bookshelf/Bookshelf/Tuple.lean

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import Mathlib.Tactic.Ring
/--
`n`-tuples are defined recursively as such:
`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
We allow empty tuples. For a `Tuple`-like type with opposite "endian", refer to
`Mathlib.Data.Vector`.
-/
inductive Tuple : (α : Type u) → (size : Nat) → Type u where
| nil : Tuple α 0
| snoc : Tuple α n → α → Tuple α (n + 1)
syntax (priority := high) "t[" term,* "]" : term
macro_rules
| `(t[]) => `(Tuple.nil)
| `(t[$x]) => `(Tuple.snoc t[] $x)
| `(t[$xs:term,*, $x]) => `(Tuple.snoc t[$xs,*] $x)
namespace Tuple
-- ========================================
-- Coercions
-- ========================================
scoped instance : CoeOut (Tuple α (min (m + n) m)) (Tuple α m) where
coe := cast (by simp)
scoped instance : Coe (Tuple α 0) (Tuple α (min n 0)) where
coe := cast (by rw [Nat.min_zero])
scoped instance : Coe (Tuple α 0) (Tuple α (min 0 n)) where
coe := cast (by rw [Nat.zero_min])
scoped instance : Coe (Tuple α n) (Tuple α (min n n)) where
coe := cast (by simp)
scoped instance : Coe (Tuple α n) (Tuple α (0 + n)) where
coe := cast (by simp)
scoped instance : Coe (Tuple α (min m n + 1)) (Tuple α (min (m + 1) (n + 1))) where
coe := cast (by rw [Nat.min_succ_succ])
scoped instance : Coe (Tuple α m) (Tuple α (min (m + n) m)) where
coe := cast (by simp)
-- ========================================
-- Equality
-- ========================================
theorem eq_nil : @Tuple.nil α = t[] := rfl
theorem eq_iff_singleton : (a = b) ↔ (t[a] = t[b]) := by
apply Iff.intro
· intro h; rw [h]
· intro h; injection h
theorem eq_iff_snoc {t₁ t₂ : Tuple α n}
: (a = b ∧ t₁ = t₂) ↔ (snoc t₁ a = snoc t₂ b) := by
apply Iff.intro
· intro ⟨h₁, h₂ ⟩; rw [h₁, h₂]
· intro h
injection h with _ h₁ h₂
exact And.intro h₂ h₁
/--
Implements decidable equality for `Tuple α m`, provided `a` has decidable
equality.
-/
protected def hasDecEq [DecidableEq α] (t₁ t₂ : Tuple α n)
: Decidable (Eq t₁ t₂) :=
match t₁, t₂ with
| t[], t[] => isTrue eq_nil
| snoc as a, snoc bs b =>
match Tuple.hasDecEq as bs with
| isFalse np => isFalse (fun h => absurd (eq_iff_snoc.mpr h).right np)
| isTrue hp =>
if hq : a = b then
isTrue (eq_iff_snoc.mp $ And.intro hq hp)
else
isFalse (fun h => absurd (eq_iff_snoc.mpr h).left hq)
instance [DecidableEq α] : DecidableEq (Tuple α n) := Tuple.hasDecEq
-- ========================================
-- Basic API
-- ========================================
/--
Returns the number of entries of the `Tuple`.
-/
def size (_ : Tuple α n) : Nat := n
/--
Returns all but the last entry of the `Tuple`.
-/
def init : (t : Tuple α (n + 1)) → Tuple α n
| snoc vs _ => vs
/--
Returns the last entry of the `Tuple`.
-/
def last : Tuple α (n + 1) → α
| snoc _ v => v
/--
Prepends an entry to the start of the `Tuple`.
-/
def cons : Tuple α n → α → Tuple α (n + 1)
| t[], a => t[a]
| snoc ts t, a => snoc (cons ts a) t
-- ========================================
-- Concatenation
-- ========================================
/--
Join two `Tuple`s together end to end.
-/
def concat : Tuple α m → Tuple α n → Tuple α (m + n)
| is, t[] => is
| is, snoc ts t => snoc (concat is ts) t
/--
Concatenating a `Tuple` with `nil` yields the original `Tuple`.
-/
theorem self_concat_nil_eq_self (t : Tuple α m) : concat t t[] = t :=
match t with
| t[] => rfl
| snoc _ _ => rfl
/--
Concatenating `nil` with a `Tuple` yields the `Tuple`.
-/
theorem nil_concat_self_eq_self (t : Tuple α m) : concat t[] t = t := by
induction t with
| nil => unfold concat; simp
| @snoc n as a ih =>
unfold concat
rw [ih]
suffices HEq (snoc (cast (_ : Tuple α n = Tuple α (0 + n)) as) a) ↑(snoc as a)
from eq_of_heq this
have h₁ := Eq.recOn
(motive := fun x h => HEq
(snoc (cast (show Tuple α n = Tuple α x by rw [h]) as) a)
(snoc as a))
(show n = 0 + n by simp)
HEq.rfl
exact Eq.recOn
(motive := fun x h => HEq
(snoc (cast (_ : Tuple α n = Tuple α (0 + n)) as) a)
(cast h (snoc as a)))
(show Tuple α (n + 1) = Tuple α (0 + (n + 1)) by simp)
h₁
/--
Concatenating a `Tuple` to a nonempty `Tuple` moves `concat` calls closer to
expression leaves.
-/
theorem concat_snoc_snoc_concat {bs : Tuple α n}
: concat as (snoc bs b) = snoc (concat as bs) b :=
rfl
/--
`snoc` is equivalent to concatenating the `init` and `last` element together.
-/
theorem snoc_eq_init_concat_last (as : Tuple α m)
: snoc as a = concat as t[a] := by
cases as with
| nil => rfl
| snoc _ _ => simp; unfold concat concat; rfl
-- ========================================
-- Initial sequences
-- ========================================
/--
Take the first `k` entries from the `Tuple` to form a new `Tuple`, or the entire
`Tuple` if `k` exceeds the number of entries.
-/
def take (t : Tuple α n) (k : Nat) : Tuple α (min n k) :=
if h : n ≤ k then
cast (by rw [min_eq_left h]) t
else
match t with
| t[] => t[]
| @snoc _ n' as a => cast (by rw [min_lt_succ_eq h]) (take as k)
where
min_lt_succ_eq {m : Nat} (h : ¬m + 1 ≤ k) : min m k = min (m + 1) k := by
have h' : k + 1 ≤ m + 1 := Nat.lt_of_not_le h
simp at h'
rw [min_eq_right h', min_eq_right (Nat.le_trans h' (Nat.le_succ m))]
/--
Taking no entries from any `Tuple` should yield an empty one.
-/
theorem self_take_zero_eq_nil (t : Tuple α n) : take t 0 = @nil α := by
induction t with
| nil => simp; rfl
| snoc as a ih => unfold take; simp; rw [ih]; simp
/--
Taking any number of entries from an empty `Tuple` should yield an empty one.
-/
theorem nil_take_zero_eq_nil (k : Nat) : (take (@nil α) k) = @nil α := by
cases k <;> (unfold take; simp)
/--
Taking `n` entries from a `Tuple` of size `n` should yield the same `Tuple`.
-/
theorem self_take_size_eq_self (t : Tuple α n) : take t n = t := by
cases t with
| nil => simp; rfl
| snoc as a => unfold take; simp
/--
Taking all but the last entry of a `Tuple` is the same result, regardless of the
value of the last entry.
-/
theorem take_subst_last {as : Tuple α n} (a₁ a₂ : α)
: take (snoc as a₁) n = take (snoc as a₂) n := by
unfold take
simp
/--
Taking `n` elements from a tuple of size `n + 1` is the same as invoking `init`.
-/
theorem init_eq_take_pred (t : Tuple α (n + 1)) : take t n = init t := by
cases t with
| snoc as a =>
unfold init take
simp
rw [self_take_size_eq_self]
simp
/--
If two `Tuple`s are equal, then any initial sequences of those two `Tuple`s are
also equal.
-/
theorem eq_tuple_eq_take {t₁ t₂ : Tuple α n}
: (t₁ = t₂) → (t₁.take k = t₂.take k) := by
intro h
rw [h]
/--
Given a `Tuple` of size `k`, concatenating an arbitrary `Tuple` and taking `k`
elements yields the original `Tuple`.
-/
theorem eq_take_concat {t₁ : Tuple α m} {t₂ : Tuple α n}
: take (concat t₁ t₂) m = t₁ := by
induction t₂ with
| nil =>
simp
rw [self_concat_nil_eq_self, self_take_size_eq_self]
| @snoc n' as a ih =>
simp
rw [concat_snoc_snoc_concat]
unfold take
simp
rw [ih]
simp
end Tuple