L∃∀N-verified Quantum Information Theory

  • RRS to QIC891, Fall 2026

  • Lecture 5, 2026-09-29

\Gamma ⊢ t : T

Where we left off

  • Equalities

    • Definitional

      • Gives tactic rfl

    • Propositional

      • Gives tactic rw through Eq.subst on Prop

    • Boolean, BEq

      • That decides or computes expressions out of propositions

  • When def and theorem define terms, what defines types?

More complex mathematical definitions

  • Compare a familiar notion of continuity with the one implemented in Mathlib

    • GlimpseOfLean.continuous_at vs Mathlib.ContinuousAt

  • Lean's class

  • Mathlib_surfing in Blackboard.lean

Structures and classes

  • Lean's structure and class are used to define

    • structure CPTPMap m n 𝕜, in QuantumInfo

    • Matrix m n α contains instances, e.g. Matrix.add

      • HermitianMat, in QuantumInfo

      • MState.instCoe

    • class InnerProductSpace, in Mathlib

      • Typeclass resolution

Function types

And the No cloning theorem

We use the no cloning theorem as it is found in Physlib/QuantumInfo/Operators/Unitary.lean to study:

  • Dependent function types,

    • and the universal quantifier ∀

  • Many more tactics,

    • and use of other definitions and theorems

  • Function_types in Blackboard.lean