Lecture 3: Lean coding

L∃∀N-verified Quantum Information Theory

RRS to QIC891, Fall 2026
Lecture 3, 2026-09-22

\Gamma ⊢ t : T

Questions we've had so far

Define: Hierarchy of sorts or types, and propositions
We write and understand x \in \mathbb{R}
- But x ∈ ℝ doesn't type-check. Why?
Does #check have a type?
What is “equality of propositions”?
- E.g. propext : (P ↔ Q) → P = Q

Working through the questions

Let's do live coding

35 minutes, ends at 11:15
Use of funext, propext, and the proof of set equality
Introduce quantifiers,
- and intro and use

theorem set_eq {α : Type u}
    (A B : Set α)
    (h : ∀ a, a ∈ A ↔ a ∈ B) : A = B := by
          funext a
          change (a ∈ A) = (a ∈ B)
          specialize h a
          apply propext
          apply h

Sorts and types

Define: Hierarchy of sorts or types, and propositions
- Sorts and types are different names for the same thing
- Type means Type 0, which is Sort 1.

#checkType : Type 1 Type
-- where `Type ≡ Type 0`

Type : Type 1
#checkType : Type 1 (Sort 1)
-- more generally, Type u ≡ Sort (u + 1)

Type : Type 1
universe u
#checkSort u : Type u (Sort u)
Sort u : Type u
#checkType u : Type (u + 1) (Type u)
Type u : Type (u + 1)

Type-checking sets and types

Let's say x \in \mathbb{R} “type-checks”. Does x ∈ ℝ type-check?
- x ∈ ℝ does not type-check.
- ℝ is not a set, and the membership symbol establishes a proposition between terms of a type and a set of that type.

-- The following will not compile
-- #check `x ∈ ℝ`
--  #check `x ∈ ℝ` failed to synthesize instance of type
--  class `Membership ℝ Type`

--  Hint: Type class instance resolution failures can be
--  inspected with the `set_option trace.Meta.synthInstance
--  true` command. #check Membership x (Set ℝ)

#checkMembership ℝ (Set ℝ) : Type Membership ℝ (Set ℝ)

Membership ℝ (Set ℝ) : Type
#checkSet.univ ℝ : Prop Set.univ ℝ

Set.univ ℝ : Prop
variable {x : ℝ}
#checkx ∈ Set.univ : Prop x ∈ (Set.univ : Set ℝ)
-- lives in Mathlib.Data.Set.Defs.lean
x ∈ Set.univ : Prop