Add length-related theorems and getters.

common/Bookshelf/Tuple.lean

@@ -3,6 +3,9 @@
 
 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
@@ -10,47 +13,81 @@ import Mathlib.Tactic.Ring
 universe u
 
 /--[1]
-An n-tuple is defined recursively as:
+An `n`-tuple is defined recursively as:
 
-  ⟨x₁, ..., xₙ₊₁⟩ = ⟨⟨x₁, ..., xₙ⟩, xₙ₊₁⟩
+  `⟨x₁, ..., xₙ₊₁⟩ = ⟨⟨x₁, ..., xₙ⟩, xₙ₊₁⟩`
 
-TODO: As [1] notes, it is useful to define ⟨x⟩ = x. Is this syntactically
-possible in Lean?
+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
-  | only : α → Tuple α 1
-  | cons : {n : Nat} → Tuple α n → α → Tuple α (n + 1)
+  | 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.only $x)
-  | `(⟨$xs:term,*, $x⟩) => `(Tuple.cons ⟨$xs,*⟩ $x)
+  | `(⟨$x⟩) => `(Tuple.snoc Tuple.nil $x)
+  | `(⟨$xs:term,*, $x⟩) => `(Tuple.snoc ⟨$xs,*⟩ $x)
 
 namespace Tuple
 
-/--
-Returns the value at the nth-index of the given tuple.
--/
-def index (t : Tuple α n) (m : Nat) : 1 ≤ m ∧ m ≤ n → α := by
-  intro h
-  cases t
-  · case only last => exact last
-  . case cons n' init last =>
-    by_cases k : m = n' + 1
+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
-    · exact index init m (And.intro h.left (by
-        have h₂ : m + 1 ≤ n' + 1 := Nat.lt_of_le_of_ne h.right k
-        norm_num at h₂
-        exact h₂))
+    · 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, 1 ≤ i ∧ i ≤ n → index t₁ i = index t₂ i) := by
+  : (t₁ = t₂) ↔ (∀ i : Nat, (p : 1 ≤ i ∧ i ≤ n) → index t₁ i p = index t₂ i p) := by
   apply Iff.intro
-  · sorry
+  · intro teq i hᵢ
+    sorry
   · sorry
 
+-/
+
 end Tuple