1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
8. 8
9. 9
10. 10
11. 11
12. 12
13. 13
14. 14
15. 15
16. 16
17. 17
18. 18
19. 19
20. 20
21. 21
22. 22
23. 23
24. 24
25. 25
26. 26
27. 27
28. 28
29. 29
30. 30
31. 31
32. 32
33. 33
34. 34
35. 35
36. 36
37. 37
38. 38
39. 39
40. 40
41. 41
42. 42
43. 43
44. 44
45. 45
46. 46
47. 47
48. 48
49. 49
50. 50
51. 51
52. 52
53. 53
54. 54
55. 55
56. 56
57. 57

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.Parity
import Mathlib.Tactic

open Nat

-- These are pieces of data.
#check 2 + 2

def f (x : ℕ) :=
  x + 3

#check f

-- These are propositions, of type `Prop`.
#check 2 + 2 = 4

def FermatLastTheorem :=
  ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n

#check FermatLastTheorem

-- These are proofs of propositions.
theorem easy : 2 + 2 = 4 :=
  rfl

#check easy

theorem hard : FermatLastTheorem :=
  sorry

#check hard

-- Here are some proofs.
example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ ↦
  have hmn : m * n = m * k + m * k := by rw [hk, mul_add]
  show ∃ l, m * n = l + l from ⟨_, hmn⟩

example : ∀ m n : Nat, Even n → Even (m * n) :=
fun m n ⟨k, hk⟩ ↦ ⟨m * k, by rw [hk, mul_add]⟩

example : ∀ m n : Nat, Even n → Even (m * n) := by
  -- say m and n are natural numbers, and assume n=2*k
  rintro m n ⟨k, hk⟩
  -- We need to prove m*n is twice a natural number. Let's show it's twice m*k.
  use m * k
  -- substitute in for n
  rw [hk]
  -- and now it's obvious
  ring

example : ∀ m n : Nat, Even n → Even (m * n) := by
  rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring

example : ∀ m n : Nat, Even n → Even (m * n) := by
  intros; simp [*, parity_simps]