bookshelf/Bookshelf/List/Basic.lean

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import Mathlib.Data.Fintype.Basic
import Mathlib.Tactic.NormNum
namespace List
-- ========================================
-- Length
-- ========================================
/--
Only the empty list has length zero.
-/
theorem length_zero_iff_self_eq_nil : length xs = 0 ↔ xs = [] := by
apply Iff.intro
· intro h
cases xs with
| nil => rfl
| cons a as => simp at h
· intro h
rw [h]
simp
/--
If the length of a list is greater than zero, it cannot be `List.nil`.
-/
theorem length_gt_zero_imp_not_nil : xs.length > 0 → xs ≠ [] := by
intro h
by_contra nh
rw [nh] at h
have : 0 > 0 := calc 0
_ = length [] := by rw [← length_zero_iff_self_eq_nil.mpr nh, nh]
_ > 0 := h
simp at this
-- ========================================
-- Membership
-- ========================================
/--
A list is nonempty if and only if it can be written as a head concatenated with
a tail.
-/
theorem self_nonempty_imp_exists_mem : xs ≠ [] ↔ (∃ a as, xs = a :: as) := by
apply Iff.intro
· intro h
cases hx : xs with
| nil => rw [hx] at h; simp at h
| cons a as => exact ⟨a, ⟨as, rfl⟩⟩
· intro ⟨a, ⟨as, hx⟩⟩
rw [hx]
simp
/--
If there exists a member of a list, the list must be nonempty.
-/
theorem nonempty_iff_mem : xs ≠ [] ↔ ∃ x, x ∈ xs := by
apply Iff.intro
· intro hx
cases xs with
| nil => simp at hx
| cons a as => exact ⟨a, by simp⟩
· intro ⟨x, hx⟩
induction hx <;> simp
/--
Getting an element `i` from a list is equivalent to `get`ting an element `i + 1`
from that list as a tail.
-/
theorem get_cons_succ_self_eq_get_tail_self
: get (x :: xs) (Fin.succ i) = get xs i := by
conv => lhs; unfold get; simp only
/--
Any value that can be retrieved via `get` must be a member of the list argument.
-/
theorem get_mem_self {xs : List α} {i : Fin xs.length} : get xs i ∈ xs := by
induction xs with
| nil => have ⟨_, hj⟩ := i; simp at hj
| cons a as ih =>
by_cases hk : i = ⟨0, by simp⟩
· -- If `i = 0`, we are `get`ting the head of our list. This element is
-- trivially a member of `xs`.
conv => lhs; unfold get; rw [hk]; simp only
simp
· -- Otherwise we are `get`ting an element in the tail. Our induction
-- hypothesis closes this case.
have ⟨k', hk'⟩ : ∃ k', i = Fin.succ k' := by
have ni : ↑i ≠ (0 : ) := fun hi => hk (Fin.ext hi)
have ⟨j, hj⟩ := Nat.exists_eq_succ_of_ne_zero ni
refine ⟨⟨j, ?_⟩, Fin.ext hj⟩
have hi : ↑i < length (a :: as) := i.2
unfold length at hi
rwa [hj, show Nat.succ j = j + 1 by rfl, add_lt_add_iff_right] at hi
conv => lhs; rw [hk', get_cons_succ_self_eq_get_tail_self]
exact mem_append_of_mem_right [a] ih
/--
`x` is a member of list `xs` if and only if there exists some index of `xs` that
`x` corresponds to.
-/
theorem mem_iff_exists_get {xs : List α}
: x ∈ xs ↔ ∃ i : Fin xs.length, xs.get i = x := by
apply Iff.intro
· intro h
induction h with
| head _ => refine ⟨⟨0, ?_⟩, ?_⟩ <;> simp
| @tail b as _ ih =>
let ⟨i, hi⟩ := ih
refine ⟨⟨i.1 + 1, ?_⟩, ?_⟩
· unfold length; simp
· simp; exact hi
· intro ⟨i, hi⟩
induction xs with
| nil => have nh := i.2; simp at nh
| cons a bs => rw [← hi]; exact get_mem_self
-- ========================================
-- Zips
-- ========================================
/--
The length of a list zipped with its tail is the length of the tail.
-/
theorem length_zip_with_self_tail_eq_length_sub_one
: length (zipWith f (a :: as) as) = length as := by
rw [length_zipWith]
simp only [length_cons, ge_iff_le, min_eq_right_iff]
show length as ≤ length as + 1
simp only [le_add_iff_nonneg_right]
/--
The result of a `zipWith` is nonempty if and only if both arguments are
nonempty.
-/
theorem zip_with_nonempty_iff_args_nonempty
: zipWith f as bs ≠ [] ↔ as ≠ [] ∧ bs ≠ [] := by
apply Iff.intro
· intro h
rw [self_nonempty_imp_exists_mem] at h
have ⟨z, ⟨zs, hzs⟩⟩ := h
refine ⟨?_, ?_⟩ <;>
· by_contra nh
rw [nh] at hzs
simp at hzs
· intro ⟨ha, hb⟩
have ⟨a', ⟨as', has⟩⟩ := self_nonempty_imp_exists_mem.mp ha
have ⟨b', ⟨bs', hbs⟩⟩ := self_nonempty_imp_exists_mem.mp hb
rw [has, hbs]
simp
private lemma fin_zip_with_imp_val_lt_length_left {i : Fin (zipWith f xs ys).length}
: i.1 < length xs := by
have hi := i.2
simp only [length_zipWith, ge_iff_le, lt_min_iff] at hi
exact hi.left
private lemma fin_zip_with_imp_val_lt_length_right {i : Fin (zipWith f xs ys).length}
: i.1 < length ys := by
have hi := i.2
simp only [length_zipWith, ge_iff_le, lt_min_iff] at hi
exact hi.right
/--
Calling `get _ i` on a zip of `xs` and `ys` is the same as applying the function
argument to each of `get xs i` and `get ys i` directly.
-/
theorem get_zip_with_apply_get_get {i : Fin (zipWith f xs ys).length}
: get (zipWith f xs ys) i = f
(get xs ⟨i.1, fin_zip_with_imp_val_lt_length_left⟩)
(get ys ⟨i.1, fin_zip_with_imp_val_lt_length_right⟩) := by
sorry
-- ========================================
-- Pairwise
-- ========================================
/--
Given a list `xs` of length `k`, produces a list of length `k - 1` where the
`i`th member of the resulting list is `f xs[i] xs[i + 1]`.
-/
def pairwise (xs : List α) (f : αα → β) : List β :=
match xs.tail? with
| none => []
| some ys => zipWith f xs ys
/--
If list `xs` is empty, then any `pairwise` operation on `xs` yields an empty
list.
-/
theorem len_pairwise_len_nil_eq_zero {xs : List α} (h : xs = [])
: (xs.pairwise f).length = 0 := by
rw [h]
unfold pairwise tail? length
simp
/--
If `List` `xs` is nonempty, then any `pairwise` operation on `xs` has length
`xs.length - 1`.
-/
theorem len_pairwise_len_cons_sub_one {xs : List α} (h : xs.length > 0)
: xs.length = (xs.pairwise f).length + 1 := by
unfold pairwise tail?
cases xs with
| nil =>
have h' := length_gt_zero_imp_not_nil h
simp at h'
| cons a bs =>
suffices length (zipWith f (a :: bs) bs) = length bs by
rw [this]
simp
rw [length_zip_with_self_tail_eq_length_sub_one]
/--
If the `pairwise` list isn't empty, then the original list must have at least
two elements.
-/
theorem mem_pairwise_imp_length_self_ge_2 {xs : List α} (h : xs.pairwise f ≠ [])
: xs.length ≥ 2 := by
unfold pairwise tail? at h
cases hx : xs with
| nil => rw [hx] at h; simp at h
| cons a bs =>
rw [hx] at h
cases hx' : bs with
| nil => rw [hx'] at h; simp at h
| cons a' bs' => unfold length length; rw [add_assoc]; norm_num
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/--
If `x` is a member of the pairwise'd list, there must exist two (adjacent)
elements of the list, say `x₁` and `x₂`, such that `x = f x₁ x₂`.
-/
theorem mem_pairwise_imp_exists {xs : List α} (h : x ∈ xs.pairwise f)
: ∃ x₁ x₂, x₁ ∈ xs ∧ x₂ ∈ xs ∧ x = f x₁ x₂ := by
unfold pairwise at h
cases hys : tail? xs with
| none => rw [hys] at h; cases h
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| some ys =>
rw [hys] at h
2023-04-24 18:59:11 +00:00
simp only at h
-- Since our `tail?` result isn't `none`, we should be able to decompose
-- `xs` into concatenation operands.
have ⟨r, hrs⟩ : ∃ r, xs = r :: ys := by
unfold tail? at hys
cases xs with
| nil => simp at hys
| cons r rs => exact ⟨r, by simp at hys; rw [hys]⟩
-- Maintain a collection of relations related to `i` and the length of `xs`.
-- Because of the proof-carrying `Fin` index, we find ourselves needing to
-- cast these values around periodically.
have ⟨i, hx⟩ := mem_iff_exists_get.mp h
have succ_i_lt_length_xs : ↑i + 1 < length xs := by
have hi := add_lt_add_right i.2 1
conv at hi => rhs; rw [hrs, length_zip_with_self_tail_eq_length_sub_one]
conv => rhs; rw [congrArg length hrs]; unfold length
exact hi
have succ_i_lt_length_cons_r_ys : ↑i + 1 < length (r :: ys) := by
have hi := i.2
conv at hi => rhs; rw [hrs, length_zip_with_self_tail_eq_length_sub_one]
exact add_lt_add_right hi 1
have i_lt_length_ys : ↑i < length ys := by
unfold length at succ_i_lt_length_cons_r_ys
exact Nat.lt_of_succ_lt_succ succ_i_lt_length_cons_r_ys
-- Choose the indices `x₁` and `x₂` that correspond to our `x` member. We
-- massage these values into the correct shape and then prove `x = f x₁ x₂`.
let x₁ := xs.get ⟨i, fin_zip_with_imp_val_lt_length_left⟩
let x₂ := xs.get ⟨i + 1, succ_i_lt_length_xs⟩
have hx₁ : x₁ = xs.get ⟨i, fin_zip_with_imp_val_lt_length_left⟩ := rfl
have hx₂ : x₂ = get (r :: ys) { val := ↑i + 1, isLt := succ_i_lt_length_cons_r_ys } := by
rw [show x₂ = xs.get _ by rfl]
congr
exact Eq.recOn
(motive := fun x h => HEq
succ_i_lt_length_xs
(cast (show (↑i + 1 < length xs) = (↑i + 1 < length x) by rw [← h])
succ_i_lt_length_xs))
(show xs = r :: ys from hrs)
HEq.rfl
refine ⟨x₁, ⟨x₂, ⟨get_mem_self, ⟨get_mem_self, ?_⟩⟩⟩⟩
have hx₂_offset_idx
: get (r :: ys) { val := ↑i + 1, isLt := succ_i_lt_length_cons_r_ys}
= get ys { val := ↑i, isLt := i_lt_length_ys } := by
conv => lhs; unfold get; simp
rw [hx₂_offset_idx] at hx₂
rw [get_zip_with_apply_get_get, ← hx₁, ← hx₂] at hx
exact Eq.symm hx
end List