A Certified Type Checker

In this example, we build a certified type checker for a simple expression language.

Remark: this example is based on an example in the book Certified Programming with Dependent Types by Adam Chlipala.

inductive Expr: Type
Expr where
  | nat: Nat → Expr
nat  : Nat: Type
Nat → Expr: Type
Expr
  | plus: Expr → Expr → Expr
plus : Expr: Type
Expr → Expr: Type
Expr → Expr: Type
Expr
  | bool: Bool → Expr
bool : Bool: Type
Bool → Expr: Type
Expr
  | and: Expr → Expr → Expr
and  : Expr: Type
Expr → Expr: Type
Expr → Expr: Type
Expr

We define a simple language of types using the inductive datatype Ty, and its typing rules using the inductive predicate HasType.

inductive Ty: Type
Ty where
  | nat: Ty
nat
  | bool: Ty
bool
  deriving DecidableEq: Sort u → Sort (max 1 u)
DecidableEq

inductive HasType: Expr → Ty → Prop
HasType : Expr: Type
Expr → Ty: Type
Ty → Prop: Type
Prop
  | nat: ∀ {v : Nat}, HasType (Expr.nat v) Ty.nat
nat  : HasType: Expr → Ty → Prop
HasType (.nat: Nat → Expr
.nat v: Nat
v) .nat: Ty
.nat
  | plus: ∀ {a b : Expr}, HasType a Ty.nat → HasType b Ty.nat → HasType (Expr.plus a b) Ty.nat
plus : HasType: Expr → Ty → Prop
HasType a: Expr
a .nat: Ty
.nat → HasType: Expr → Ty → Prop
HasType b: Expr
b .nat: Ty
.nat → HasType: Expr → Ty → Prop
HasType (.plus: Expr → Expr → Expr
.plus a: Expr
a b: Expr
b) .nat: Ty
.nat
  | bool: ∀ {v : Bool}, HasType (Expr.bool v) Ty.bool
bool : HasType: Expr → Ty → Prop
HasType (.bool: Bool → Expr
.bool v: Bool
v) .bool: Ty
.bool
  | and: ∀ {a b : Expr}, HasType a Ty.bool → HasType b Ty.bool → HasType (Expr.and a b) Ty.bool
and  : HasType: Expr → Ty → Prop
HasType a: Expr
a .bool: Ty
.bool → HasType: Expr → Ty → Prop
HasType b: Expr
b .bool: Ty
.bool → HasType: Expr → Ty → Prop
HasType (.and: Expr → Expr → Expr
.and a: Expr
a b: Expr
b) .bool: Ty
.bool

We can easily show that if e has type t₁ and type t₂, then t₁ and t₂ must be equal by using the the cases tactic. This tactic creates a new subgoal for every constructor, and automatically discharges unreachable cases. The tactic combinator tac₁ <;> tac₂ applies tac₂ to each subgoal produced by tac₁. Then, the tactic rfl is used to close all produced goals using reflexivity.

theorem HasType.det: ∀ {e : Expr} {t₁ t₂ : Ty}, HasType e t₁ → HasType e t₂ → t₁ = t₂
HasType.det (h₁: HasType e t₁
h₁ : HasType: Expr → Ty → Prop
HasType e: Expr
e t₁: Ty
t₁) (h₂: HasType e t₂
h₂ : HasType: Expr → Ty → Prop
HasType e: Expr
e t₂: Ty
t₂) : t₁: Ty
t₁ = t₂: Ty
t₂ := byGoals accomplished! 🐙
  cases h₁: HasType e t₁
h₁t₂: Ty
v✝: Nat
h₂: HasType (Expr.nat v✝) t₂
natTy.nat = t₂
t₂: Ty
a✝², b✝: Expr
a✝¹: HasType a✝² Ty.nat
a✝: HasType b✝ Ty.nat
h₂: HasType (Expr.plus a✝² b✝) t₂
plusTy.nat = t₂
t₂: Ty
v✝: Bool
h₂: HasType (Expr.bool v✝) t₂
boolTy.bool = t₂
t₂: Ty
a✝², b✝: Expr
a✝¹: HasType a✝² Ty.bool
a✝: HasType b✝ Ty.bool
h₂: HasType (Expr.and a✝² b✝) t₂
andTy.bool = t₂ <;>t₂: Ty
v✝: Nat
h₂: HasType (Expr.nat v✝) t₂
natTy.nat = t₂
t₂: Ty
a✝², b✝: Expr
a✝¹: HasType a✝² Ty.nat
a✝: HasType b✝ Ty.nat
h₂: HasType (Expr.plus a✝² b✝) t₂
plusTy.nat = t₂
t₂: Ty
v✝: Bool
h₂: HasType (Expr.bool v✝) t₂
boolTy.bool = t₂
t₂: Ty
a✝², b✝: Expr
a✝¹: HasType a✝² Ty.bool
a✝: HasType b✝ Ty.bool
h₂: HasType (Expr.and a✝² b✝) t₂
andTy.bool = t₂ cases h₂: HasType (Expr.nat v✝) t₂
h₂a✝⁴, b✝: Expr
a✝³: HasType a✝⁴ Ty.bool
a✝²: HasType b✝ Ty.bool
a✝¹: HasType a✝⁴ Ty.bool
a✝: HasType b✝ Ty.bool
and.andTy.bool = Ty.bool <;>v✝: Nat
nat.natTy.nat = Ty.nat
a✝⁴, b✝: Expr
a✝³: HasType a✝⁴ Ty.nat
a✝²: HasType b✝ Ty.nat
a✝¹: HasType a✝⁴ Ty.nat
a✝: HasType b✝ Ty.nat
plus.plusTy.nat = Ty.nat
v✝: Bool
bool.boolTy.bool = Ty.bool
a✝⁴, b✝: Expr
a✝³: HasType a✝⁴ Ty.bool
a✝²: HasType b✝ Ty.bool
a✝¹: HasType a✝⁴ Ty.bool
a✝: HasType b✝ Ty.bool
and.andTy.bool = Ty.bool rflGoals accomplished! 🐙

The inductive type Maybe p has two contructors: found a h and unknown. The former contains an element a : α and a proof that a satisfies the predicate p. The constructor unknown is used to encode "failure".

inductive Maybe: {α : Sort u_1} → (α → Prop) → Sort (max 1 u_1)
Maybe (p: α → Prop
p : α: Sort u_1
α → Prop: Type
Prop) where
  | found: {α : Sort u_1} → {p : α → Prop} → (a : α) → p a → Maybe p
found : (a: α
a : α: Sort u_1
α) → p: α → Prop
p a: α
a → Maybe: {α : Sort u_1} → (α → Prop) → Sort (max 1 u_1)
Maybe p: α → Prop
p
  | unknown: {α : Sort u_1} → {p : α → Prop} → Maybe p
unknown

We define a notation for Maybe that is similar to the builtin notation for the Lean builtin type Subtype.

notation "{{ " x: Lean.TSyntax `term
x " | " p: Lean.TSyntax `term
p " }}" => Maybe: {α : Sort u_1} → (α → Prop) → Sort (max 1 u_1)
Maybe (fun x: Lean.TSyntax `term
x => p: Lean.TSyntax `term
p)

The function Expr.typeCheck e returns a type ty and a proof that e has type ty, or unknown. Recall that, def Expr.typeCheck ... in Lean is notation for namespace Expr def typeCheck ... end Expr. The term .found .nat .nat is sugar for Maybe.found Ty.nat HasType.nat. Lean can infer the namespaces using the expected types.

def Expr.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
Expr.typeCheck (e: Expr
e : Expr: Type
Expr) : {{ ty: Ty
ty | HasType: Expr → Ty → Prop
HasType e: Expr
e ty: Ty
ty }} :=
  match e: Expr
e with
  | nat: Nat → Expr
nat ..   => .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .nat: Ty
.nat .nat: ∀ {v : Nat}, HasType (nat v) Ty.nat
.nat
  | bool: Bool → Expr
bool ..  => .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .bool: Ty
.bool .bool: ∀ {v : Bool}, HasType (bool v) Ty.bool
.bool
  | plus: Expr → Expr → Expr
plus a: Expr
a b: Expr
b =>
    match a: Expr
a.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck, b: Expr
b.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck with
    | .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .nat: Ty
.nat h₁: HasType a Ty.nat
h₁, .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .nat: Ty
.nat h₂: HasType b Ty.nat
h₂ => .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .nat: Ty
.nat (.plus: ∀ {a b : Expr}, HasType a Ty.nat → HasType b Ty.nat → HasType (plus a b) Ty.nat
.plus h₁: HasType a Ty.nat
h₁ h₂: HasType b Ty.nat
h₂)
    | _, _ => .unknown: {α : Type} → {p : α → Prop} → Maybe p
.unknown
  | and: Expr → Expr → Expr
and a: Expr
a b: Expr
b =>
    match a: Expr
a.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck, b: Expr
b.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck with
    | .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .bool: Ty
.bool h₁: HasType a Ty.bool
h₁, .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .bool: Ty
.bool h₂: HasType b Ty.bool
h₂ => .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found .bool: Ty
.bool (.and: ∀ {a b : Expr}, HasType a Ty.bool → HasType b Ty.bool → HasType (and a b) Ty.bool
.and h₁: HasType a Ty.bool
h₁ h₂: HasType b Ty.bool
h₂)
    | _, _ => .unknown: {α : Type} → {p : α → Prop} → Maybe p
.unknown

theorem Expr.typeCheck_correct: ∀ {e : Expr} {ty : Ty} {h : HasType e ty}, HasType e ty → typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
Expr.typeCheck_correct (h₁: HasType e ty
h₁ : HasType: Expr → Ty → Prop
HasType e: Expr
e ty: Ty
ty) (h₂: typeCheck e ≠ Maybe.unknown
h₂ : e: Expr
e.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck ≠ .unknown: {α : Sort ?u.23451} → {p : α → Prop} → Maybe p
.unknown)
        : e: Expr
e.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck = .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found ty: Ty
ty h: HasType e ty
h := byGoals accomplished! 🐙
  revert h₂: typeCheck e ≠ Maybe.unknown
h₂e: Expr
ty: Ty
h, h₁: HasType e ty
typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
  cases typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck e: Expr
e withe: Expr
ty: Ty
h, h₁: HasType e ty
x✝: {{ ty | HasType e ty }}
x✝ ≠ Maybe.unknown → x✝ = Maybe.found ty h
  | found: {α : Sort u_1} → {p : α → Prop} → (a : α) → p a → Maybe p
found ty': Ty
ty' h': HasType e ty'
h' =>e: Expr
ty: Ty
h, h₁: HasType e ty
ty': Ty
h': HasType e ty'
foundMaybe.found ty' h' ≠ Maybe.unknown → Maybe.found ty' h' = Maybe.found ty h introe: Expr
ty: Ty
h, h₁: HasType e ty
ty': Ty
h': HasType e ty'
h₂✝: Maybe.found ty' h' ≠ Maybe.unknown
foundMaybe.found ty' h' = Maybe.found ty h; have := HasType.det: ∀ {e : Expr} {t₁ t₂ : Ty}, HasType e t₁ → HasType e t₂ → t₁ = t₂
HasType.det h₁: HasType e ty
h₁ h': HasType e ty'
h'e: Expr
ty: Ty
h, h₁: HasType e ty
ty': Ty
h': HasType e ty'
h₂✝: Maybe.found ty' h' ≠ Maybe.unknown
this: ty = ty'
foundMaybe.found ty' h' = Maybe.found ty h; subst this: ty = ty'
thise: Expr
ty: Ty
h, h₁, h': HasType e ty
h₂✝: Maybe.found ty h' ≠ Maybe.unknown
foundMaybe.found ty h' = Maybe.found ty h; rflGoals accomplished! 🐙
  | unknown: {α : Sort u_1} → {p : α → Prop} → Maybe p
unknown =>e: Expr
ty: Ty
h, h₁: HasType e ty
unknownMaybe.unknown ≠ Maybe.unknown → Maybe.unknown = Maybe.found ty h introse: Expr
ty: Ty
h, h₁: HasType e ty
h₂✝: Maybe.unknown ≠ Maybe.unknown
unknownMaybe.unknown = Maybe.found ty h; contradictionGoals accomplished! 🐙

Now, we prove that if Expr.typeCheck e returns Maybe.unknown, then forall ty, HasType e ty does not hold. The notation e.typeCheck is sugar for Expr.typeCheck e. Lean can infer this because we explicitly said that e has type Expr. The proof is by induction on e and case analysis. The tactic rename_i is used to to rename "inaccessible" variables. We say a variable is inaccessible if it is introduced by a tactic (e.g., cases) or has been shadowed by another variable introduced by the user. Note that the tactic simp [typeCheck] is applied to all goal generated by the induction tactic, and closes the cases corresponding to the constructors Expr.nat and Expr.bool.

theorem Expr.typeCheck_complete: ∀ {ty : Ty} {e : Expr}, typeCheck e = Maybe.unknown → ¬HasType e ty
Expr.typeCheck_complete {e: Expr
e : Expr: Type
Expr} : e: Expr
e.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck = .unknown: {α : Type} → {p : α → Prop} → Maybe p
.unknown → ¬ HasType: Expr → Ty → Prop
HasType e: Expr
e ty: Ty
ty := byGoals accomplished! 🐙
  induction e: Expr
e withty: Ty
e: Expr
typeCheck e = Maybe.unknown → ¬HasType e ty simp [typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck]Goals accomplished! 🐙
  | plus: Expr → Expr → Expr
plus a: Expr
a b: Expr
b iha: typeCheck a = Maybe.unknown → ¬HasType a ty
iha ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ihb =>ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
plus(match typeCheck a, typeCheck b with
    | Maybe.found Ty.nat h₁, Maybe.found Ty.nat h₂ => Maybe.found Ty.nat (_ : HasType (plus a b) Ty.nat)
    | x, x_1 => Maybe.unknown) =
    Maybe.unknown →
  ¬HasType (plus a b) ty
    splitty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.nat
h₂✝: HasType b Ty.nat
heq✝¹: typeCheck a = Maybe.found Ty.nat h₁✝
heq✝: typeCheck b = Maybe.found Ty.nat h₂✝
plus.h_1Maybe.found Ty.nat (_ : HasType (plus a b) Ty.nat) = Maybe.unknown → ¬HasType (plus a b) ty
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
plus.h_2Maybe.unknown = Maybe.unknown → ¬HasType (plus a b) ty
    next =>ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.nat
h₂✝: HasType b Ty.nat
heq✝¹: typeCheck a = Maybe.found Ty.nat h₁✝
heq✝: typeCheck b = Maybe.found Ty.nat h₂✝
plus.h_1Maybe.found Ty.nat (_ : HasType (plus a b) Ty.nat) = Maybe.unknown → ¬HasType (plus a b) ty
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
plus.h_2Maybe.unknown = Maybe.unknown → ¬HasType (plus a b) ty introsty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.nat
h₂✝: HasType b Ty.nat
heq✝¹: typeCheck a = Maybe.found Ty.nat h₁✝
heq✝: typeCheck b = Maybe.found Ty.nat h₂✝
a✝: Maybe.found Ty.nat (_ : HasType (plus a b) Ty.nat) = Maybe.unknown
¬HasType (plus a b) ty; contradictionGoals accomplished! 🐙
    next ra: {{ ty | HasType a ty }}
ra rb: {{ ty | HasType b ty }}
rb hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
hnp =>
      -- Recall that `hnp` is a hypothesis generated by the `split` tactic
      -- that asserts the previous case was not takenty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
plus.h_2Maybe.unknown = Maybe.unknown → ¬HasType (plus a b) ty
      introty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
Maybe.unknown = Maybe.unknown → ¬HasType (plus a b) ty h: Maybe.unknown = Maybe.unknown
hWarning: unused variable `h` [linter.unusedVariables]
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
Maybe.unknown = Maybe.unknown → ¬HasType (plus a b) ty ht: HasType (plus a b) tyht: HasType (plus a b) ty
htty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
h: Maybe.unknown = Maybe.unknown
ht: HasType (plus a b) ty
False
      cases ht: HasType (plus a b) ty
ht withty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
h: Maybe.unknown = Maybe.unknown
ht: HasType (plus a b) ty
False
      | plus: ∀ {a b : Expr}, HasType a Ty.nat → HasType b Ty.nat → HasType (plus a b) Ty.nat
plus h₁: HasType a✝ Ty.nat
h₁ h₂: HasType b✝ Ty.nat
h₂ =>a, b: Expr
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
h: Maybe.unknown = Maybe.unknown
h₁: HasType a Ty.nat
h₂: HasType b Ty.nat
iha: typeCheck a = Maybe.unknown → ¬HasType a Ty.nat
ihb: typeCheck b = Maybe.unknown → ¬HasType b Ty.nat
plusFalse exact hnp: ∀ (h₁ : HasType a Ty.nat) (h₂ : HasType b Ty.nat),
  typeCheck a = Maybe.found Ty.nat h₁ → typeCheck b = Maybe.found Ty.nat h₂ → False
hnp h₁: HasType a Ty.nat
h₁ h₂: HasType b Ty.nat
h₂ (typeCheck_correct: ∀ {e : Expr} {ty : Ty} {h : HasType e ty}, HasType e ty → typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
typeCheck_correct h₁: HasType a Ty.nat
h₁ (iha: typeCheck a = Maybe.unknown → ¬HasType a Ty.nat
iha · h₁: HasType a Ty.nat
h₁)) (typeCheck_correct: ∀ {e : Expr} {ty : Ty} {h : HasType e ty}, HasType e ty → typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
typeCheck_correct h₂: HasType b Ty.nat
h₂ (ihb: typeCheck b = Maybe.unknown → ¬HasType b Ty.nat
ihb · h₂: HasType b Ty.nat
h₂))Goals accomplished! 🐙
  | and: Expr → Expr → Expr
and a: Expr
a b: Expr
b iha: typeCheck a = Maybe.unknown → ¬HasType a ty
iha ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ihb =>ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
and(match typeCheck a, typeCheck b with
    | Maybe.found Ty.bool h₁, Maybe.found Ty.bool h₂ => Maybe.found Ty.bool (_ : HasType (and a b) Ty.bool)
    | x, x_1 => Maybe.unknown) =
    Maybe.unknown →
  ¬HasType (and a b) ty
    splitty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.bool
h₂✝: HasType b Ty.bool
heq✝¹: typeCheck a = Maybe.found Ty.bool h₁✝
heq✝: typeCheck b = Maybe.found Ty.bool h₂✝
and.h_1Maybe.found Ty.bool (_ : HasType (and a b) Ty.bool) = Maybe.unknown → ¬HasType (and a b) ty
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
and.h_2Maybe.unknown = Maybe.unknown → ¬HasType (and a b) ty
    next =>ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.bool
h₂✝: HasType b Ty.bool
heq✝¹: typeCheck a = Maybe.found Ty.bool h₁✝
heq✝: typeCheck b = Maybe.found Ty.bool h₂✝
and.h_1Maybe.found Ty.bool (_ : HasType (and a b) Ty.bool) = Maybe.unknown → ¬HasType (and a b) ty
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
and.h_2Maybe.unknown = Maybe.unknown → ¬HasType (and a b) ty introsty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝¹: {{ ty | HasType a ty }}
x✝: {{ ty | HasType b ty }}
h₁✝: HasType a Ty.bool
h₂✝: HasType b Ty.bool
heq✝¹: typeCheck a = Maybe.found Ty.bool h₁✝
heq✝: typeCheck b = Maybe.found Ty.bool h₂✝
a✝: Maybe.found Ty.bool (_ : HasType (and a b) Ty.bool) = Maybe.unknown
¬HasType (and a b) ty; contradictionGoals accomplished! 🐙
    next ra: {{ ty | HasType a ty }}
ra rb: {{ ty | HasType b ty }}
rb hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
hnp =>ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
x✝²: {{ ty | HasType a ty }}
x✝¹: {{ ty | HasType b ty }}
x✝: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
and.h_2Maybe.unknown = Maybe.unknown → ¬HasType (and a b) ty
      introty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
Maybe.unknown = Maybe.unknown → ¬HasType (and a b) ty h: Maybe.unknown = Maybe.unknown
hWarning: unused variable `h` [linter.unusedVariables]
ty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
Maybe.unknown = Maybe.unknown → ¬HasType (and a b) ty ht: HasType (and a b) tyht: HasType (and a b) ty
htty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
h: Maybe.unknown = Maybe.unknown
ht: HasType (and a b) ty
False
      cases ht: HasType (and a b) ty
ht withty: Ty
a, b: Expr
iha: typeCheck a = Maybe.unknown → ¬HasType a ty
ihb: typeCheck b = Maybe.unknown → ¬HasType b ty
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
h: Maybe.unknown = Maybe.unknown
ht: HasType (and a b) ty
False
      | and: ∀ {a b : Expr}, HasType a Ty.bool → HasType b Ty.bool → HasType (and a b) Ty.bool
and h₁: HasType a✝ Ty.bool
h₁ h₂: HasType b✝ Ty.bool
h₂ =>a, b: Expr
ra: {{ ty | HasType a ty }}
rb: {{ ty | HasType b ty }}
hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
h: Maybe.unknown = Maybe.unknown
h₁: HasType a Ty.bool
h₂: HasType b Ty.bool
iha: typeCheck a = Maybe.unknown → ¬HasType a Ty.bool
ihb: typeCheck b = Maybe.unknown → ¬HasType b Ty.bool
andFalse  exact hnp: ∀ (h₁ : HasType a Ty.bool) (h₂ : HasType b Ty.bool),
  typeCheck a = Maybe.found Ty.bool h₁ → typeCheck b = Maybe.found Ty.bool h₂ → False
hnp h₁: HasType a Ty.bool
h₁ h₂: HasType b Ty.bool
h₂ (typeCheck_correct: ∀ {e : Expr} {ty : Ty} {h : HasType e ty}, HasType e ty → typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
typeCheck_correct h₁: HasType a Ty.bool
h₁ (iha: typeCheck a = Maybe.unknown → ¬HasType a Ty.bool
iha · h₁: HasType a Ty.bool
h₁)) (typeCheck_correct: ∀ {e : Expr} {ty : Ty} {h : HasType e ty}, HasType e ty → typeCheck e ≠ Maybe.unknown → typeCheck e = Maybe.found ty h
typeCheck_correct h₂: HasType b Ty.bool
h₂ (ihb: typeCheck b = Maybe.unknown → ¬HasType b Ty.bool
ihb · h₂: HasType b Ty.bool
h₂))Goals accomplished! 🐙

Finally, we show that type checking for e can be decided using Expr.typeCheck.

instance: (e : Expr) → (t : Ty) → Decidable (HasType e t)
instance (e: Expr
e : Expr: Type
Expr) (t: Ty
t : Ty: Type
Ty) : Decidable: Prop → Type
Decidable (HasType: Expr → Ty → Prop
HasType e: Expr
e t: Ty
t) :=
  match h': Expr.typeCheck e = Maybe.unknown
h' : e: Expr
e.typeCheck: (e : Expr) → {{ ty | HasType e ty }}
typeCheck with
  | .found: {α : Type} → {p : α → Prop} → (a : α) → p a → Maybe p
.found t': Ty
t' ht': HasType e t'
ht' =>
    if heq: t = t'
heq : t: Ty
t = t': Ty
t' then
      isTrue: {p : Prop} → p → Decidable p
isTrue (heq: t = t'
heq ▸ ht': HasType e t'
ht')
    else
      isFalse: {p : Prop} → ¬p → Decidable p
isFalse fun ht: HasType e t
ht => heq: ¬t = t'
heq (HasType.det: ∀ {e : Expr} {t₁ t₂ : Ty}, HasType e t₁ → HasType e t₂ → t₁ = t₂
HasType.det ht: HasType e t
ht ht': HasType e t'
ht')
  | .unknown: {α : Type} → {p : α → Prop} → Maybe p
.unknown => isFalse: {p : Prop} → ¬p → Decidable p
isFalse (Expr.typeCheck_complete: ∀ {ty : Ty} {e : Expr}, Expr.typeCheck e = Maybe.unknown → ¬HasType e ty
Expr.typeCheck_complete h': Expr.typeCheck e = Maybe.unknown
h')

Lean Manual

A Certified Type Checker