94 lines
2.6 KiB
Plaintext
94 lines
2.6 KiB
Plaintext
/-
|
||
# References
|
||
|
||
1. Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego:
|
||
Harcourt/Academic Press, 2001.
|
||
2. Axler, Sheldon. Linear Algebra Done Right. Undergraduate Texts in
|
||
Mathematics. Cham: Springer International Publishing, 2015.
|
||
https://doi.org/10.1007/978-3-319-11080-6.
|
||
-/
|
||
|
||
import Mathlib.Tactic.Ring
|
||
|
||
universe u
|
||
|
||
/--[1]
|
||
An `n`-tuple is defined recursively as:
|
||
|
||
`⟨x₁, ..., xₙ₊₁⟩ = ⟨⟨x₁, ..., xₙ⟩, xₙ₊₁⟩`
|
||
|
||
As [1] notes, it is useful to define `⟨x⟩ = x`. It is not clear this would be
|
||
possible in Lean though.
|
||
|
||
Though [1] does not describe a notion of an empty tuple, [2] does (though under
|
||
the name of a "list").
|
||
--/
|
||
inductive Tuple : (α : Type u) → Nat → Type u where
|
||
| nil : Tuple α 0
|
||
| snoc : {n : Nat} → Tuple α n → α → Tuple α (n + 1)
|
||
|
||
syntax (priority := high) "⟨" term,+ "⟩" : term
|
||
|
||
-- Notice the ambiguity this syntax introduces. For example, pattern `⟨a, b⟩`
|
||
-- could refer to a `2`-tuple or an `n`-tuple, where `a` is an `(n-1)`-tuple.
|
||
macro_rules
|
||
| `(⟨$x⟩) => `(Tuple.snoc Tuple.nil $x)
|
||
| `(⟨$xs:term,*, $x⟩) => `(Tuple.snoc ⟨$xs,*⟩ $x)
|
||
|
||
namespace Tuple
|
||
|
||
def length : Tuple α n → Nat
|
||
| Tuple.nil => 0
|
||
| Tuple.snoc init _ => length init + 1
|
||
|
||
theorem nil_length_zero : length (@Tuple.nil α) = 0 :=
|
||
rfl
|
||
|
||
theorem snoc_length_succ : length (Tuple.snoc init last) = length init + 1 :=
|
||
rfl
|
||
|
||
theorem tuple_length {n : Nat} (t : Tuple α n) : length t = n :=
|
||
Tuple.recOn t nil_length_zero
|
||
fun _ _ ih => by
|
||
rw [snoc_length_succ]
|
||
norm_num
|
||
exact ih
|
||
|
||
def head : {n : Nat} → Tuple α n → n ≥ 1 → α
|
||
| n + 1, Tuple.snoc init last, h => by
|
||
by_cases k : 0 = n
|
||
· exact last
|
||
· have h' : 0 ≤ n := Nat.le_of_succ_le_succ h
|
||
exact head init (Nat.lt_of_le_of_ne h' k)
|
||
|
||
def last : Tuple α n → n ≥ 1 → α
|
||
| Tuple.snoc _ last, _ => last
|
||
|
||
def index : {n : Nat} → Tuple α n → (k : Nat) → 1 ≤ k ∧ k ≤ n → α
|
||
| 0, _, m, h => by
|
||
have ff : 1 ≤ 0 := Nat.le_trans h.left h.right
|
||
ring_nf at ff
|
||
exact False.elim ff
|
||
| n + 1, Tuple.snoc init last, k, h => by
|
||
by_cases hₖ : k = n + 1
|
||
· exact last
|
||
· exact index init k $ And.intro
|
||
h.left
|
||
(Nat.le_of_lt_succ $ Nat.lt_of_le_of_ne h.right hₖ)
|
||
|
||
/-
|
||
|
||
-- TODO: Prove `eq_by_index`.
|
||
-- TODO: Prove Lemma 0A [1].
|
||
|
||
theorem eq_by_index (t₁ t₂ : Tuple α n)
|
||
: (t₁ = t₂) ↔ (∀ i : Nat, (p : 1 ≤ i ∧ i ≤ n) → index t₁ i p = index t₂ i p) := by
|
||
apply Iff.intro
|
||
· intro teq i hᵢ
|
||
sorry
|
||
· sorry
|
||
|
||
-/
|
||
|
||
end Tuple
|