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Div.lean
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Div.lean
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/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.WF
import Init.WFTactics
import Init.Data.Nat.Basic
namespace Nat
/--
Divisibility of natural numbers. `a ∣ b` (typed as `\|`) says that
there is some `c` such that `b = a * c`.
-/
instance : Dvd Nat where
dvd a b := Exists (fun c => b = a * c)
theorem div_rec_lemma {x y : Nat} : 0 < y ∧ y ≤ x → x - y < x :=
fun ⟨ypos, ylex⟩ => sub_lt (Nat.lt_of_lt_of_le ypos ylex) ypos
@[extern "lean_nat_div"]
protected def div (x y : @& Nat) : Nat :=
if 0 < y ∧ y ≤ x then
Nat.div (x - y) y + 1
else
0
decreasing_by apply div_rec_lemma; assumption
instance instDiv : Div Nat := ⟨Nat.div⟩
theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 else 0 := by
show Nat.div x y = _
rw [Nat.div]
rfl
def div.inductionOn.{u}
{motive : Nat → Nat → Sort u}
(x y : Nat)
(ind : ∀ x y, 0 < y ∧ y ≤ x → motive (x - y) y → motive x y)
(base : ∀ x y, ¬(0 < y ∧ y ≤ x) → motive x y)
: motive x y :=
if h : 0 < y ∧ y ≤ x then
ind x y h (inductionOn (x - y) y ind base)
else
base x y h
decreasing_by apply div_rec_lemma; assumption
theorem div_le_self (n k : Nat) : n / k ≤ n := by
induction n using Nat.strongInductionOn with
| ind n ih =>
rw [div_eq]
-- Note: manual split to avoid Classical.em which is not yet defined
cases (inferInstance : Decidable (0 < k ∧ k ≤ n)) with
| isFalse h => simp [h]
| isTrue h =>
suffices (n - k) / k + 1 ≤ n by simp [h, this]
have ⟨hK, hKN⟩ := h
have hSub : n - k < n := sub_lt (Nat.lt_of_lt_of_le hK hKN) hK
have : (n - k) / k ≤ n - k := ih (n - k) hSub
exact succ_le_of_lt (Nat.lt_of_le_of_lt this hSub)
theorem div_lt_self {n k : Nat} (hLtN : 0 < n) (hLtK : 1 < k) : n / k < n := by
rw [div_eq]
cases (inferInstance : Decidable (0 < k ∧ k ≤ n)) with
| isFalse h => simp [hLtN, h]
| isTrue h =>
suffices (n - k) / k + 1 < n by simp [h, this]
have ⟨_, hKN⟩ := h
have : (n - k) / k ≤ n - k := div_le_self (n - k) k
have := Nat.add_le_of_le_sub hKN this
exact Nat.lt_of_lt_of_le (Nat.add_lt_add_left hLtK _) this
@[extern "lean_nat_mod"]
protected def modCore (x y : @& Nat) : Nat :=
if 0 < y ∧ y ≤ x then
Nat.modCore (x - y) y
else
x
decreasing_by apply div_rec_lemma; assumption
@[extern "lean_nat_mod"]
protected def mod : @& Nat → @& Nat → Nat
/-
Nat.modCore is defined by well-founded recursion and thus irreducible. Nevertheless it is
desireable if trivial `Nat.mod` calculations, namely
* `Nat.mod 0 m` for all `m`
* `Nat.mod n (m+n)` for concrete literals `n`
reduce definitionally.
This property is desirable for `Fin n` literals, as it means `(ofNat 0 : Fin n).val = 0` by
definition.
-/
| 0, _ => 0
| n@(_ + 1), m =>
if m ≤ n -- NB: if n < m does not reduce as well as `m ≤ n`!
then Nat.modCore n m
else n
instance instMod : Mod Nat := ⟨Nat.mod⟩
protected theorem modCore_eq_mod (n m : Nat) : Nat.modCore n m = n % m := by
show Nat.modCore n m = Nat.mod n m
match n, m with
| 0, _ =>
rw [Nat.modCore]
exact if_neg fun ⟨hlt, hle⟩ => Nat.lt_irrefl _ (Nat.lt_of_lt_of_le hlt hle)
| (_ + 1), _ =>
rw [Nat.mod]; dsimp
refine iteInduction (fun _ => rfl) (fun h => ?false) -- cannot use `split` this early yet
rw [Nat.modCore]
exact if_neg fun ⟨_hlt, hle⟩ => h hle
theorem mod_eq (x y : Nat) : x % y = if 0 < y ∧ y ≤ x then (x - y) % y else x := by
rw [←Nat.modCore_eq_mod, ←Nat.modCore_eq_mod, Nat.modCore]
def mod.inductionOn.{u}
{motive : Nat → Nat → Sort u}
(x y : Nat)
(ind : ∀ x y, 0 < y ∧ y ≤ x → motive (x - y) y → motive x y)
(base : ∀ x y, ¬(0 < y ∧ y ≤ x) → motive x y)
: motive x y :=
div.inductionOn x y ind base
@[simp] theorem mod_zero (a : Nat) : a % 0 = a :=
have : (if 0 < 0 ∧ 0 ≤ a then (a - 0) % 0 else a) = a :=
have h : ¬ (0 < 0 ∧ 0 ≤ a) := fun ⟨h₁, _⟩ => absurd h₁ (Nat.lt_irrefl _)
if_neg h
(mod_eq a 0).symm ▸ this
theorem mod_eq_of_lt {a b : Nat} (h : a < b) : a % b = a :=
have : (if 0 < b ∧ b ≤ a then (a - b) % b else a) = a :=
have h' : ¬(0 < b ∧ b ≤ a) := fun ⟨_, h₁⟩ => absurd h₁ (Nat.not_le_of_gt h)
if_neg h'
(mod_eq a b).symm ▸ this
theorem mod_eq_sub_mod {a b : Nat} (h : a ≥ b) : a % b = (a - b) % b :=
match eq_zero_or_pos b with
| Or.inl h₁ => h₁.symm ▸ (Nat.sub_zero a).symm ▸ rfl
| Or.inr h₁ => (mod_eq a b).symm ▸ if_pos ⟨h₁, h⟩
theorem mod_lt (x : Nat) {y : Nat} : y > 0 → x % y < y := by
induction x, y using mod.inductionOn with
| base x y h₁ =>
intro h₂
have h₁ : ¬ 0 < y ∨ ¬ y ≤ x := Iff.mp (Decidable.not_and_iff_or_not _ _) h₁
match h₁ with
| Or.inl h₁ => exact absurd h₂ h₁
| Or.inr h₁ =>
have hgt : y > x := gt_of_not_le h₁
have heq : x % y = x := mod_eq_of_lt hgt
rw [← heq] at hgt
exact hgt
| ind x y h h₂ =>
intro h₃
have ⟨_, h₁⟩ := h
rw [mod_eq_sub_mod h₁]
exact h₂ h₃
theorem mod_le (x y : Nat) : x % y ≤ x := by
match Nat.lt_or_ge x y with
| Or.inl h₁ => rw [mod_eq_of_lt h₁]; apply Nat.le_refl
| Or.inr h₁ => match eq_zero_or_pos y with
| Or.inl h₂ => rw [h₂, Nat.mod_zero x]; apply Nat.le_refl
| Or.inr h₂ => exact Nat.le_trans (Nat.le_of_lt (mod_lt _ h₂)) h₁
@[simp] theorem zero_mod (b : Nat) : 0 % b = 0 := by
rw [mod_eq]
have : ¬ (0 < b ∧ b = 0) := by
intro ⟨h₁, h₂⟩
simp_all
simp [this]
@[simp] theorem mod_self (n : Nat) : n % n = 0 := by
rw [mod_eq_sub_mod (Nat.le_refl _), Nat.sub_self, zero_mod]
theorem mod_one (x : Nat) : x % 1 = 0 := by
have h : x % 1 < 1 := mod_lt x (by decide)
have : (y : Nat) → y < 1 → y = 0 := by
intro y
cases y with
| zero => intro _; rfl
| succ y => intro h; apply absurd (Nat.lt_of_succ_lt_succ h) (Nat.not_lt_zero y)
exact this _ h
theorem div_add_mod (m n : Nat) : n * (m / n) + m % n = m := by
rw [div_eq, mod_eq]
have h : Decidable (0 < n ∧ n ≤ m) := inferInstance
cases h with
| isFalse h => simp [h]
| isTrue h =>
simp [h]
have ih := div_add_mod (m - n) n
rw [Nat.left_distrib, Nat.mul_one, Nat.add_assoc, Nat.add_left_comm, ih, Nat.add_comm, Nat.sub_add_cancel h.2]
decreasing_by apply div_rec_lemma; assumption
theorem div_eq_sub_div (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 := by
rw [div_eq a, if_pos]; constructor <;> assumption
theorem mod_add_div (m k : Nat) : m % k + k * (m / k) = m := by
induction m, k using mod.inductionOn with rw [div_eq, mod_eq]
| base x y h => simp [h]
| ind x y h IH => simp [h]; rw [Nat.mul_succ, ← Nat.add_assoc, IH, Nat.sub_add_cancel h.2]
@[simp] protected theorem div_one (n : Nat) : n / 1 = n := by
have := mod_add_div n 1
rwa [mod_one, Nat.zero_add, Nat.one_mul] at this
@[simp] protected theorem div_zero (n : Nat) : n / 0 = 0 := by
rw [div_eq]; simp [Nat.lt_irrefl]
@[simp] protected theorem zero_div (b : Nat) : 0 / b = 0 :=
(div_eq 0 b).trans <| if_neg <| And.rec Nat.not_le_of_gt
theorem le_div_iff_mul_le (k0 : 0 < k) : x ≤ y / k ↔ x * k ≤ y := by
induction y, k using mod.inductionOn generalizing x with
(rw [div_eq]; simp [h]; cases x with | zero => simp [zero_le] | succ x => ?_)
| base y k h =>
simp only [add_one, succ_mul, false_iff, Nat.not_le]
refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_left ..)
exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩
| ind y k h IH =>
rw [Nat.add_le_add_iff_right, IH k0, succ_mul,
← Nat.add_sub_cancel (x*k) k, Nat.sub_le_sub_iff_right h.2, Nat.add_sub_cancel]
protected theorem div_div_eq_div_mul (m n k : Nat) : m / n / k = m / (n * k) := by
cases eq_zero_or_pos k with
| inl k0 => rw [k0, Nat.mul_zero, Nat.div_zero, Nat.div_zero] | inr kpos => ?_
cases eq_zero_or_pos n with
| inl n0 => rw [n0, Nat.zero_mul, Nat.div_zero, Nat.zero_div] | inr npos => ?_
apply Nat.le_antisymm
apply (le_div_iff_mul_le (Nat.mul_pos npos kpos)).2
rw [Nat.mul_comm n k, ← Nat.mul_assoc]
apply (le_div_iff_mul_le npos).1
apply (le_div_iff_mul_le kpos).1
(apply Nat.le_refl)
apply (le_div_iff_mul_le kpos).2
apply (le_div_iff_mul_le npos).2
rw [Nat.mul_assoc, Nat.mul_comm n k]
apply (le_div_iff_mul_le (Nat.mul_pos kpos npos)).1
apply Nat.le_refl
theorem div_mul_le_self : ∀ (m n : Nat), m / n * n ≤ m
| m, 0 => by simp
| m, n+1 => (le_div_iff_mul_le (Nat.succ_pos _)).1 (Nat.le_refl _)
theorem div_lt_iff_lt_mul (Hk : 0 < k) : x / k < y ↔ x < y * k := by
rw [← Nat.not_le, ← Nat.not_le]; exact not_congr (le_div_iff_mul_le Hk)
@[simp] theorem add_div_right (x : Nat) {z : Nat} (H : 0 < z) : (x + z) / z = (x / z) + 1 := by
rw [div_eq_sub_div H (Nat.le_add_left _ _), Nat.add_sub_cancel]
@[simp] theorem add_div_left (x : Nat) {z : Nat} (H : 0 < z) : (z + x) / z = (x / z) + 1 := by
rw [Nat.add_comm, add_div_right x H]
theorem add_mul_div_left (x z : Nat) {y : Nat} (H : 0 < y) : (x + y * z) / y = x / y + z := by
induction z with
| zero => rw [Nat.mul_zero, Nat.add_zero, Nat.add_zero]
| succ z ih => rw [mul_succ, ← Nat.add_assoc, add_div_right _ H, ih]; rfl
theorem add_mul_div_right (x y : Nat) {z : Nat} (H : 0 < z) : (x + y * z) / z = x / z + y := by
rw [Nat.mul_comm, add_mul_div_left _ _ H]
@[simp] theorem add_mod_right (x z : Nat) : (x + z) % z = x % z := by
rw [mod_eq_sub_mod (Nat.le_add_left ..), Nat.add_sub_cancel]
@[simp] theorem add_mod_left (x z : Nat) : (x + z) % x = z % x := by
rw [Nat.add_comm, add_mod_right]
@[simp] theorem add_mul_mod_self_left (x y z : Nat) : (x + y * z) % y = x % y := by
match z with
| 0 => rw [Nat.mul_zero, Nat.add_zero]
| succ z => rw [mul_succ, ← Nat.add_assoc, add_mod_right, add_mul_mod_self_left (z := z)]
@[simp] theorem add_mul_mod_self_right (x y z : Nat) : (x + y * z) % z = x % z := by
rw [Nat.mul_comm, add_mul_mod_self_left]
@[simp] theorem mul_mod_right (m n : Nat) : (m * n) % m = 0 := by
rw [← Nat.zero_add (m * n), add_mul_mod_self_left, zero_mod]
@[simp] theorem mul_mod_left (m n : Nat) : (m * n) % n = 0 := by
rw [Nat.mul_comm, mul_mod_right]
protected theorem div_eq_of_lt_le (lo : k * n ≤ m) (hi : m < (k + 1) * n) : m / n = k :=
have npos : 0 < n := (eq_zero_or_pos _).resolve_left fun hn => by
rw [hn, Nat.mul_zero] at hi lo; exact absurd lo (Nat.not_le_of_gt hi)
Nat.le_antisymm
(le_of_lt_succ ((Nat.div_lt_iff_lt_mul npos).2 hi))
((Nat.le_div_iff_mul_le npos).2 lo)
theorem sub_mul_div (x n p : Nat) (h₁ : n*p ≤ x) : (x - n*p) / n = x / n - p := by
match eq_zero_or_pos n with
| .inl h₀ => rw [h₀, Nat.div_zero, Nat.div_zero, Nat.zero_sub]
| .inr h₀ => induction p with
| zero => rw [Nat.mul_zero, Nat.sub_zero, Nat.sub_zero]
| succ p IH =>
have h₂ : n * p ≤ x := Nat.le_trans (Nat.mul_le_mul_left _ (le_succ _)) h₁
have h₃ : x - n * p ≥ n := by
apply Nat.le_of_add_le_add_right
rw [Nat.sub_add_cancel h₂, Nat.add_comm]
rw [mul_succ] at h₁
exact h₁
rw [sub_succ, ← IH h₂, div_eq_sub_div h₀ h₃]
simp [Nat.pred_succ, mul_succ, Nat.sub_sub]
theorem mul_sub_div (x n p : Nat) (h₁ : x < n*p) : (n * p - (x + 1)) / n = p - ((x / n) + 1) := by
have npos : 0 < n := (eq_zero_or_pos _).resolve_left fun n0 => by
rw [n0, Nat.zero_mul] at h₁; exact not_lt_zero _ h₁
apply Nat.div_eq_of_lt_le
focus
rw [Nat.mul_sub_right_distrib, Nat.mul_comm]
exact Nat.sub_le_sub_left ((div_lt_iff_lt_mul npos).1 (lt_succ_self _)) _
focus
show succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n
rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h₁),
fun h => succ_pred_eq_of_pos (Nat.sub_pos_of_lt h)] -- TODO: why is the function needed?
focus
rw [Nat.mul_sub_right_distrib, Nat.mul_comm]
exact Nat.sub_le_sub_left (div_mul_le_self ..) _
focus
rwa [div_lt_iff_lt_mul npos, Nat.mul_comm]
theorem mul_mod_mul_left (z x y : Nat) : (z * x) % (z * y) = z * (x % y) :=
if y0 : y = 0 then by
rw [y0, Nat.mul_zero, mod_zero, mod_zero]
else if z0 : z = 0 then by
rw [z0, Nat.zero_mul, Nat.zero_mul, Nat.zero_mul, mod_zero]
else by
induction x using Nat.strongInductionOn with
| _ n IH =>
have y0 : y > 0 := Nat.pos_of_ne_zero y0
have z0 : z > 0 := Nat.pos_of_ne_zero z0
cases Nat.lt_or_ge n y with
| inl yn => rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]
| inr yn =>
rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),
← Nat.mul_sub_left_distrib]
exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)
theorem div_eq_of_lt (h₀ : a < b) : a / b = 0 := by
rw [div_eq a, if_neg]
intro h₁
apply Nat.not_le_of_gt h₀ h₁.right
protected theorem mul_div_cancel (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
let t := add_mul_div_right 0 m H
rwa [Nat.zero_add, Nat.zero_div, Nat.zero_add] at t
protected theorem mul_div_cancel_left (m : Nat) {n : Nat} (H : 0 < n) : n * m / n = m := by
rw [Nat.mul_comm, Nat.mul_div_cancel _ H]
protected theorem div_le_of_le_mul {m n : Nat} : ∀ {k}, m ≤ k * n → m / k ≤ n
| 0, _ => by simp [Nat.div_zero, n.zero_le]
| succ k, h => by
suffices succ k * (m / succ k) ≤ succ k * n from
Nat.le_of_mul_le_mul_left this (zero_lt_succ _)
have h1 : succ k * (m / succ k) ≤ m % succ k + succ k * (m / succ k) := Nat.le_add_left _ _
have h2 : m % succ k + succ k * (m / succ k) = m := by rw [mod_add_div]
have h3 : m ≤ succ k * n := h
rw [← h2] at h3
exact Nat.le_trans h1 h3
@[simp] theorem mul_div_right (n : Nat) {m : Nat} (H : 0 < m) : m * n / m = n := by
induction n <;> simp_all [mul_succ]
@[simp] theorem mul_div_left (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
rw [Nat.mul_comm, mul_div_right _ H]
protected theorem div_self (H : 0 < n) : n / n = 1 := by
let t := add_div_right 0 H
rwa [Nat.zero_add, Nat.zero_div] at t
protected theorem div_eq_of_eq_mul_left (H1 : 0 < n) (H2 : m = k * n) : m / n = k :=
by rw [H2, Nat.mul_div_cancel _ H1]
protected theorem div_eq_of_eq_mul_right (H1 : 0 < n) (H2 : m = n * k) : m / n = k :=
by rw [H2, Nat.mul_div_cancel_left _ H1]
protected theorem mul_div_mul_left {m : Nat} (n k : Nat) (H : 0 < m) :
m * n / (m * k) = n / k := by rw [← Nat.div_div_eq_div_mul, Nat.mul_div_cancel_left _ H]
protected theorem mul_div_mul_right {m : Nat} (n k : Nat) (H : 0 < m) :
n * m / (k * m) = n / k := by rw [Nat.mul_comm, Nat.mul_comm k, Nat.mul_div_mul_left _ _ H]
theorem mul_div_le (m n : Nat) : n * (m / n) ≤ m := by
match n, Nat.eq_zero_or_pos n with
| _, Or.inl rfl => rw [Nat.zero_mul]; exact m.zero_le
| n, Or.inr h => rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le h]; exact Nat.le_refl _
end Nat