lean-pl-tutorials/tutorial-01-basics/03-propositions-and-proofs.org

• Unit 3 — Propositions and Proofs
  • Goals
  • Source
  • Concepts
  • Exercises
    • Exercise 3.1 — Implicational logic
    • Exercise 3.2 — Conjunction and disjunction
    • Exercise 3.3 — Negation

Unit 3 — Propositions and Proofs

Tutorial 1: Lean 4 Fundamentals · ← Back to README

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

-- (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

-- (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

-- (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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