => true
  | _ => false

def isNodeOf (stx : Syntax) (k : SyntaxNodeKind) (n : Nat) : Bool :=
  and (stx.isOfKind k) (beq stx.getNumArgs n)

def isIdent : Syntax → Bool
  | ident _ _ _ _ => true
  | _             => false

def getId : Syntax → Name
  | ident _ _ val _ => val
  | _               => Name.anonymous

def matchesNull (stx : Syntax) (n : Nat) : Bool :=
  isNodeOf stx nullKind n

def matchesIdent (stx : Syntax) (id : Name) : Bool :=
  and stx.isIdent (beq stx.getId id)

def matchesLit (stx : Syntax) (k : SyntaxNodeKind) (val : String) : Bool :=
  match stx with
  | Syntax.node _ k' args => and (beq k k') (match args.getD 0 Syntax.missing with
    | Syntax.atom _ val' => beq val val'
    | _                  => false)
  | _                     => false

def setArgs (stx : Syntax) (args : Array Syntax) : Syntax :=
  match stx with
  | node info k _ => node info k args
  | stx           => stx

def setArg (stx : Syntax) (i : Nat) (arg : Syntax) : Syntax :=
  match stx with
  | node info k args => node info k (args.setD i arg)
  | stx              => stx

/-- Retrieve the left-most node or leaf's info in the Syntax tree. -/
partial def getHeadInfo? : Syntax → Option SourceInfo
  | atom info _   => some info
  | ident info .. => some info
  | node SourceInfo.none _ args   =>
    let rec loop (i : Nat) : Option SourceInfo :=
      match decide (LT.lt i args.size) with
      | true => match getHeadInfo? (args.get! i) with
         | some info => some info
         | none      => loop (hAdd i 1)
      | false => none
    loop 0
  | node info _ _ => some info
  | _             => none

/-- Retrieve the left-most leaf's info in the Syntax tree, or `none` if there is no token. -/
partial def getHeadInfo (stx : Syntax) : SourceInfo :=
  match stx.getHeadInfo? with
  | some info => info
  | none      => SourceInfo.none

def getPos? (stx : Syntax) (originalOnly := false) : Option String.Pos :=
  stx.getHeadInfo.getPos? originalOnly

partial def getTailPos? (stx : Syntax) (originalOnly := false) : Option String.Pos :=
  match stx, originalOnly with
  | atom (SourceInfo.original (endPos := pos) ..) ..,    _    => some pos
  | atom (SourceInfo.synthetic (endPos := pos) ..) _,  false  => some pos
  | ident (SourceInfo.original (endPos := pos) ..) .., _      => some pos
  | ident (SourceInfo.synthetic (endPos := pos) ..) .., false => some pos
  | node (SourceInfo.original (endPos := pos) ..) ..,    _    => some pos
  | node (SourceInfo.synthetic (endPos := pos) ..) .., false  => some pos
  | node _ _ args,                                        _     =>
    let rec loop (i : Nat) : Option String.Pos :=
      match decide (LT.lt i args.size) with
      | true => match getTailPos? (args.get! ((args.size.sub i).sub 1)) originalOnly with
         | some info => some info
         | none      => loop (hAdd i 1)
      | false => none
    loop 0
  | _, _ => none

/--
  An array of syntax elements interspersed with separators. Can be coerced to/from `Array Syntax` to automatically
  remove/insert the separators. -/
structure SepArray (sep : String) where
  elemsAndSeps : Array Syntax

end Syntax

def SourceInfo.fromRef (ref : Syntax) : SourceInfo :=
  match ref.getPos?, ref.getTailPos? with
  | some pos, some tailPos => SourceInfo.synthetic pos tailPos
  | _,        _            => SourceInfo.none

def mkAtom (val : String) : Syntax :=
  Syntax.atom SourceInfo.none val

def mkAtomFrom (src : Syntax) (val : String) : Syntax :=
  Syntax.atom (SourceInfo.fromRef src) val

/- Parser descriptions -/

inductive ParserDescr where
  | const  (name : Name)
  | unary  (name : Name) (p : ParserDescr)
  | binary (name : Name) (p₁ p₂ : ParserDescr)
  | node (kind : SyntaxNodeKind) (prec : Nat) (p : ParserDescr)
  | trailingNode (kind : SyntaxNodeKind) (prec lhsPrec : Nat) (p : ParserDescr)
  | symbol (val : String)
  | nonReservedSymbol (val : String) (includeIdent : Bool)
  | cat (catName : Name) (rbp : Nat)
  | parser (declName : Name)
  | nodeWithAntiquot (name : String) (kind : SyntaxNodeKind) (p : ParserDescr)
  | sepBy  (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)
  | sepBy1 (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)

instance : Inhabited ParserDescr where
  default := ParserDescr.symbol ""

abbrev TrailingParserDescr := ParserDescr

/-
Runtime support for making quotation terms auto-hygienic, by mangling identifiers
introduced by them with a "macro scope" supplied by the context. Details to appear in a
paper soon.
-/

abbrev MacroScope := Nat
/-- Macro scope used internally. It is not available for our frontend. -/
def reservedMacroScope := 0
/-- First macro scope available for our frontend -/
def firstFrontendMacroScope := hAdd reservedMacroScope 1

class MonadRef (m : Type → Type) where
  getRef      : m Syntax
  withRef {α} : Syntax → m α → m α

export MonadRef (getRef)

instance (m n : Type → Type) [MonadLift m n] [MonadFunctor m n] [MonadRef m] : MonadRef n where
  getRef        := liftM (getRef : m _)
  withRef ref x := monadMap (m := m) (MonadRef.withRef ref) x

def replaceRef (ref : Syntax) (oldRef : Syntax) : Syntax :=
  match ref.getPos? with
  | some _ => ref
  | _      => oldRef

@[inline] def withRef {m : Type → Type} [Monad m] [MonadRef m] {α} (ref : Syntax) (x : m α) : m α :=
  bind getRef fun oldRef =>
  let ref := replaceRef ref oldRef
  MonadRef.withRef ref x

/-- A monad that supports syntax quotations. Syntax quotations (in term
    position) are monadic values that when executed retrieve the current "macro
    scope" from the monad and apply it to every identifier they introduce
    (independent of whether this identifier turns out to be a reference to an
    existing declaration, or an actually fresh binding during further
    elaboration). We also apply the position of the result of `getRef` to each
    introduced symbol, which results in better error positions than not applying
    any position. -/
class MonadQuotation (m : Type → Type) extends MonadRef m where
  -- Get the fresh scope of the current macro invocation
  getCurrMacroScope : m MacroScope
  getMainModule     : m Name
  /- Execute action in a new macro invocation context. This transformer should be
     used at all places that morally qualify as the beginning of a "macro call",
     e.g. `elabCommand` and `elabTerm` in the case of the elaborator. However, it
     can also be used internally inside a "macro" if identifiers introduced by
     e.g. different recursive calls should be independent and not collide. While
     returning an intermediate syntax tree that will recursively be expanded by
     the elaborator can be used for the same effect, doing direct recursion inside
     the macro guarded by this transformer is often easier because one is not
     restricted to passing a single syntax tree. Modelling this helper as a
     transformer and not just a monadic action ensures that the current macro
     scope before the recursive call is restored after it, as expected. -/
  withFreshMacroScope {α : Type} : m α → m α

export MonadQuotation (getCurrMacroScope getMainModule withFreshMacroScope)

def MonadRef.mkInfoFromRefPos [Monad m] [MonadRef m] : m SourceInfo :=
  return SourceInfo.fromRef (← getRef)

instance {m n : Type → Type} [MonadFunctor m n] [MonadLift m n] [MonadQuotation m] : MonadQuotation n where
  getCurrMacroScope   := liftM (m := m) getCurrMacroScope
  getMainModule       := liftM (m := m) getMainModule
  withFreshMacroScope := monadMap (m := m) withFreshMacroScope

/-
We represent a name with macro scopes as
```
<actual name>._@.(<module_name>.<scopes>)*.<module_name>._hyg.<scopes>
```
Example: suppose the module name is `Init.Data.List.Basic`, and name is `foo.bla`, and macroscopes [2, 5]
```
foo.bla._@.Init.Data.List.Basic._hyg.2.5
```

We may have to combine scopes from different files/modules.
The main modules being processed is always the right most one.
This situation may happen when we execute a macro generated in
an imported file in the current file.
```
foo.bla._@.Init.Data.List.Basic.2.1.Init.Lean.Expr_hyg.4
```

The delimiter `_hyg` is used just to improve the `hasMacroScopes` performance.
-/

def Name.hasMacroScopes : Name → Bool
  | str _ s _   => beq s "_hyg"
  | num p _   _ => hasMacroScopes p
  | _           => false

private def eraseMacroScopesAux : Name → Name
  | Name.str p s _   => match beq s "_@" with
    | true  => p
    | false => eraseMacroScopesAux p
  | Name.num p _ _   => eraseMacroScopesAux p
  | Name.anonymous   => Name.anonymous

@[export lean_erase_macro_scopes]
def Name.eraseMacroScopes (n : Name) : Name :=
  match n.hasMacroScopes with
  | true  => eraseMacroScopesAux n
  | false => n

private def simpMacroScopesAux : Name → Name
  | Name.num p i _ => Name.mkNum (simpMacroScopesAux p) i
  | n              => eraseMacroScopesAux n

/- Helper function we use to create binder names that do not need to be unique. -/
@[export lean_simp_macro_scopes]
def Name.simpMacroScopes (n : Name) : Name :=
  match n.hasMacroScopes with
  | true  => simpMacroScopesAux n
  | false => n

structure MacroScopesView where
  name       : Name
  imported   : Name
  mainModule : Name
  scopes     : List MacroScope

instance : Inhabited MacroScopesView where
  default := ⟨default, default, default, default⟩

def MacroScopesView.review (view : MacroScopesView) : Name :=
  match view.scopes with
  | List.nil      => view.name
  | List.cons _ _ =>
    let base := (Name.mkStr (hAppend (hAppend (Name.mkStr view.name "_@") view.imported) view.mainModule) "_hyg")
    view.scopes.foldl Name.mkNum base

private def assembleParts : List Name → Name → Name
  | List.nil,                      acc => acc
  | List.cons (Name.str _ s _) ps, acc => assembleParts ps (Name.mkStr acc s)
  | List.cons (Name.num _ n _) ps, acc => assembleParts ps (Name.mkNum acc n)
  | _,                             acc => panic "Error: unreachable @ assembleParts"

private def extractImported (scps : List MacroScope) (mainModule : Name) : Name → List Name → MacroScopesView
  | n@(Name.str p str _), parts =>
    match beq str "_@" with
    | true  => { name := p, mainModule := mainModule, imported := assembleParts parts Name.anonymous, scopes := scps }
    | false => extractImported scps mainModule p (List.cons n parts)
  | n@(Name.num p str _), parts => extractImported scps mainModule p (List.cons n parts)
  | _,                    _     => panic "Error: unreachable @ extractImported"

private def extractMainModule (scps : List MacroScope) : Name → List Name → MacroScopesView
  | n@(Name.str p str _), parts =>
    match beq str "_@" with
    | true  => { name := p, mainModule := assembleParts parts Name.anonymous, imported := Name.anonymous, scopes := scps }
    | false => extractMainModule scps p (List.cons n parts)
  | n@(Name.num p num _), acc => extractImported scps (assembleParts acc Name.anonymous) n List.nil
  | _,                    _   => panic "Error: unreachable @ extractMainModule"

private def extractMacroScopesAux : Name → List MacroScope → MacroScopesView
  | Name.num p scp _, acc => extractMacroScopesAux p (List.cons scp acc)
  | Name.str p str _, acc => extractMainModule acc p List.nil -- str must be "_hyg"
  | _,                _   => panic "Error: unreachable @ extractMacroScopesAux"

/--
  Revert all `addMacroScope` calls. `v = extractMacroScopes n → n = v.review`.
  This operation is useful for analyzing/transforming the original identifiers, then adding back
  the scopes (via `MacroScopesView.review`). -/
def extractMacroScopes (n : Name) : MacroScopesView :=
  match n.hasMacroScopes with
  | true  => extractMacroScopesAux n List.nil
  | false => { name := n, scopes := List.nil, imported := Name.anonymous, mainModule := Name.anonymous }

def addMacroScope (mainModule : Name) (n : Name) (scp : MacroScope) : Name :=
  match n.hasMacroScopes with
  | true =>
    let view := extractMacroScopes n
    match beq view.mainModule mainModule with
    | true  => Name.mkNum n scp
    | false =>
      { view with
        imported   := view.scopes.foldl Name.mkNum (hAppend view.imported view.mainModule)
        mainModule := mainModule
        scopes     := List.cons scp List.nil
      }.review
  | false =>
    Name.mkNum (Name.mkStr (hAppend (Name.mkStr n "_@") mainModule) "_hyg") scp

@[inline] def MonadQuotation.addMacroScope {m : Type → Type} [MonadQuotation m] [Monad m] (n : Name) : m Name :=
  bind getMainModule     fun mainModule =>
  bind getCurrMacroScope fun scp =>
  pure (Lean.addMacroScope mainModule n scp)

def defaultMaxRecDepth := 512

def maxRecDepthErrorMessage : String :=
  "maximum recursion depth has been reached (use `set_option maxRecDepth <num>` to increase limit)"

namespace Macro

/- References -/
private constant MethodsRefPointed : NonemptyType.{0}

private def MethodsRef : Type := MethodsRefPointed.type

instance : Nonempty MethodsRef := MethodsRefPointed.property

structure Context where
  methods        : MethodsRef
  mainModule     : Name
  currMacroScope : MacroScope
  currRecDepth   : Nat := 0
  maxRecDepth    : Nat := defaultMaxRecDepth
  ref            : Syntax

inductive Exception where
  | error             : Syntax → String → Exception
  | unsupportedSyntax : Exception

structure State where
  macroScope : MacroScope
  traceMsgs  : List (Prod Name String) := List.nil
  deriving Inhabited

end Macro

abbrev MacroM := ReaderT Macro.Context (EStateM Macro.Exception Macro.State)

abbrev Macro := Syntax → MacroM Syntax

namespace Macro

instance : MonadRef MacroM where
  getRef     := bind read fun ctx => pure ctx.ref
  withRef    := fun ref x => withReader (fun ctx => { ctx with ref := ref }) x

def addMacroScope (n : Name) : MacroM Name :=
  bind read fun ctx =>
  pure (Lean.addMacroScope ctx.mainModule n ctx.currMacroScope)

def throwUnsupported {α} : MacroM α :=
  throw Exception.unsupportedSyntax

def throwError {α} (msg : String) : MacroM α :=
  bind getRef fun ref =>
  throw (Exception.error ref msg)

def throwErrorAt {α} (ref : Syntax) (msg : String) : MacroM α :=
  withRef ref (throwError msg)

@[inline] protected def withFreshMacroScope {α} (x : MacroM α) : MacroM α :=
  bind (modifyGet (fun s => (s.macroScope, { s with macroScope := hAdd s.macroScope 1 }))) fun fresh =>
  withReader (fun ctx => { ctx with currMacroScope := fresh }) x

@[inline] def withIncRecDepth {α} (ref : Syntax) (x : MacroM α) : MacroM α :=
  bind read fun ctx =>
  match beq ctx.currRecDepth ctx.maxRecDepth with
  | true  => throw (Exception.error ref maxRecDepthErrorMessage)
  | false => withReader (fun ctx => { ctx with currRecDepth := hAdd ctx.currRecDepth 1 }) x

instance : MonadQuotation MacroM where
  getCurrMacroScope ctx := pure ctx.currMacroScope
  getMainModule     ctx := pure ctx.mainModule
  withFreshMacroScope   := Macro.withFreshMacroScope

structure Methods where
  expandMacro?      : Syntax → MacroM (Option Syntax)
  getCurrNamespace  : MacroM Name
  hasDecl           : Name → MacroM Bool
  resolveNamespace? : Name → MacroM (Option Name)
  resolveGlobalName : Name → MacroM (List (Prod Name (List String)))
  deriving Inhabited

unsafe def mkMethodsImp (methods : Methods) : MethodsRef :=
  unsafeCast methods

@[implementedBy mkMethodsImp]
constant mkMethods (methods : Methods) : MethodsRef

instance : Inhabited MethodsRef where
  default := mkMethods default

unsafe def getMethodsImp : MacroM Methods :=
  bind read fun ctx => pure (unsafeCast (ctx.methods))

@[implementedBy getMethodsImp] constant getMethods : MacroM Methods

/-- `expandMacro? stx` return `some stxNew` if `stx` is a macro, and `stxNew` is its expansion. -/
def expandMacro? (stx : Syntax) : MacroM (Option Syntax) := do
  (← getMethods).expandMacro? stx

/-- Return `true` if the environment contains a declaration with name `declName` -/
def hasDecl (declName : Name) : MacroM Bool := do
  (← getMethods).hasDecl declName

def getCurrNamespace : MacroM Name := do
  (← getMethods).getCurrNamespace

def resolveNamespace? (n : Name) : MacroM (Option Name) := do
  (← getMethods).resolveNamespace? n

def resolveGlobalName (n : Name) : MacroM (List (Prod Name (List String))) := do
  (← getMethods).resolveGlobalName n

def trace (clsName : Name) (msg : String) : MacroM Unit := do
  modify fun s => { s with traceMsgs := List.cons (Prod.mk clsName msg) s.traceMsgs }

end Macro

export Macro (expandMacro?)

namespace PrettyPrinter

abbrev UnexpandM := ReaderT Syntax (EStateM Unit Unit)

/--
  Function that tries to reverse macro expansions as a post-processing step of delaboration.
  While less general than an arbitrary delaborator, it can be declared without importing `Lean`.
  Used by the `[appUnexpander]` attribute. -/
-- a `kindUnexpander` could reasonably be added later
abbrev Unexpander := Syntax → UnexpandM Syntax

instance : MonadQuotation UnexpandM where
  getRef              := read
  withRef ref x       := withReader (fun _ => ref) x
  -- unexpanders should not need to introduce new names
  getCurrMacroScope   := pure 0
  getMainModule       := pure `_fakeMod
  withFreshMacroScope := id

end PrettyPrinter

end Lean


/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Chris Hughes
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.zsqrtd.basic
import Mathlib.data.complex.basic
import Mathlib.ring_theory.principal_ideal_domain
import Mathlib.number_theory.quadratic_reciprocity
import Mathlib.PostPort

universes u_1

namespace Mathlib

/-!
# Gaussian integers

The Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both
integers.

## Main definitions

The Euclidean domain structure on `ℤ[i]` is defined in this file.

The homomorphism `to_complex` into the complex numbers is also defined in this file.

## Main statements

`prime_iff_mod_four_eq_three_of_nat_prime`
A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`

## Notations

This file uses the local notation `ℤ[i]` for `gaussian_int`

## Implementation notes

Gaussian integers are implemented using the more general definition `zsqrtd`, the type of integers
adjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties
and definitions about `zsqrtd` can easily be used.
-/

def gaussian_int := ℤ√(-1)

namespace gaussian_int

protected instance has_repr : has_repr gaussian_int :=
  has_repr.mk
    fun (x : gaussian_int) =>
      string.str string.empty
                (char.of_nat
                  (bit0
                    (bit0
                      (bit0
                        (bit1
                          (bit0 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit0 (bit0 1)))))))))))))) ++
              repr (zsqrtd.re x) ++
            string.str (string.str string.empty (char.of_nat (bit0 (bit0 (bit1 (bit1 (bit0 1)))))))
              (char.of_nat (bit0 (bit0 (bit0 (bit0 (bit0 1)))))) ++
          repr (zsqrtd.im x) ++
        string.str string.empty
          (char.of_nat
            (bit1
              (bit0
                (bit0 (bit1 (bit0 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit0 (bit0 1))))))))))))))

protected instance comm_ring : comm_ring gaussian_int := zsqrtd.comm_ring

/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/
def to_complex : gaussian_int →+* ℂ := coe_fn zsqrtd.lift { val := complex.I, property := sorry }

protected instance complex.has_coe : has_coe gaussian_int ℂ := has_coe.mk ⇑to_complex

theorem to_complex_def (x : gaussian_int) : ↑x = ↑(zsqrtd.re x) + ↑(zsqrtd.im x) * complex.I := rfl

theorem to_complex_def' (x : ℤ) (y : ℤ) : ↑(zsqrtd.mk x y) = ↑x + ↑y * complex.I := sorry

theorem to_complex_def₂ (x : gaussian_int) : ↑x = complex.mk ↑(zsqrtd.re x) ↑(zsqrtd.im x) := sorry

@[simp] theorem to_real_re (x : gaussian_int) : ↑(zsqrtd.re x) = complex.re ↑x := sorry

@[simp] theorem to_real_im (x : gaussian_int) : ↑(zsqrtd.im x) = complex.im ↑x := sorry

@[simp] theorem to_complex_re (x : ℤ) (y : ℤ) : complex.re ↑(zsqrtd.mk x y) = ↑x := sorry

@[simp] theorem to_complex_im (x : ℤ) (y : ℤ) : complex.im ↑(zsqrtd.mk x y) = ↑y := sorry

@[simp] theorem to_complex_add (x : gaussian_int) (y : gaussian_int) : ↑(x + y) = ↑x + ↑y :=
  ring_hom.map_add to_complex x y

@[simp] theorem to_complex_mul (x : gaussian_int) (y : gaussian_int) : ↑(x * y) = ↑x * ↑y :=
  ring_hom.map_mul to_complex x y

@[simp] theorem to_complex_one : ↑1 = 1 := ring_hom.map_one to_complex

@[simp] theorem to_complex_zero : ↑0 = 0 := ring_hom.map_zero to_complex

@[simp] theorem to_complex_neg (x : gaussian_int) : ↑(-x) = -↑x := ring_hom.map_neg to_complex x

@[simp] theorem to_complex_sub (x : gaussian_int) (y : gaussian_int) : ↑(x - y) = ↑x - ↑y :=
  ring_hom.map_sub to_complex x y

@[simp] theorem to_complex_inj {x : gaussian_int} {y : gaussian_int} : ↑x = ↑y ↔ x = y := sorry

@[simp] theorem to_complex_eq_zero {x : gaussian_int} : ↑x = 0 ↔ x = 0 :=
  eq.mpr (id (Eq._oldrec (Eq.refl (↑x = 0 ↔ x = 0)) (Eq.symm to_complex_zero)))
    (eq.mpr (id (Eq._oldrec (Eq.refl (↑x = ↑0 ↔ x = 0)) (propext to_complex_inj)))
      (iff.refl (x = 0)))

@[simp] theorem nat_cast_real_norm (x : gaussian_int) :
    ↑(zsqrtd.norm x) = coe_fn complex.norm_sq ↑x :=
  sorry

@[simp] theorem nat_cast_complex_norm (x : gaussian_int) :
    ↑(zsqrtd.norm x) = ↑(coe_fn complex.norm_sq ↑x) :=
  sorry

theorem norm_nonneg (x : gaussian_int) : 0 ≤ zsqrtd.norm x :=
  zsqrtd.norm_nonneg
    (eq.mpr (id (eq_true_intro (neg_nonpos_of_nonneg (norm_num.nonneg_pos 1 zero_lt_one'))))
      trivial)
    x

@[simp] theorem norm_eq_zero {x : gaussian_int} : zsqrtd.norm x = 0 ↔ x = 0 := sorry

theorem norm_pos {x : gaussian_int} : 0 < zsqrtd.norm x ↔ x ≠ 0 := sorry

@[simp] theorem coe_nat_abs_norm (x : gaussian_int) :
    ↑(int.nat_abs (zsqrtd.norm x)) = zsqrtd.norm x :=
  int.nat_abs_of_nonneg (norm_nonneg x)

@[simp] theorem nat_cast_nat_abs_norm {α : Type u_1} [ring α] (x : gaussian_int) :
    ↑(int.nat_abs (zsqrtd.norm x)) = ↑(zsqrtd.norm x) :=
  sorry

theorem nat_abs_norm_eq (x : gaussian_int) :
    int.nat_abs (zsqrtd.norm x) =
        int.nat_abs (zsqrtd.re x) * int.nat_abs (zsqrtd.re x) +
          int.nat_abs (zsqrtd.im x) * int.nat_abs (zsqrtd.im x) :=
  sorry

protected def div (x : gaussian_int) (y : gaussian_int) : gaussian_int :=
  let n : ℚ := rat.of_int (zsqrtd.norm y)⁻¹;
  let c : ℤ√(-1) := zsqrtd.conj y;
  zsqrtd.mk (round (rat.of_int (zsqrtd.re (x * c)) * n))
    (round (rat.of_int (zsqrtd.im (x * c)) * n))

protected instance has_div : Div gaussian_int := { div := gaussian_int.div }

theorem div_def (x : gaussian_int) (y : gaussian_int) :
    x / y =
        zsqrtd.mk (round (↑(zsqrtd.re (x * zsqrtd.conj y)) / ↑(zsqrtd.norm y)))
          (round (↑(zsqrtd.im (x * zsqrtd.conj y)) / ↑(zsqrtd.norm y))) :=
  sorry

theorem to_complex_div_re (x : gaussian_int) (y : gaussian_int) :
    complex.re ↑(x / y) = ↑(round (complex.re (↑x / ↑y))) :=
  sorry

theorem to_complex_div_im (x : gaussian_int) (y : gaussian_int) :
    complex.im ↑(x / y) = ↑(round (complex.im (↑x / ↑y))) :=
  sorry

theorem norm_sq_le_norm_sq_of_re_le_of_im_le {x : ℂ} {y : ℂ}
    (hre : abs (complex.re x) ≤ abs (complex.re y))
    (him : abs (complex.im x) ≤ abs (complex.im y)) :
    coe_fn complex.norm_sq x ≤ coe_fn complex.norm_sq y :=
  sorry

theorem norm_sq_div_sub_div_lt_one (x : gaussian_int) (y : gaussian_int) :
    coe_fn complex.norm_sq (↑x / ↑y - ↑(x / y)) < 1 :=
  sorry

protected def mod (x : gaussian_int) (y : gaussian_int) : gaussian_int := x - y * (x / y)

protected instance has_mod : Mod gaussian_int := { mod := gaussian_int.mod }

theorem mod_def (x : gaussian_int) (y : gaussian_int) : x % y = x - y * (x / y) := rfl

theorem norm_mod_lt (x : gaussian_int) {y : gaussian_int} (hy : y ≠ 0) :
    zsqrtd.norm (x % y) < zsqrtd.norm y :=
  sorry

theorem nat_abs_norm_mod_lt (x : gaussian_int) {y : gaussian_int} (hy : y ≠ 0) :
    int.nat_abs (zsqrtd.norm (x % y)) < int.nat_abs (zsqrtd.norm y) :=
  sorry

theorem norm_le_norm_mul_left (x : gaussian_int) {y : gaussian_int} (hy : y ≠ 0) :
    int.nat_abs (zsqrtd.norm x) ≤ int.nat_abs (zsqrtd.norm (x * y)) :=
  sorry

protected instance nontrivial : nontrivial gaussian_int :=
  nontrivial.mk (Exists.intro 0 (Exists.intro 1 (of_as_true trivial)))

protected instance euclidean_domain : euclidean_domain gaussian_int :=
  euclidean_domain.mk comm_ring.add comm_ring.add_assoc comm_ring.zero comm_ring.zero_add
    comm_ring.add_zero comm_ring.neg comm_ring.sub comm_ring.add_left_neg comm_ring.add_comm
    comm_ring.mul comm_ring.mul_assoc comm_ring.one comm_ring.one_mul comm_ring.mul_one
    comm_ring.left_distrib comm_ring.right_distrib comm_ring.mul_comm nontrivial.exists_pair_ne
    Div.div sorry Mod.mod sorry (measure (int.nat_abs ∘ zsqrtd.norm)) sorry nat_abs_norm_mod_lt
    sorry

theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : fact (nat.prime p)] (hpi : prime ↑p) :
    p % bit0 (bit0 1) = bit1 1 :=
  sorry

theorem sum_two_squares_of_nat_prime_of_not_irreducible (p : ℕ) [hp : fact (nat.prime p)]
    (hpi : ¬irreducible ↑p) : ∃ (a : ℕ), ∃ (b : ℕ), a ^ bit0 1 + b ^ bit0 1 = p :=
  sorry

theorem prime_of_nat_prime_of_mod_four_eq_three (p : ℕ) [hp : fact (nat.prime p)]
    (hp3 : p % bit0 (bit0 1) = bit1 1) : prime ↑p :=
  sorry

/-- A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4` -/
theorem prime_iff_mod_four_eq_three_of_nat_prime (p : ℕ) [hp : fact (nat.prime p)] :
    prime ↑p ↔ p % bit0 (bit0 1) = bit1 1 :=
  { mp := mod_four_eq_three_of_nat_prime_of_prime p,
    mpr := prime_of_nat_prime_of_mod_four_eq_three p }

end Mathlib


/- 2018-10-12T21:52:34 Locally nameless formulas, built off Floris' preterms
---Jesse

I wanted to avoid the headache over shifts (at the cost of needing to define a well-formedness predicate on terms); my implementation of locally nameless formulas is based off Chargueraud's paper (thanks for the pointer Jeremy!).

TODO:
 - finish writing the coercion for variable lists
 - finish writing the substitution functions
 - add a well-formedness predicate
-/

structure Language :=
(relations : Π n : nat, Type) (functions : Π  n : nat, Type)
section
parameter L : Language

/- preterm n is a partially applied term. if applied to n terms, it becomes a term -/
inductive preterm  : ℕ → Type
| bvar : ℕ → preterm 0
| fvar : string → preterm 0
| func : ∀ {n : nat}, L.functions n → preterm n
| apply : ∀ {n : nat}, preterm (n + 1) → preterm 0 → preterm n

open preterm
def term := preterm 0

-- /- raise_depth_term _ t n m raises variables in t which are at least m by n -/
-- def raise_depth_term : ∀ {l}, preterm l → ℕ → ℕ → preterm l
-- | _ (var L k) n m := if m ≤ k then var (k+m) else var k
-- | _ (func f) n m := func f
-- | _ (apply t1 t2) n m := apply (raise_depth_term t1 n m) (raise_depth_term t2 n m)

-- /- substitute_term t s n substitutes s for (var n) and reduces the level of all variables above n by 1 -/
-- def substitute_term : ∀ {l}, preterm l → term → ℕ → preterm l
-- | _ (var L k) s n := if k < n then var k else if k > n then var (k-1) else s
-- | _ (func f) s n := func f
-- | _ (apply t1 t2) s n := apply (substitute_term t1 s n) (substitute_term t2 s n)

/-- Given a preterm, return a list of free variables which occur in it--/
def free_vars_preterm : Π n : ℕ, preterm n → list string
| _ (bvar L k) := []
| _ (fvar L s) := [s]
| _ (@func L _ f) := []
| _ (@apply L n t1 t2) := (free_vars_preterm (n+1) t1 ∪ free_vars_preterm 0 t2)

def free_vars_term : term → list string := free_vars_preterm 0

lemma free_var_preterm_coercionl (n : ℕ) (t1 : preterm (n+1)) (t2 : term) : {x : string // x ∈ (free_vars_preterm (n+1) t1)} → {x : string // x ∈ (free_vars_preterm n (apply t1 t2))} := sorry

def substitute_preterm : Π n : ℕ, Π (t : preterm n), term → {x : string // x ∈ (free_vars_preterm n t)} → preterm n
| _ (bvar L k) t x := (bvar k)
| _ (fvar L s) t x := sorry
| _ (@func L _ f) t x := (func f)
| _ (@apply L n t1 t2) t x := sorry
end

section

parameter L : Language
/- preformula n is a partially applied formula. if applied to n terms, it becomes a formula -/
inductive preformula : ℕ → Type
| true : preformula 0
| false : preformula 0
| equal : (term L)  → (term L) → preformula 0
| rel : ∀ {n : nat}, L.relations n → preformula n
| apprel : ∀ {n : nat}, preformula (n + 1) → (term L) → preformula n
| imp : preformula 0 → preformula 0 → preformula 0
| all : preformula 0 → preformula 0
open preformula
def formula := preformula 0

def free_vars_preformula : Π n : ℕ, preformula n → list string
| _ (true L) := []
| _ (false L) := []
| _ (equal t1 t2) := free_vars_term L t1 ∪ free_vars_term L t2
| _ (@rel L n R) := []
| _ (@apprel L n ψ t) := free_vars_preformula _ ψ ∪ free_vars_term L t
| _ (imp ϕ ψ) := free_vars_preformula _ ϕ ∪ free_vars_preformula _ ψ
| _ (all ψ) := free_vars_preformula _ ψ

def free_vars_formula : formula → list string := free_vars_preformula 0

-- def raise_depth_formula : ∀ {l}, preformula l → ℕ → ℕ → preformula l
-- | _ (true L) n m := true
-- | _ (false L) n m := false
-- | _ (equal t1 t2) n m := equal (raise_depth_term t1 n m) (raise_depth_term t2 n m)
-- | _ (rel R) n m := rel R
-- | _ (apprel f t) n m := apprel (raise_depth_formula f n m) (raise_depth_term t n m)
-- | _ (imp f1 f2) n m := imp (raise_depth_formula f1 n m) (raise_depth_formula f2 n m)
-- | _ (all f) n m := all (raise_depth_formula f n (m+1))

def substitute_formula : ∀ {l}, preformula l → (term L) → ℕ → preformula l
| _ (true L) s n := true
| _ (false L) s n := false
| _ (equal t1 t2) s n := equal (substitute_term L t1 s n) (substitute_term L t2 s n)
| _ (rel R) s n := rel R
| _ (apprel f t) s n := apprel (substitute_formula f s n) (substitute_term t s n)
| _ (imp f1 f2) s n := imp (substitute_formula f1 s n) (substitute_formula f2 s n)
| _ (all f) s n := all (substitute_formula f s (n+1))

def substitute_preformula : Π n : ℕ, Π (ψ : preformula n), term → {x : string // x ∈ (free_vars_preformula _ ψ)} → preformula n
| _ (true L) t x := true
| _ (false L) t x := false
| _ (equal t1 t2) := equal (substitute_preformula _ _ _ _ t1) (substitute_preformula _ _ _ _ t2)
end


/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison
-/
import tactic.reassoc_axiom
import tactic.monotonicity
import category_theory.category.basic

/-!
# Functors

Defines a functor between categories, extending a `prefunctor` between quivers.

Introduces notation `C ⥤ D` for the type of all functors from `C` to `D`.
(Unfortunately the `⇒` arrow (`\functor`) is taken by core,
but in mathlib4 we should switch to this.)
-/

namespace category_theory

-- declare the `v`'s first; see `category_theory.category` for an explanation
universes v v₁ v₂ v₃ u u₁ u₂ u₃

section

set_option old_structure_cmd true

/--
`functor C D` represents a functor between categories `C` and `D`.

To apply a functor `F` to an object use `F.obj X`, and to a morphism use `F.map f`.

The axiom `map_id` expresses preservation of identities, and
`map_comp` expresses functoriality.

See https://stacks.math.columbia.edu/tag/001B.
-/
structure functor (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D]
  extends prefunctor C D : Type (max v₁ v₂ u₁ u₂) :=
(map_id'   : ∀ (X : C), map (𝟙 X) = 𝟙 (obj X) . obviously)
(map_comp' : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = (map f) ≫ (map g) . obviously)

/-- The prefunctor between the underlying quivers. -/
add_decl_doc functor.to_prefunctor

end

-- A functor is basically a function, so give ⥤ a similar precedence to → (25).
-- For example, `C × D ⥤ E` should parse as `(C × D) ⥤ E` not `C × (D ⥤ E)`.
infixr ` ⥤ `:26 := functor       -- type as \func --

restate_axiom functor.map_id'
attribute [simp] functor.map_id
restate_axiom functor.map_comp'
attribute [reassoc, simp] functor.map_comp

namespace functor

section
variables (C : Type u₁) [category.{v₁} C]

/-- `𝟭 C` is the identity functor on a category `C`. -/
protected def id : C ⥤ C :=
{ obj := λ X, X,
  map := λ _ _ f, f }

notation `𝟭` := functor.id -- Type this as `\sb1`

instance : inhabited (C ⥤ C) := ⟨functor.id C⟩

variable {C}

@[simp] lemma id_obj (X : C) : (𝟭 C).obj X = X := rfl
@[simp] lemma id_map {X Y : C} (f : X ⟶ Y) : (𝟭 C).map f = f := rfl
end

section
variables {C : Type u₁} [category.{v₁} C]
          {D : Type u₂} [category.{v₂} D]
          {E : Type u₃} [category.{v₃} E]

/--
`F ⋙ G` is the composition of a functor `F` and a functor `G` (`F` first, then `G`).
-/
def comp (F : C ⥤ D) (G : D ⥤ E) : C ⥤ E :=
{ obj := λ X, G.obj (F.obj X),
  map := λ _ _ f, G.map (F.map f) }

infixr ` ⋙ `:80 := comp

@[simp] lemma comp_obj (F : C ⥤ D) (G : D ⥤ E) (X : C) : (F ⋙ G).obj X = G.obj (F.obj X) := rfl
@[simp] lemma comp_map (F : C ⥤ D) (G : D ⥤ E) {X Y : C} (f : X ⟶ Y) :
  (F ⋙ G).map f = G.map (F.map f) := rfl

-- These are not simp lemmas because rewriting along equalities between functors
-- is not necessarily a good idea.
-- Natural isomorphisms are also provided in `whiskering.lean`.
protected lemma comp_id (F : C ⥤ D) : F ⋙ (𝟭 D) = F := by cases F; refl
protected lemma id_comp (F : C ⥤ D) : (𝟭 C) ⋙ F = F := by cases F; refl

@[simp] lemma map_dite (F : C ⥤ D) {X Y : C} {P : Prop} [decidable P]
  (f : P → (X ⟶ Y)) (g : ¬P → (X ⟶ Y)) :
  F.map (if h : P then f h else g h) = if h : P then F.map (f h) else F.map (g h) :=
by { split_ifs; refl, }

end

end functor

end category_theory