Initial commit: 15-unit Lean 4 + PL semantics tutorial curriculum

tutorial-01-basics/03-propositions-and-proofs.md

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+# 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
+```
+
+---
+
+← [Previous: Unit 2](02-inductive-types.md) · Next: [Unit 4 — Quantifiers and Equality](04-quantifiers-and-equality.md)