Hermes Agent
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Converted with pandoc 3.7 (markdown → org), all 17 files: - README, references, 15 tutorial units - Internal file links updated from .md to .org - Source code blocks (#+begin_src lean) preserved - Tables, math notation, links intact For the Emacs workflow.
93 lines
2.9 KiB
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93 lines
2.9 KiB
Org Mode
* Unit 8 --- Representing Syntax
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:PROPERTIES:
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:CUSTOM_ID: unit-8-representing-syntax
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:END:
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*Tutorial 2: PL Semantics in Lean* · [[../README.org][← Back to README]]
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** Goals
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:PROPERTIES:
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:CUSTOM_ID: goals
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:END:
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- Encode lambda calculus terms as an inductive type
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- Understand three binding representations:
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1. *Named* (strings --- simple, but α-equiv isn't definitional)
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2. *de Bruijn indices* (numbers --- α-equiv is free, shifting is painful)
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3. *Locally nameless* (free vars named, bound vars indexed --- compromise)
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We'll use de Bruijn indices (the "heavy lifter") for the rest of this tutorial,
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with locally nameless for comparison.
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** Sources
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:PROPERTIES:
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:CUSTOM_ID: sources
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:END:
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- syndikos/lean4-stlc =Syntax.lean=: https://github.com/syndikos/lean4-stlc
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- Chris Henson 2025: https://chrishenson.net/posts/2025-05-10-formalized_lambda_calculus.html
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- chenson2018/LeanScratch: https://github.com/chenson2018/LeanScratch
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- Software Foundations Vol.2: https://softwarefoundations.cis.upenn.edu/
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** Exercises
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:PROPERTIES:
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:CUSTOM_ID: exercises
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:END:
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#+begin_src lean
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-- 8.1 — Named representation
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inductive NamedTerm where
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| var (x : String)
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| lam (x : String) (body : NamedTerm)
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| app (f arg : NamedTerm)
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deriving Repr
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-- The Church encoding of identity: λx. x
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def idNamed : NamedTerm := NamedTerm.lam "x" (NamedTerm.var "x")
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-- Encode λx. λy. x (K combinator)
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def kNamed : NamedTerm :=
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sorry
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-- Encode λf. λx. f (f x) (Church numeral 2)
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def twoNamed : NamedTerm :=
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sorry
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-- 8.2 — de Bruijn representation
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-- Variables are numbers: 0 = nearest binder, 1 = next, etc.
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inductive DBTerm where
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| var (idx : Nat) -- variable reference by binding distance
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| lam (body : DBTerm) -- λ. body (no name needed!)
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| app (f arg : DBTerm)
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deriving Repr
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-- λ. λ. 1 (= λx. λy. x in named form)
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def kDB : DBTerm := DBTerm.lam (DBTerm.lam (DBTerm.var 1))
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-- λ. 0 (= λx. x in named form)
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def idDB : DBTerm := DBTerm.lam (DBTerm.var 0)
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-- Encode λf. λx. f (f x) (Church 2)
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def twoDB : DBTerm :=
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sorry
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-- 8.3 — Locally nameless
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-- Free variables are strings, bound variables are de Bruijn indices
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-- (You don't need to implement this fully — just understand the idea)
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inductive LNTerm where
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| fvar (x : String) -- free variable
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| bvar (idx : Nat) -- bound variable (de Bruijn)
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| lam (body : LNTerm) -- binder
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| app (f arg : LNTerm)
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deriving Repr
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#+end_src
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*** Key insight for PL semantics
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:PROPERTIES:
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:CUSTOM_ID: key-insight-for-pl-semantics
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:END:
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When we encode *typing contexts* =Γ = x₁:τ₁, x₂:τ₂, ...=, de Bruijn indices
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give us "index into the context" for free. The last binding is index 0, the
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second-last is index 1, etc. This makes the typing rules elegant in Lean ---
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no name-clash avoidance needed.
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--------------
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← [[../tutorial-01-basics/07-dependent-types.org][Tutorial 1 --- Unit 7]] · Next: [[file:09-substitution.org][Unit 9 --- Substitution]]
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