* Unit 3 --- Propositions and Proofs
:PROPERTIES:
:CUSTOM_ID: unit-3-propositions-and-proofs
:END:
*Tutorial 1: Lean 4 Fundamentals* · [[../README.org][← Back to README]]

** Goals
:PROPERTIES:
:CUSTOM_ID: goals
:END:
- Understand propositions as types (Curry-Howard)
- Write basic proofs using =example= and =theorem=
- Use the tactic language: =intro=, =apply=, =exact=, =have=

** Source
:PROPERTIES:
:CUSTOM_ID: source
:END:
/Theorem Proving in Lean 4/ (TPIL), Chapters 2--3
→ https://leanprover.github.io/theorem_proving_in_lean4/

** Concepts
:PROPERTIES:
:CUSTOM_ID: concepts
:END:
| Concept             | Lean                                                        |
|---------------------+-------------------------------------------------------------|
| Proposition         | =P : Prop= --- a type whose terms are proofs                |
| Implication =P → Q= | Function from proofs of =P= to proofs of =Q=                |
| Conjunction =P ∧ Q= | =And.intro= (constructor), =.left= / =.right= (projections) |
| Disjunction =P ∨ Q= | =Or.inl= / =Or.inr=                                         |
| Negation =¬ P=      | =P → False=                                                 |
| Tactics             | =intro=, =apply=, =exact=, =have=, =assumption=             |

** Exercises
:PROPERTIES:
:CUSTOM_ID: exercises
:END:
*** Exercise 3.1 --- Implicational logic
:PROPERTIES:
:CUSTOM_ID: exercise-3.1-implicational-logic
:END:
#+begin_src lean
-- (a) Identity
theorem id_prop (A : Prop) : A → A :=
  by
    intro h
    exact h

-- (b) Composition
theorem compose (A B C : Prop) : (A → B) → (B → C) → (A → C) :=
  by
    sorry

-- (c) Currying
theorem curry (A B C : Prop) : (A → B → C) → (A ∧ B → C) :=
  by
    sorry
#+end_src

*** Exercise 3.2 --- Conjunction and disjunction
:PROPERTIES:
:CUSTOM_ID: exercise-3.2-conjunction-and-disjunction
:END:
#+begin_src lean
-- (a) Swap conjuncts
theorem and_comm (A B : Prop) : A ∧ B → B ∧ A :=
  by
    sorry

-- (b) Disjunction is symmetric
theorem or_comm (A B : Prop) : A ∨ B → B ∨ A :=
  by
    sorry

-- (c) Forward reasoning with `have`
theorem forward_example (A B C : Prop) (h1 : A → B) (h2 : B → C) (ha : A) : C :=
  by
    have hb : B := h1 ha
    sorry
#+end_src

*** Exercise 3.3 --- Negation
:PROPERTIES:
:CUSTOM_ID: exercise-3.3-negation
:END:
#+begin_src lean
-- (a) Contradiction
theorem contrapositive (A B : Prop) : (A → B) → (¬ B → ¬ A) :=
  by
    sorry

-- (b) Double negation introduction
theorem double_neg_intro (A : Prop) : A → ¬ ¬ A :=
  by
    sorry

-- (c) Ex falso quodlibet
theorem ex_falso (A : Prop) : False → A :=
  by
    sorry
#+end_src

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