def SequenceHasLimit (u : ℕ → ℝ) (l : ℝ) : Prop

The sequence u of real numbers converges to l.

Equations

• SequenceHasLimit u l = ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, |u n - l| < ε

def ContinuousAtPoint (f : ℝ → ℝ) (x₀ : ℝ) : Prop

The functionf : ℝ → ℝ is continuous at x₀.

Equations

• ContinuousAtPoint f x₀ = ∀ ε > 0, ∃ δ > 0, ∀ (x : ℝ), |x - x₀| < δ → |f x - f x₀| < ε

theorem first_proof (f : ℝ → ℝ) (u : ℕ → ℝ) (x₀ : ℝ) (u_lim : SequenceHasLimit u x₀) (f_cont : ContinuousAtPoint f x₀) : SequenceHasLimit (f ∘ u) (f x₀)

How does Lean help you?

• You can use Lean to verify all the details of a proof.
• Lean helps you during a proof by
  • displaying all information in the tactic state
  • keeping a proof organized
  • proving parts automatically using AI
• You can explore mathematics using Lean's mathematical library Mathlib

General context

Proof assistants software exist since the early 70s.

There is currently a lot of momentum in formalized mathematics, especially Lean:

• AlphaProof
• Terrence Tao has started a few formalization projects
• A proof by Peter Scholze in condensed mathematics was verified in Lean.

Lean exists since 2013, and its mathematical library Mathlib since 2017.

def fib : ℕ → ℕ

Equations

• fib 0 = 1
• fib 1 = 1
• fib n.succ.succ = fib n + fib (n + 1)

def Statement1 : Prop

Equations

• Statement1 = ∃ (p : ℕ), Prime p ∧ Prime (p + 2) ∧ Prime (p + 4)

def Statement2 : Prop

Equations

• Statement2 = ∀ (n : ℕ), ∃ p ≥ n, Prime p ∧ Prime (p + 2) ∧ Prime (p + 4)

def Statement3 : Prop

Equations

• Statement3 = ∀ (n : ℕ), ∃ p ≥ n, Prime p ∧ Prime (p + 2)

def add_complex_i (y : ℂ) : ℂ

Equations

• add_complex_i y = y + Complex.I

def less_than_pi (x : ℝ) : Prop

Equations

• less_than_pi x = (x < Real.pi)

def less_than (x : ℝ) (y : ℝ) : Prop

Equations

• less_than x y = (x < y)

def differentiable (f : ℝ → ℝ) : Prop

Equations

• differentiable f = Differentiable ℝ f