Hermes Agent
6e2914b06e
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.
74 lines
2.4 KiB
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74 lines
2.4 KiB
Org Mode
* Unit 10 --- Small-Step Operational Semantics
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:PROPERTIES:
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:CUSTOM_ID: unit-10-small-step-operational-semantics
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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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- Define single-step reduction =→= for call-by-value λ-calculus
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- Define reflexive-transitive closure =→*=
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- Prove determinism: =t → t₁ ∧ t → t₂ ⇒ t₁ = t₂=
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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 =Reduction.lean=, =Semantics.lean=
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- Software Foundations Vol.2 "Smallstep"
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- PLFA Part 2
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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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open DBTerm
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-- 10.1 — Single-step cbv reduction
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-- Call-by-value: reduce the *argument* before substituting into the function body.
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-- A value is a λ-abstraction (in pure STLC).
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inductive Value : DBTerm → Prop where
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| lam : Value (lam b)
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inductive Step : DBTerm → DBTerm → Prop where
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-- β-reduction: (λ. t) v → subst t 0 v (where v is a value)
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| beta (t v : DBTerm) (hv : Value v) :
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Step (app (lam t) v) (subst t 0 v)
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-- Congruence: if f → f', then f a → f' a
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| appL (f f' a : DBTerm) (hstep : Step f f') :
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Step (app f a) (app f' a)
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-- Congruence: if v is a value and a → a', then v a → v a'
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| appR (v a a' : DBTerm) (hv : Value v) (hstep : Step a a') :
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Step (app v a) (app v a')
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-- 10.2 — Multi-step (reflexive-transitive closure)
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inductive StepStar : DBTerm → DBTerm → Prop where
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| refl (t : DBTerm) : StepStar t t
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| step (t₁ t₂ t₃ : DBTerm) (hstep : Step t₁ t₂) (hstar : StepStar t₂ t₃) :
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StepStar t₁ t₃
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-- 10.3 — Determinism
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theorem step_deterministic {t t₁ t₂ : DBTerm} (h₁ : Step t t₁) (h₂ : Step t t₂) : t₁ = t₂ :=
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by
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sorry
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-- Hint: induction on h₁, then cases on h₂. The tricky case:
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-- when one side is (lam t) and the other is a congruence rule that also
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-- produces (lam t) — they match.
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-- 10.4 — Stuck terms
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-- A term is "stuck" if it's not a value and can't take a step.
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-- Type soundness = "well-typed terms never get stuck".
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def Stuck (t : DBTerm) : Prop :=
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¬ Value t ∧ ¬ ∃ t', Step t t'
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-- Find a stuck term in the untyped calculus
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def stuckExample : DBTerm :=
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app (var 0) (var 1) -- a free variable applied to another free variable
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-- Show it's stuck (you'll prove something similar in Unit 12 with types)
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#+end_src
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