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.
107 lines
2.8 KiB
Org Mode
107 lines
2.8 KiB
Org Mode
* Unit 3 --- Propositions and Proofs
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:PROPERTIES:
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:CUSTOM_ID: unit-3-propositions-and-proofs
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:END:
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*Tutorial 1: Lean 4 Fundamentals* · [[../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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- Understand propositions as types (Curry-Howard)
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- Write basic proofs using =example= and =theorem=
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- Use the tactic language: =intro=, =apply=, =exact=, =have=
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** Source
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:PROPERTIES:
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:CUSTOM_ID: source
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:END:
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/Theorem Proving in Lean 4/ (TPIL), Chapters 2--3
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→ https://leanprover.github.io/theorem_proving_in_lean4/
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** Concepts
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:PROPERTIES:
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:CUSTOM_ID: concepts
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:END:
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| Concept | Lean |
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|---------------------+-------------------------------------------------------------|
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| Proposition | =P : Prop= --- a type whose terms are proofs |
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| Implication =P → Q= | Function from proofs of =P= to proofs of =Q= |
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| Conjunction =P ∧ Q= | =And.intro= (constructor), =.left= / =.right= (projections) |
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| Disjunction =P ∨ Q= | =Or.inl= / =Or.inr= |
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| Negation =¬ P= | =P → False= |
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| Tactics | =intro=, =apply=, =exact=, =have=, =assumption= |
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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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*** Exercise 3.1 --- Implicational logic
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:PROPERTIES:
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:CUSTOM_ID: exercise-3.1-implicational-logic
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:END:
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#+begin_src lean
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-- (a) Identity
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theorem id_prop (A : Prop) : A → A :=
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by
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intro h
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exact h
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-- (b) Composition
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theorem compose (A B C : Prop) : (A → B) → (B → C) → (A → C) :=
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by
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sorry
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-- (c) Currying
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theorem curry (A B C : Prop) : (A → B → C) → (A ∧ B → C) :=
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by
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sorry
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#+end_src
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*** Exercise 3.2 --- Conjunction and disjunction
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:PROPERTIES:
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:CUSTOM_ID: exercise-3.2-conjunction-and-disjunction
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:END:
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#+begin_src lean
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-- (a) Swap conjuncts
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theorem and_comm (A B : Prop) : A ∧ B → B ∧ A :=
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by
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sorry
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-- (b) Disjunction is symmetric
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theorem or_comm (A B : Prop) : A ∨ B → B ∨ A :=
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by
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sorry
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-- (c) Forward reasoning with `have`
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theorem forward_example (A B C : Prop) (h1 : A → B) (h2 : B → C) (ha : A) : C :=
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by
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have hb : B := h1 ha
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sorry
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#+end_src
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*** Exercise 3.3 --- Negation
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:PROPERTIES:
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:CUSTOM_ID: exercise-3.3-negation
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:END:
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#+begin_src lean
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-- (a) Contradiction
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theorem contrapositive (A B : Prop) : (A → B) → (¬ B → ¬ A) :=
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by
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sorry
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-- (b) Double negation introduction
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theorem double_neg_intro (A : Prop) : A → ¬ ¬ A :=
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by
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sorry
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-- (c) Ex falso quodlibet
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theorem ex_falso (A : Prop) : False → A :=
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by
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sorry
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#+end_src
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--------------
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← [[file:02-inductive-types.org][Previous: Unit 2]] · Next: [[file:04-quantifiers-and-equality.org][Unit 4 --- Quantifiers and Equality]]
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