# Unit 3 — Propositions and Proofs

**Tutorial 1: Lean 4 Fundamentals** · [← Back to README](../README.md)

## Goals

- Understand propositions as types (Curry-Howard)
- Write basic proofs using `example` and `theorem`
- Use the tactic language: `intro`, `apply`, `exact`, `have`

## Source

*Theorem Proving in Lean 4* (TPIL), Chapters 2–3
→ https://leanprover.github.io/theorem_proving_in_lean4/

## Concepts

| 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

### Exercise 3.1 — Implicational logic

```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
```

### Exercise 3.2 — Conjunction and disjunction

```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
```

### Exercise 3.3 — Negation

```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
```

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