lemmas.lean

/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
prelude
import init.data.nat.basic init.data.nat.div init.meta init.algebra.functions
universes u

namespace nat
attribute [pre_smt] nat_zero_eq_zero

protected lemma add_comm : ∀ n m : ℕ, n + m = m + n
| n 0     := eq.symm (nat.zero_add n)
| n (m+1) :=
  suffices succ (n + m) = succ (m + n), from
    eq.symm (succ_add m n) ▸ this,
  congr_arg succ (add_comm n m)

protected lemma add_assoc : ∀ n m k : ℕ, (n + m) + k = n + (m + k)
| n m 0        := rfl
| n m (succ k) := by rw [add_succ, add_succ, add_assoc]; refl

protected lemma add_left_comm : ∀ (n m k : ℕ), n + (m + k) = m + (n + k) :=
left_comm nat.add nat.add_comm nat.add_assoc

protected lemma add_left_cancel : ∀ {n m k : ℕ}, n + m = n + k → m = k
| 0        m k := by simp [nat.zero_add] {contextual := tt}
| (succ n) m k := λ h,
  have n+m = n+k, by { simp [succ_add] at h, assumption },
  add_left_cancel this

protected lemma add_right_cancel {n m k : ℕ} (h : n + m = k + m) : n = k :=
have m + n = m + k, by rwa [nat.add_comm n m, nat.add_comm k m] at h,
nat.add_left_cancel this

lemma succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
assume h, nat.no_confusion h

lemma succ_ne_self : ∀ n : ℕ, succ n ≠ n
| 0     h := absurd h (nat.succ_ne_zero 0)
| (n+1) h := succ_ne_self n (nat.no_confusion h (λ h, h))

protected lemma one_ne_zero : 1 ≠ (0 : ℕ) :=
assume h, nat.no_confusion h

protected lemma zero_ne_one : 0 ≠ (1 : ℕ) :=
assume h, nat.no_confusion h

protected lemma eq_zero_of_add_eq_zero_right : ∀ {n m : ℕ}, n + m = 0 → n = 0
| 0     m := by simp [nat.zero_add]
| (n+1) m := λ h,
  begin
    exfalso,
    rw [add_one, succ_add] at h,
    apply succ_ne_zero _ h
  end

protected lemma eq_zero_of_add_eq_zero_left {n m : ℕ} (h : n + m = 0) : m = 0 :=
@nat.eq_zero_of_add_eq_zero_right m n (nat.add_comm n m ▸ h)

@[simp]
lemma pred_zero : pred 0 = 0 :=
rfl

@[simp]
lemma pred_succ (n : ℕ) : pred (succ n) = n :=
rfl

protected lemma mul_zero (n : ℕ) : n * 0 = 0 :=
rfl

lemma mul_succ (n m : ℕ) : n * succ m = n * m + n :=
rfl

protected theorem zero_mul : ∀ (n : ℕ), 0 * n = 0
| 0        := rfl
| (succ n) := by rw [mul_succ, zero_mul]

private meta def sort_add :=
`[simp [nat.add_assoc, nat.add_comm, nat.add_left_comm]]

lemma succ_mul : ∀ (n m : ℕ), (succ n) * m = (n * m) + m
| n 0        := rfl
| n (succ m) :=
  begin
    simp [mul_succ, add_succ, succ_mul n m],
    sort_add
  end

protected lemma right_distrib : ∀ (n m k : ℕ), (n + m) * k = n * k + m * k
| n m 0        := rfl
| n m (succ k) :=
  begin simp [mul_succ, right_distrib n m k], sort_add end

protected lemma left_distrib : ∀ (n m k : ℕ), n * (m + k) = n * m + n * k
| 0        m k := by simp [nat.zero_mul]
| (succ n) m k :=
  begin simp [succ_mul, left_distrib n m k], sort_add end

protected lemma mul_comm : ∀ (n m : ℕ), n * m = m * n
| n 0        := by rw [nat.zero_mul, nat.mul_zero]
| n (succ m) := by simp [mul_succ, succ_mul, mul_comm n m]

protected lemma mul_assoc : ∀ (n m k : ℕ), (n * m) * k = n * (m * k)
| n m 0        := rfl
| n m (succ k) := by simp [mul_succ, nat.left_distrib, mul_assoc n m k]

protected lemma mul_one : ∀ (n : ℕ), n * 1 = n := nat.zero_add

protected lemma one_mul (n : ℕ) : 1 * n = n :=
by rw [nat.mul_comm, nat.mul_one]

/- properties of inequality -/

protected lemma le_of_eq {n m : ℕ} (p : n = m) : n ≤ m :=
p ▸ less_than_or_equal.refl

lemma le_succ_of_le {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
nat.le_trans h (le_succ m)

lemma le_of_succ_le {n m : ℕ} (h : succ n ≤ m) : n ≤ m :=
nat.le_trans (le_succ n) h

protected lemma le_of_lt {n m : ℕ} (h : n < m) : n ≤ m :=
le_of_succ_le h

lemma lt.step {n m : ℕ} : n < m → n < succ m := less_than_or_equal.step

protected lemma eq_zero_or_pos (n : ℕ) : n = 0 ∨ 0 < n :=
by {cases n, exact or.inl rfl, exact or.inr (succ_pos _)}

protected lemma pos_of_ne_zero {n : nat} : n ≠ 0 → 0 < n :=
or.resolve_left n.eq_zero_or_pos

protected lemma lt_trans {n m k : ℕ} (h₁ : n < m) : m < k → n < k :=
nat.le_trans (less_than_or_equal.step h₁)

protected lemma lt_of_le_of_lt {n m k : ℕ} (h₁ : n ≤ m) : m < k → n < k :=
nat.le_trans (succ_le_succ h₁)

lemma lt.base (n : ℕ) : n < succ n := nat.le_refl (succ n)

lemma lt_succ_self (n : ℕ) : n < succ n := lt.base n

protected lemma le_antisymm {n m : ℕ} (h₁ : n ≤ m) : m ≤ n → n = m :=
less_than_or_equal.cases_on h₁ (λ a, rfl) (λ a b c, absurd (nat.lt_of_le_of_lt b c) (nat.lt_irrefl n))

protected lemma lt_or_ge : ∀ (a b : ℕ), a < b ∨ b ≤ a
| a 0     := or.inr a.zero_le
| a (b+1) :=
  match lt_or_ge a b with
  | or.inl h := or.inl (le_succ_of_le h)
  | or.inr h :=
    match nat.eq_or_lt_of_le h with
    | or.inl h1 := or.inl (h1 ▸ lt_succ_self b)
    | or.inr h1 := or.inr h1
    end
  end

protected lemma le_total {m n : ℕ} : m ≤ n ∨ n ≤ m :=
or.imp_left nat.le_of_lt (nat.lt_or_ge m n)

protected lemma lt_of_le_and_ne {m n : ℕ} (h1 : m ≤ n) : m ≠ n → m < n :=
or.resolve_right (or.swap (nat.eq_or_lt_of_le h1))

protected lemma lt_iff_le_not_le {m n : ℕ} : m < n ↔ (m ≤ n ∧ ¬ n ≤ m) :=
⟨λ hmn, ⟨nat.le_of_lt hmn, λ hnm, nat.lt_irrefl _ (nat.lt_of_le_of_lt hnm hmn)⟩,
 λ ⟨hmn, hnm⟩, nat.lt_of_le_and_ne hmn (λ heq, hnm (heq ▸ nat.le_refl _))⟩

instance : linear_order ℕ :=
{ le := nat.less_than_or_equal,
  le_refl := @nat.le_refl,
  le_trans := @nat.le_trans,
  le_antisymm := @nat.le_antisymm,
  le_total := @nat.le_total,
  lt := nat.lt,
  lt_iff_le_not_le := @nat.lt_iff_le_not_le,
  decidable_lt               := nat.decidable_lt,
  decidable_le               := nat.decidable_le,
  decidable_eq               := nat.decidable_eq }

protected lemma eq_zero_of_le_zero {n : nat} (h : n ≤ 0) : n = 0 :=
le_antisymm h n.zero_le

lemma succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
succ_le_succ

lemma lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
le_of_succ_le

lemma lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
le_of_succ_le_succ

lemma pred_lt_pred : ∀ {n m : ℕ}, n ≠ 0 → n < m → pred n < pred m
| 0         _       h₁ h := absurd rfl h₁
| n         0       h₁ h := absurd h n.not_lt_zero
| (succ n) (succ m) _  h := lt_of_succ_lt_succ h

lemma lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h

lemma succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h

protected lemma le_add_right : ∀ (n k : ℕ), n ≤ n + k
| n 0     := nat.le_refl n
| n (k+1) := le_succ_of_le (le_add_right n k)

protected lemma le_add_left (n m : ℕ): n ≤ m + n :=
nat.add_comm n m ▸ n.le_add_right m

lemma le.dest : ∀ {n m : ℕ}, n ≤ m → ∃ k, n + k = m
| n _ less_than_or_equal.refl := ⟨0, rfl⟩
| n _ (less_than_or_equal.step h) :=
  match le.dest h with
  | ⟨w, hw⟩ := ⟨succ w, hw ▸ add_succ n w⟩
  end

protected lemma le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
h ▸ n.le_add_right k

protected lemma add_le_add_left {n m : ℕ} (h : n ≤ m) (k : ℕ) : k + n ≤ k + m :=
match le.dest h with
| ⟨w, hw⟩ := @le.intro _ _ w begin rw [nat.add_assoc, hw] end
end

protected lemma add_le_add_right {n m : ℕ} (h : n ≤ m) (k : ℕ) : n + k ≤ m + k :=
begin rw [nat.add_comm n k, nat.add_comm m k], apply nat.add_le_add_left h end

protected lemma le_of_add_le_add_left {k n m : ℕ} (h : k + n ≤ k + m) : n ≤ m :=
match le.dest h with
| ⟨w, hw⟩ := @le.intro _ _ w
  begin
    rw [nat.add_assoc] at hw,
    apply nat.add_left_cancel hw
  end
end

protected lemma le_of_add_le_add_right {k n m : ℕ} : n + k ≤ m + k → n ≤ m :=
begin
  rw [nat.add_comm _ k, nat.add_comm _ k],
  apply nat.le_of_add_le_add_left
end

protected lemma add_le_add_iff_le_right (k n m : ℕ) : n + k ≤ m + k ↔ n ≤ m :=
  ⟨ nat.le_of_add_le_add_right , assume h, nat.add_le_add_right h _ ⟩

protected theorem lt_of_add_lt_add_left {k n m : ℕ} (h : k + n < k + m) : n < m :=
let h' := nat.le_of_lt h in
nat.lt_of_le_and_ne
  (nat.le_of_add_le_add_left h')
  (λ heq, nat.lt_irrefl (k + m) begin rw heq at h, assumption end)

protected lemma lt_of_add_lt_add_right {a b c : ℕ} (h : a + b < c + b) : a < c :=
nat.lt_of_add_lt_add_left $
show b + a < b + c, by rwa [nat.add_comm b a, nat.add_comm b c]

protected lemma add_lt_add_left {n m : ℕ} (h : n < m) (k : ℕ) : k + n < k + m :=
lt_of_succ_le (add_succ k n ▸ nat.add_le_add_left (succ_le_of_lt h) k)

protected lemma add_lt_add_right {n m : ℕ} (h : n < m) (k : ℕ) : n + k < m + k :=
nat.add_comm k m ▸ nat.add_comm k n ▸ nat.add_lt_add_left h k

protected lemma lt_add_of_pos_right {n k : ℕ} (h : 0 < k) : n < n + k :=
nat.add_lt_add_left h n

protected lemma lt_add_of_pos_left {n k : ℕ} (h : 0 < k) : n < k + n :=
by rw nat.add_comm; exact nat.lt_add_of_pos_right h

protected lemma add_lt_add {a b c d : ℕ} (h₁ : a < b) (h₂ : c < d) : a + c < b + d :=
lt_trans (nat.add_lt_add_right h₁ c) (nat.add_lt_add_left h₂ b)

protected lemma add_le_add {a b c d : ℕ} (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d :=
le_trans (nat.add_le_add_right h₁ c) (nat.add_le_add_left h₂ b)

protected lemma zero_lt_one : 0 < (1:nat) :=
zero_lt_succ 0

protected lemma mul_le_mul_left {n m : ℕ} (k : ℕ) (h : n ≤ m) : k * n ≤ k * m :=
match le.dest h with
| ⟨l, hl⟩ :=
  have k * n + k * l = k * m, by rw [← nat.left_distrib, hl],
  le.intro this
end

protected lemma mul_le_mul_right {n m : ℕ} (k : ℕ) (h : n ≤ m) : n * k ≤ m * k :=
nat.mul_comm k m ▸ nat.mul_comm k n ▸ k.mul_le_mul_left h

protected lemma mul_lt_mul_of_pos_left {n m k : ℕ} (h : n < m) (hk : 0 < k) : k * n < k * m :=
nat.lt_of_lt_of_le (nat.lt_add_of_pos_right hk)
  (mul_succ k n ▸ nat.mul_le_mul_left k (succ_le_of_lt h))

protected lemma mul_lt_mul_of_pos_right {n m k : ℕ} (h : n < m) (hk : 0 < k) : n * k < m * k :=
nat.mul_comm k m ▸ nat.mul_comm k n ▸ nat.mul_lt_mul_of_pos_left h hk

protected lemma le_of_mul_le_mul_left {a b c : ℕ} (h : c * a ≤ c * b) (hc : 0 < c) : a ≤ b :=
not_lt.1
  (assume h1 : b < a,
   have h2 : c * b < c * a, from nat.mul_lt_mul_of_pos_left h1 hc,
   not_le_of_gt h2 h)

lemma le_of_lt_succ {m n : nat} : m < succ n → m ≤ n :=
le_of_succ_le_succ

protected theorem eq_of_mul_eq_mul_left {m k n : ℕ} (Hn : 0 < n) (H : n * m = n * k) : m = k :=
le_antisymm (nat.le_of_mul_le_mul_left (le_of_eq H) Hn)
            (nat.le_of_mul_le_mul_left (le_of_eq H.symm) Hn)

/- sub properties -/

@[simp] protected lemma zero_sub : ∀ a : ℕ, 0 - a = 0
| 0     := rfl
| (a+1) := congr_arg pred (zero_sub a)

lemma sub_lt_succ (a b : ℕ) : a - b < succ a :=
lt_succ_of_le (a.sub_le b)

protected theorem sub_le_sub_right {n m : ℕ} (h : n ≤ m) : ∀ k, n - k ≤ m - k
| 0        := h
| (succ z) := pred_le_pred (sub_le_sub_right z)

/- bit0/bit1 properties -/

protected lemma bit1_eq_succ_bit0 (n : ℕ) : bit1 n = succ (bit0 n) :=
rfl

protected lemma bit1_succ_eq (n : ℕ) : bit1 (succ n) = succ (succ (bit1 n)) :=
eq.trans (nat.bit1_eq_succ_bit0 (succ n)) (congr_arg succ (nat.bit0_succ_eq n))

protected lemma bit1_ne_one : ∀ {n : ℕ}, n ≠ 0 → bit1 n ≠ 1
| 0     h h1 := absurd rfl h
| (n+1) h h1 := nat.no_confusion h1 (λ h2, absurd h2 (succ_ne_zero _))

protected lemma bit0_ne_one : ∀ n : ℕ, bit0 n ≠ 1
| 0     h := absurd h (ne.symm nat.one_ne_zero)
| (n+1) h :=
  have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
  nat.no_confusion h1
    (λ h2, absurd h2 (succ_ne_zero (n + n)))

protected lemma add_self_ne_one : ∀ (n : ℕ), n + n ≠ 1
| 0     h := nat.no_confusion h
| (n+1) h :=
  have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
  nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero (n + n)))

protected lemma bit1_ne_bit0 : ∀ (n m : ℕ), bit1 n ≠ bit0 m
| 0     m     h := absurd h (ne.symm (nat.add_self_ne_one m))
| (n+1) 0     h :=
  have h1 : succ (bit0 (succ n)) = 0, from h,
  absurd h1 (nat.succ_ne_zero _)
| (n+1) (m+1) h :=
  have h1 : succ (succ (bit1 n)) = succ (succ (bit0 m)), from
    nat.bit0_succ_eq m ▸ nat.bit1_succ_eq n ▸ h,
  have h2 : bit1 n = bit0 m, from
    nat.no_confusion h1 (λ h2', nat.no_confusion h2' (λ h2'', h2'')),
  absurd h2 (bit1_ne_bit0 n m)

protected lemma bit0_ne_bit1 : ∀ (n m : ℕ), bit0 n ≠ bit1 m :=
λ n m : nat, ne.symm (nat.bit1_ne_bit0 m n)

protected lemma bit0_inj : ∀ {n m : ℕ}, bit0 n = bit0 m → n = m
| 0     0     h := rfl
| 0     (m+1) h := by contradiction
| (n+1) 0     h := by contradiction
| (n+1) (m+1) h :=
  have succ (succ (n + n)) = succ (succ (m + m)),
  by { unfold bit0 at h, simp [add_one, add_succ, succ_add] at h,
       have aux : n + n = m + m := h, rw aux },
  have n + n = m + m, by iterate { injection this with this },
  have n = m, from bit0_inj this,
  by rw this

protected lemma bit1_inj : ∀ {n m : ℕ}, bit1 n = bit1 m → n = m :=
λ n m h,
have succ (bit0 n) = succ (bit0 m), begin simp [nat.bit1_eq_succ_bit0] at h, rw h end,
have bit0 n = bit0 m, by injection this,
nat.bit0_inj this

protected lemma bit0_ne {n m : ℕ} : n ≠ m → bit0 n ≠ bit0 m :=
λ h₁ h₂, absurd (nat.bit0_inj h₂) h₁

protected lemma bit1_ne {n m : ℕ} : n ≠ m → bit1 n ≠ bit1 m :=
λ h₁ h₂, absurd (nat.bit1_inj h₂) h₁

protected lemma zero_ne_bit0 {n : ℕ} : n ≠ 0 → 0 ≠ bit0 n :=
λ h, ne.symm (nat.bit0_ne_zero h)

protected lemma zero_ne_bit1 (n : ℕ) : 0 ≠ bit1 n :=
ne.symm (nat.bit1_ne_zero n)

protected lemma one_ne_bit0 (n : ℕ) : 1 ≠ bit0 n :=
ne.symm (nat.bit0_ne_one n)

protected lemma one_ne_bit1 {n : ℕ} : n ≠ 0 → 1 ≠ bit1 n :=
λ h, ne.symm (nat.bit1_ne_one h)

protected lemma one_lt_bit1 : ∀ {n : nat}, n ≠ 0 → 1 < bit1 n
| 0        h := by contradiction
| (succ n) h :=
  begin
    rw nat.bit1_succ_eq,
    apply succ_lt_succ,
    apply zero_lt_succ
  end

protected lemma one_lt_bit0 : ∀ {n : nat}, n ≠ 0 → 1 < bit0 n
| 0        h := by contradiction
| (succ n) h :=
  begin
    rw nat.bit0_succ_eq,
    apply succ_lt_succ,
    apply zero_lt_succ
  end

protected lemma bit0_lt {n m : nat} (h : n < m) : bit0 n < bit0 m :=
nat.add_lt_add h h

protected lemma bit1_lt {n m : nat} (h : n < m) : bit1 n < bit1 m :=
succ_lt_succ (nat.add_lt_add h h)

protected lemma bit0_lt_bit1 {n m : nat} (h : n ≤ m) : bit0 n < bit1 m :=
lt_succ_of_le (nat.add_le_add h h)

protected lemma bit1_lt_bit0 : ∀ {n m : nat}, n < m → bit1 n < bit0 m
| n 0        h := absurd h n.not_lt_zero
| n (succ m) h :=
  have n ≤ m, from le_of_lt_succ h,
  have succ (n + n) ≤ succ (m + m), from succ_le_succ (nat.add_le_add this this),
  have succ (n + n) ≤ succ m + m, {rw succ_add, assumption},
  show succ (n + n) < succ (succ m + m), from lt_succ_of_le this

protected lemma one_le_bit1 (n : ℕ) : 1 ≤ bit1 n :=
show 1 ≤ succ (bit0 n), from
succ_le_succ (bit0 n).zero_le

protected lemma one_le_bit0 : ∀ (n : ℕ), n ≠ 0 → 1 ≤ bit0 n
| 0     h := absurd rfl h
| (n+1) h :=
  suffices 1 ≤ succ (succ (bit0 n)), from
    eq.symm (nat.bit0_succ_eq n) ▸ this,
  succ_le_succ (bit0 n).succ.zero_le

/- subtraction -/
@[simp]
protected theorem sub_zero (n : ℕ) : n - 0 = n :=
rfl

theorem sub_succ (n m : ℕ) : n - succ m = pred (n - m) :=
rfl

theorem succ_sub_succ (n m : ℕ) : succ n - succ m = n - m :=
succ_sub_succ_eq_sub n m

protected theorem sub_self : ∀ (n : ℕ), n - n = 0
| 0        := by rw nat.sub_zero
| (succ n) := by rw [succ_sub_succ, sub_self n]

/- TODO(Leo): remove the following ematch annotations as soon as we have
   arithmetic theory in the smt_stactic -/
@[ematch_lhs]
protected theorem add_sub_add_right : ∀ (n k m : ℕ), (n + k) - (m + k) = n - m
| n 0        m := by rw [nat.add_zero, nat.add_zero]
| n (succ k) m := by rw [add_succ, add_succ, succ_sub_succ, add_sub_add_right n k m]

@[ematch_lhs]
protected theorem add_sub_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
by rw [nat.add_comm k n, nat.add_comm k m, nat.add_sub_add_right]

@[ematch_lhs]
protected theorem add_sub_cancel (n m : ℕ) : n + m - m = n :=
suffices n + m - (0 + m) = n, from
  by rwa [nat.zero_add] at this,
by rw [nat.add_sub_add_right, nat.sub_zero]

@[ematch_lhs]
protected theorem add_sub_cancel_left (n m : ℕ) : n + m - n = m :=
show n + m - (n + 0) = m, from
by rw [nat.add_sub_add_left, nat.sub_zero]

protected theorem sub_sub : ∀ (n m k : ℕ), n - m - k = n - (m + k)
| n m 0        := by rw [nat.add_zero, nat.sub_zero]
| n m (succ k) := by rw [add_succ, nat.sub_succ, nat.sub_succ, sub_sub n m k]

protected theorem le_of_le_of_sub_le_sub_right {n m k : ℕ}
  (h₀ : k ≤ m)
  (h₁ : n - k ≤ m - k)
: n ≤ m :=
begin
  revert k m,
  induction n with n ; intros k m h₀ h₁,
  { exact m.zero_le },
  { cases k with k,
    { apply h₁ },
    cases m with m,
    { cases not_succ_le_zero _ h₀ },
    { simp [succ_sub_succ] at h₁,
      apply succ_le_succ,
      apply n_ih _ h₁,
      apply le_of_succ_le_succ h₀ }, }
end

protected theorem sub_le_sub_right_iff (n m k : ℕ)
  (h : k ≤ m)
: n - k ≤ m - k ↔ n ≤ m :=
⟨ nat.le_of_le_of_sub_le_sub_right h , assume h, nat.sub_le_sub_right h k ⟩

protected theorem sub_self_add (n m : ℕ) : n - (n + m) = 0 :=
show (n + 0) - (n + m) = 0, from
by rw [nat.add_sub_add_left, nat.zero_sub]

protected theorem add_le_to_le_sub (x : ℕ) {y k : ℕ}
  (h : k ≤ y)
: x + k ≤ y ↔ x ≤ y - k :=
by rw [← nat.add_sub_cancel x k, nat.sub_le_sub_right_iff _ _ _ h, nat.add_sub_cancel]

protected lemma sub_lt_of_pos_le (a b : ℕ) (h₀ : 0 < a) (h₁ : a ≤ b)
: b - a < b :=
begin
  apply nat.sub_lt _ h₀,
  apply lt_of_lt_of_le h₀ h₁
end

protected theorem sub_one (n : ℕ) : n - 1 = pred n :=
rfl

theorem succ_sub_one (n : ℕ) : succ n - 1 = n :=
rfl

theorem succ_pred_eq_of_pos : ∀ {n : ℕ}, 0 < n → succ (pred n) = n
| 0 h        := absurd h (lt_irrefl 0)
| (succ k) h := rfl

protected theorem sub_eq_zero_of_le {n m : ℕ} (h : n ≤ m) : n - m = 0 :=
exists.elim (nat.le.dest h)
  (assume k, assume hk : n + k = m, by rw [← hk, nat.sub_self_add])

protected theorem le_of_sub_eq_zero : ∀{n m : ℕ}, n - m = 0 → n ≤ m
| n 0 H := begin rw [nat.sub_zero] at H, simp [H] end
| 0 (m+1) H := (m + 1).zero_le
| (n+1) (m+1) H := nat.add_le_add_right
  (le_of_sub_eq_zero begin simp [nat.add_sub_add_right] at H, exact H end) _

protected theorem sub_eq_zero_iff_le {n m : ℕ} : n - m = 0 ↔ n ≤ m :=
⟨nat.le_of_sub_eq_zero, nat.sub_eq_zero_of_le⟩

protected theorem add_sub_of_le {n m : ℕ} (h : n ≤ m) : n + (m - n) = m :=
exists.elim (nat.le.dest h)
  (assume k, assume hk : n + k = m,
    by rw [← hk, nat.add_sub_cancel_left])

protected theorem sub_add_cancel {n m : ℕ} (h : m ≤ n) : n - m + m = n :=
by rw [nat.add_comm, nat.add_sub_of_le h]

protected theorem add_sub_assoc {m k : ℕ} (h : k ≤ m) (n : ℕ) : n + m - k = n + (m - k) :=
exists.elim (nat.le.dest h)
  (assume l, assume hl : k + l = m,
    by rw [← hl, nat.add_sub_cancel_left, nat.add_comm k, ← nat.add_assoc, nat.add_sub_cancel])

protected lemma sub_eq_iff_eq_add {a b c : ℕ} (ab : b ≤ a) : a - b = c ↔ a = c + b :=
⟨assume c_eq, begin rw [c_eq.symm, nat.sub_add_cancel ab] end,
  assume a_eq, begin rw [a_eq, nat.add_sub_cancel] end⟩

protected lemma lt_of_sub_eq_succ {m n l : ℕ} (H : m - n = nat.succ l) : n < m :=
not_le.1
  (assume (H' : n ≥ m), begin simp [nat.sub_eq_zero_of_le H'] at H, contradiction end)

protected lemma zero_min (a : ℕ) : min 0 a = 0 :=
min_eq_left a.zero_le

protected lemma min_zero (a : ℕ) : min a 0 = 0 :=
min_eq_right a.zero_le

-- Distribute succ over min
theorem min_succ_succ (x y : ℕ) : min (succ x) (succ y) = succ (min x y) :=
have f : x ≤ y → min (succ x) (succ y) = succ (min x y), from λp,
  calc min (succ x) (succ y)
              = succ x         : if_pos (succ_le_succ p)
          ... = succ (min x y) : congr_arg succ (eq.symm (if_pos p)),
have g : ¬ (x ≤ y) → min (succ x) (succ y) = succ (min x y), from λp,
  calc min (succ x) (succ y)
              = succ y         : if_neg (λeq, p (pred_le_pred eq))
          ... = succ (min x y) : congr_arg succ (eq.symm (if_neg p)),
decidable.by_cases f g

theorem sub_eq_sub_min (n m : ℕ) : n - m = n - min n m :=
if h : n ≥ m then by rewrite [min_eq_right h]
else by rewrite [nat.sub_eq_zero_of_le (le_of_not_ge h), min_eq_left (le_of_not_ge h), nat.sub_self]

@[simp] protected theorem sub_add_min_cancel (n m : ℕ) : n - m + min n m = n :=
by rw [sub_eq_sub_min, nat.sub_add_cancel (min_le_left n m)]

/- TODO(Leo): sub + inequalities -/

protected def strong_rec_on {p : nat → Sort u} (n : nat) (h : ∀ n, (∀ m, m < n → p m) → p n) : p n :=
suffices ∀ n m, m < n → p m, from this (succ n) n (lt_succ_self _),
begin
  intros n, induction n with n ih,
    {intros m h₁, exact absurd h₁ m.not_lt_zero},
    {intros m h₁,
      apply or.by_cases (decidable.lt_or_eq_of_le (le_of_lt_succ h₁)),
        {intros, apply ih, assumption},
        {intros, subst m, apply h _ ih}}
end

protected lemma strong_induction_on {p : nat → Prop} (n : nat) (h : ∀ n, (∀ m, m < n → p m) → p n) : p n :=
nat.strong_rec_on n h

protected lemma case_strong_induction_on {p : nat → Prop} (a : nat)
  (hz : p 0)
  (hi : ∀ n, (∀ m, m ≤ n → p m) → p (succ n)) : p a :=
nat.strong_induction_on a $ λ n,
  match n with
  | 0     := λ _, hz
  | (n+1) := λ h₁, hi n (λ m h₂, h₁ _ (lt_succ_of_le h₂))
  end

/- mod -/

private lemma mod_core_congr {x y f1 f2} (h1 : x ≤ f1) (h2 : x ≤ f2) :
  nat.mod_core y f1 x = nat.mod_core y f2 x :=
begin
  cases y, { cases f1; cases f2; refl },
  induction f1 with f1 ih generalizing x f2, { cases h1, cases f2; refl },
  cases x, { cases f1; cases f2; refl },
  cases f2, { cases h2 },
  refine if_congr iff.rfl _ rfl,
  simp only [succ_sub_succ],
  exact ih
    (le_trans (nat.sub_le _ _) (le_of_succ_le_succ h1))
    (le_trans (nat.sub_le _ _) (le_of_succ_le_succ h2))
end

lemma mod_def (x y : nat) : x % y = if 0 < y ∧ y ≤ x then (x - y) % y else x :=
begin
  cases x, { cases y; refl },
  cases y, { refl },
  refine if_congr iff.rfl (mod_core_congr _ _) rfl; simp [nat.sub_le]
end

@[simp] lemma mod_zero (a : nat) : a % 0 = a :=
begin
  rw mod_def,
  have h : ¬ (0 < 0 ∧ 0 ≤ a),
  simp [lt_irrefl],
  simp [if_neg, h]
end

lemma mod_eq_of_lt {a b : nat} (h : a < b) : a % b = a :=
begin
  rw mod_def,
  have h' : ¬(0 < b ∧ b ≤ a),
  simp [not_le_of_gt h],
  simp [if_neg, h']
end

@[simp] lemma zero_mod (b : nat) : 0 % b = 0 :=
begin
  rw mod_def,
  have h : ¬(0 < b ∧ b ≤ 0),
  {intro hn, cases hn with l r, exact absurd (lt_of_lt_of_le l r) (lt_irrefl 0)},
  simp [if_neg, h]
end

lemma mod_eq_sub_mod {a b : nat} (h : b ≤ a) : a % b = (a - b) % b :=
or.elim b.eq_zero_or_pos
  (λb0, by rw [b0, nat.sub_zero])
  (λh₂, by rw [mod_def, if_pos (and.intro h₂ h)])

lemma mod_lt (x : nat) {y : nat} (h : 0 < y) : x % y < y :=
begin
  induction x using nat.case_strong_induction_on with x ih,
  { rw zero_mod, assumption },
  { by_cases h₁ : succ x < y,
    { rwa [mod_eq_of_lt h₁] },
    { have h₁ : succ x % y = (succ x - y) % y := mod_eq_sub_mod (not_lt.1 h₁),
      have : succ x - y ≤ x := le_of_lt_succ (nat.sub_lt (succ_pos x) h),
      have h₂ : (succ x - y) % y < y := ih _ this,
      rwa [← h₁] at h₂ } }
end

@[simp] theorem mod_self (n : nat) : n % n = 0 :=
by rw [mod_eq_sub_mod (le_refl _), nat.sub_self, zero_mod]

@[simp] lemma mod_one (n : ℕ) : n % 1 = 0 :=
have n % 1 < 1, from (mod_lt n) (succ_pos 0),
nat.eq_zero_of_le_zero (le_of_lt_succ this)

lemma mod_two_eq_zero_or_one (n : ℕ) : n % 2 = 0 ∨ n % 2 = 1 :=
match n % 2, @nat.mod_lt n 2 dec_trivial with
| 0,   _ := or.inl rfl
| 1,   _ := or.inr rfl
| k+2, h := absurd h dec_trivial
end

/- div & mod -/

private lemma div_core_congr {x y f1 f2} (h1 : x ≤ f1) (h2 : x ≤ f2) :
  nat.div_core y f1 x = nat.div_core y f2 x :=
begin
  cases y, { cases f1; cases f2; refl },
  induction f1 with f1 ih generalizing x f2, { cases h1, cases f2; refl },
  cases x, { cases f1; cases f2; refl },
  cases f2, { cases h2 },
  refine if_congr iff.rfl _ rfl,
  simp only [succ_sub_succ],
  refine congr_arg (+1) _,
  exact ih
    (le_trans (nat.sub_le _ _) (le_of_succ_le_succ h1))
    (le_trans (nat.sub_le _ _) (le_of_succ_le_succ h2))
end

lemma div_def (x y : nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 else 0 :=
begin
  cases x, { cases y; refl },
  cases y, { refl },
  refine if_congr iff.rfl (congr_arg (+1) _) rfl,
  refine div_core_congr _ _; simp [nat.sub_le]
end

lemma mod_add_div (m k : ℕ)
: m % k + k * (m / k) = m :=
begin
  apply nat.strong_induction_on m,
  clear m,
  intros m IH,
  cases decidable.em (0 < k ∧ k ≤ m) with h h',
  -- 0 < k ∧ k ≤ m
  { have h' : m - k < m,
    { apply nat.sub_lt _ h.left,
      apply lt_of_lt_of_le h.left h.right },
    rw [div_def, mod_def, if_pos h, if_pos h],
    simp [nat.left_distrib, IH _ h', nat.add_comm, nat.add_left_comm],
    rw [nat.add_comm, ← nat.add_sub_assoc h.right, nat.mul_one, nat.add_sub_cancel_left] },
  -- ¬ (0 < k ∧ k ≤ m)
  { rw [div_def, mod_def, if_neg h', if_neg h', nat.mul_zero, nat.add_zero] },
end

/- div -/

@[simp] protected lemma div_one (n : ℕ) : n / 1 = n :=
have n % 1 + 1 * (n / 1) = n, from mod_add_div _ _,
by { rwa [mod_one, nat.zero_add, nat.one_mul] at this }

@[simp] protected lemma div_zero (n : ℕ) : n / 0 = 0 :=
begin rw [div_def], simp [lt_irrefl] end

@[simp] protected lemma zero_div (b : ℕ) : 0 / b = 0 :=
eq.trans (div_def 0 b) $ if_neg (and.rec not_le_of_gt)

protected lemma div_le_of_le_mul {m n : ℕ} : ∀ {k}, m ≤ k * n → m / k ≤ n
| 0        h := by simp [nat.div_zero, n.zero_le]
| (succ k) h :=
  suffices succ k * (m / succ k) ≤ succ k * n, from nat.le_of_mul_le_mul_left this (zero_lt_succ _),
  calc
    succ k * (m / succ k) ≤ m % succ k + succ k * (m / succ k) : nat.le_add_left _ _
                      ... = m                                  : by rw mod_add_div
                      ... ≤ succ k * n                         : h

protected lemma div_le_self : ∀ (m n : ℕ), m / n ≤ m
| m 0        := by simp [nat.div_zero, m.zero_le]
| m (succ n) :=
  have m ≤ succ n * m, from calc
    m  = 1 * m      : by rw nat.one_mul
   ... ≤ succ n * m : m.mul_le_mul_right (succ_le_succ n.zero_le),
  nat.div_le_of_le_mul this

lemma div_eq_sub_div {a b : nat} (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 :=
begin
  rw [div_def a, if_pos],
  split ; assumption
end

lemma div_eq_of_lt {a b : ℕ} (h₀ : a < b) : a / b = 0 :=
begin
  rw [div_def a, if_neg],
  intro h₁,
  apply not_le_of_gt h₀ h₁.right
end

-- this is a Galois connection
--   f x ≤ y ↔ x ≤ g y
-- with
--   f x = x * k
--   g y = y / k
theorem le_div_iff_mul_le (x y : ℕ) {k : ℕ} (Hk : 0 < k) : x ≤ y / k ↔ x * k ≤ y :=
begin
  -- Hk is needed because, despite div being made total, y / 0 := 0
  --     x * 0 ≤ y ↔ x ≤ y / 0
  --   ↔ 0 ≤ y ↔ x ≤ 0
  --   ↔ true ↔ x = 0
  --   ↔ x = 0
  revert x,
  apply nat.strong_induction_on y _,
  clear y,
  intros y IH x,
  cases lt_or_le y k with h h,
  -- base case: y < k
  { rw [div_eq_of_lt h],
    cases x with x,
    { simp [nat.zero_mul, y.zero_le] },
    { simp [succ_mul, not_succ_le_zero, nat.add_comm],
      apply lt_of_lt_of_le h,
      apply nat.le_add_right } },
  -- step: k ≤ y
  { rw [div_eq_sub_div Hk h],
    cases x with x,
    { simp [nat.zero_mul, nat.zero_le] },
    { have Hlt : y - k < y,
      { apply nat.sub_lt_of_pos_le ; assumption },
      rw [ ← add_one
         , nat.add_le_add_iff_le_right
         , IH (y - k) Hlt x
         , add_one
         , succ_mul, nat.add_le_to_le_sub _ h ]
     } }
end

theorem div_lt_iff_lt_mul (x y : ℕ) {k : ℕ} (Hk : 0 < k) : x / k < y ↔ x < y * k :=
begin
  simp [← not_le],
  apply not_iff_not_of_iff,
  apply le_div_iff_mul_le _ _ Hk
end

def iterate {α : Sort u} (op : α → α) : ℕ → α → α
 | 0        a := a
 | (succ k) a := iterate k (op a)

notation f`^[`n`]` := iterate f n

/- successor and predecessor -/

theorem add_one_ne_zero (n : ℕ) : n + 1 ≠ 0 := succ_ne_zero _

theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
by cases n; simp

theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃k : ℕ, n = succ k :=
⟨_, (eq_zero_or_eq_succ_pred _).resolve_left H⟩

def discriminate {B : Sort u} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
by induction h : n; [exact H1 h, exact H2 _ h]

theorem one_succ_zero : 1 = succ 0 := rfl

def two_step_induction {P : ℕ → Sort u} (H1 : P 0) (H2 : P 1)
    (H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : Π (a : ℕ), P a
| 0               := H1
| 1               := H2
| (succ (succ n)) := H3 _ (two_step_induction _) (two_step_induction _)

def sub_induction {P : ℕ → ℕ → Sort u} (H1 : ∀m, P 0 m)
   (H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : Π (n m : ℕ), P n m
| 0        m        := H1 _
| (succ n) 0        := H2 _
| (succ n) (succ m) := H3 _ _ (sub_induction n m)

/- addition -/

theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m :=
by simp [succ_add, add_succ]

-- theorem one_add (n : ℕ) : 1 + n = succ n := by simp [add_comm]

protected theorem add_right_comm : ∀ (n m k : ℕ), n + m + k = n + k + m :=
right_comm nat.add nat.add_comm nat.add_assoc

theorem eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
⟨nat.eq_zero_of_add_eq_zero_right H, nat.eq_zero_of_add_eq_zero_left H⟩

theorem eq_zero_of_mul_eq_zero : ∀ {n m : ℕ}, n * m = 0 → n = 0 ∨ m = 0
| 0        m := λ h, or.inl rfl
| (succ n) m :=
  begin
    rw succ_mul, intro h,
    exact or.inr (nat.eq_zero_of_add_eq_zero_left h)
  end

/- properties of inequality -/

theorem le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
nat.cases_on n less_than_or_equal.step (λ a, succ_le_succ)

theorem le_lt_antisymm {n m : ℕ} (h₁ : n ≤ m) (h₂ : m < n) : false :=
nat.lt_irrefl n (nat.lt_of_le_of_lt h₁ h₂)

theorem lt_le_antisymm {n m : ℕ} (h₁ : n < m) (h₂ : m ≤ n) : false :=
le_lt_antisymm h₂ h₁

protected theorem lt_asymm {n m : ℕ} (h₁ : n < m) : ¬ m < n :=
le_lt_antisymm (nat.le_of_lt h₁)

protected def lt_ge_by_cases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C :=
decidable.by_cases h₁ (λ h, h₂ (or.elim (nat.lt_or_ge a b) (λ a, absurd a h) (λ a, a)))

protected def lt_by_cases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : a = b → C)
  (h₃ : b < a → C) : C :=
nat.lt_ge_by_cases h₁ (λ h₁,
  nat.lt_ge_by_cases h₃ (λ h, h₂ (nat.le_antisymm h h₁)))

protected theorem lt_trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
nat.lt_by_cases (λ h, or.inl h) (λ h, or.inr (or.inl h)) (λ h, or.inr (or.inr h))

protected theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
(nat.lt_trichotomy a b).resolve_left hnlt

theorem lt_succ_of_lt {a b : nat} (h : a < b) : a < succ b := le_succ_of_le h

lemma one_pos : 0 < 1 := nat.zero_lt_one

/- subtraction -/

protected theorem sub_le_sub_left {n m : ℕ} (k) (h : n ≤ m) : k - m ≤ k - n :=
by induction h; [refl, exact le_trans (pred_le _) h_ih]

theorem succ_sub_sub_succ (n m k : ℕ) : succ n - m - succ k = n - m - k :=
by rw [nat.sub_sub, nat.sub_sub, add_succ, succ_sub_succ]

protected theorem sub.right_comm (m n k : ℕ) : m - n - k = m - k - n :=
by rw [nat.sub_sub, nat.sub_sub, nat.add_comm]

theorem mul_pred_left : ∀ (n m : ℕ), pred n * m = n * m - m
| 0        m := by simp [nat.zero_sub, pred_zero, nat.zero_mul]
| (succ n) m := by rw [pred_succ, succ_mul, nat.add_sub_cancel]

theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n :=
by rw [nat.mul_comm, mul_pred_left, nat.mul_comm]

protected theorem mul_sub_right_distrib : ∀ (n m k : ℕ), (n - m) * k = n * k - m * k
| n 0        k := by simp [nat.sub_zero, nat.zero_mul]
| n (succ m) k := by rw [nat.sub_succ, mul_pred_left, mul_sub_right_distrib, succ_mul, nat.sub_sub]

protected theorem mul_sub_left_distrib (n m k : ℕ) : n * (m - k) = n * m - n * k :=
by rw [nat.mul_comm, nat.mul_sub_right_distrib, nat.mul_comm m n, nat.mul_comm n k]

protected theorem mul_self_sub_mul_self_eq (a b : nat) : a * a - b * b = (a + b) * (a - b) :=
by rw [nat.mul_sub_left_distrib, nat.right_distrib, nat.right_distrib, nat.mul_comm b a, nat.add_comm (a*a) (a*b),
       nat.add_sub_add_left]

theorem succ_mul_succ_eq (a b : nat) : succ a * succ b = a*b + a + b + 1 :=
begin
  rw [← add_one, ← add_one],
  simp [nat.right_distrib, nat.left_distrib, nat.add_left_comm, nat.mul_one, nat.one_mul, nat.add_assoc],
end

theorem succ_sub {m n : ℕ} (h : n ≤ m) : succ m - n = succ (m - n) :=
exists.elim (nat.le.dest h)
  (assume k, assume hk : n + k = m,
    by rw [← hk, nat.add_sub_cancel_left, ← add_succ, nat.add_sub_cancel_left])

protected theorem sub_pos_of_lt {m n : ℕ} (h : m < n) : 0 < n - m :=
have 0 + m < n - m + m, begin rw [nat.zero_add, nat.sub_add_cancel (le_of_lt h)], exact h end,
nat.lt_of_add_lt_add_right this

protected theorem sub_sub_self {n m : ℕ} (h : m ≤ n) : n - (n - m) = m :=
(nat.sub_eq_iff_eq_add (nat.sub_le _ _)).2 (nat.add_sub_of_le h).symm

protected theorem sub_add_comm {n m k : ℕ} (h : k ≤ n) : n + m - k = n - k + m :=
(nat.sub_eq_iff_eq_add (nat.le_trans h (nat.le_add_right _ _))).2
  (by rwa [nat.add_right_comm, nat.sub_add_cancel])

theorem sub_one_sub_lt {n i} (h : i < n) : n - 1 - i < n := begin
  rw nat.sub_sub,
  apply nat.sub_lt,
  apply lt_of_lt_of_le (nat.zero_lt_succ _) h,
  rw nat.add_comm,
  apply nat.zero_lt_succ
end

theorem pred_inj : ∀ {a b : nat}, 0 < a → 0 < b → nat.pred a = nat.pred b → a = b
| (succ a) (succ b) ha hb h := have a = b, from h, by rw this
| (succ a) 0        ha hb h := absurd hb (lt_irrefl _)
| 0        (succ b) ha hb h := absurd ha (lt_irrefl _)
| 0        0        ha hb h := rfl

/- find -/

section find
parameter {p : ℕ → Prop}

private def lbp (m n : ℕ) : Prop := m = n + 1 ∧ ∀ k ≤ n, ¬p k

parameters [decidable_pred p] (H : ∃n, p n)

private def wf_lbp : well_founded lbp :=
⟨let ⟨n, pn⟩ := H in
suffices ∀m k, n ≤ k + m → acc lbp k, from λa, this _ _ (nat.le_add_left _ _),
λm, nat.rec_on m
  (λk kn, ⟨_, λy r, match y, r with ._, ⟨rfl, a⟩ := absurd pn (a _ kn) end⟩)
  (λm IH k kn, ⟨_, λy r, match y, r with ._, ⟨rfl, a⟩ := IH _ (by rw nat.add_right_comm; exact kn) end⟩)⟩

protected def find_x : {n // p n ∧ ∀m < n, ¬p m} :=
@well_founded.fix _ (λk, (∀n < k, ¬p n) → {n // p n ∧ ∀m < n, ¬p m}) lbp wf_lbp
(λm IH al, if pm : p m then ⟨m, pm, al⟩ else
    have ∀ n ≤ m, ¬p n, from λn h, or.elim (decidable.lt_or_eq_of_le h) (al n) (λe, by rw e; exact pm),
    IH _ ⟨rfl, this⟩ (λn h, this n $ nat.le_of_succ_le_succ h))
0 (λn h, absurd h (nat.not_lt_zero _))

/--
If `p` is a (decidable) predicate on `ℕ` and `hp : ∃ (n : ℕ), p n` is a proof that
there exists some natural number satisfying `p`, then `nat.find hp` is the
smallest natural number satisfying `p`. Note that `nat.find` is protected,
meaning that you can't just write `find`, even if the `nat` namespace is open.

The API for `nat.find` is:

* `nat.find_spec` is the proof that `nat.find hp` satisfies `p`.
* `nat.find_min` is the proof that if `m < nat.find hp` then `m` does not satisfy `p`.
* `nat.find_min'` is the proof that if `m` does satisfy `p` then `nat.find hp ≤ m`.
-/
protected def find : ℕ := nat.find_x.1

protected theorem find_spec : p nat.find := nat.find_x.2.left

protected theorem find_min : ∀ {m : ℕ}, m < nat.find → ¬p m := nat.find_x.2.right

protected theorem find_min' {m : ℕ} (h : p m) : nat.find ≤ m :=
le_of_not_lt (λ l, find_min l h)

end find

/- mod -/

theorem mod_le (x y : ℕ) : x % y ≤ x :=
or.elim (lt_or_le x y)
  (λxlty, by rw mod_eq_of_lt xlty; refl)
  (λylex, or.elim y.eq_zero_or_pos
    (λy0, by rw [y0, mod_zero]; refl)
    (λypos, le_trans (le_of_lt (mod_lt _ ypos)) ylex))

@[simp] theorem add_mod_right (x z : ℕ) : (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 : ℕ) : (x + z) % x = z % x :=
by rw [nat.add_comm, add_mod_right]

@[simp] theorem add_mul_mod_self_left (x y z : ℕ) : (x + y * z) % y = x % y :=
by {induction z with z ih, rw [nat.mul_zero, nat.add_zero], rw [mul_succ, ← nat.add_assoc, add_mod_right, ih]}

@[simp] theorem add_mul_mod_self_right (x y z : ℕ) : (x + y * z) % z = x % z :=
by rw [nat.mul_comm, add_mul_mod_self_left]

@[simp] theorem mul_mod_right (m n : ℕ) : (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 : ℕ) : (m * n) % n = 0 :=
by rw [nat.mul_comm, mul_mod_right]

theorem mul_mod_mul_left (z x y : ℕ) : (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 x.strong_induction_on $ λn IH,
  have y0 : y > 0, from nat.pos_of_ne_zero y0,
  have z0 : z > 0, from nat.pos_of_ne_zero z0,
  or.elim (le_or_lt y n)
    (λyn, by rw [
        mod_eq_sub_mod yn,
        mod_eq_sub_mod (nat.mul_le_mul_left z yn),
        ← nat.mul_sub_left_distrib];
      exact IH _ (nat.sub_lt (lt_of_lt_of_le y0 yn) y0))
    (λyn, by rw [mod_eq_of_lt yn, mod_eq_of_lt (nat.mul_lt_mul_of_pos_left yn z0)])

theorem mul_mod_mul_right (z x y : ℕ) : (x * z) % (y * z) = (x % y) * z :=
by rw [nat.mul_comm x z, nat.mul_comm y z, nat.mul_comm (x % y) z]; apply mul_mod_mul_left

theorem cond_to_bool_mod_two (x : ℕ) [d : decidable (x % 2 = 1)]
: cond (@to_bool (x % 2 = 1) d) 1 0 = x % 2 :=
begin
  by_cases h : x % 2 = 1,
  { simp! [*] },
  { cases mod_two_eq_zero_or_one x; simp! [*, nat.zero_ne_one] }
end

theorem sub_mul_mod (x k n : ℕ) (h₁ : n*k ≤ x) : (x - n*k) % n = x % n :=
begin
  induction k with k,
  { rw [nat.mul_zero, nat.sub_zero] },
  { have h₂ : n * k ≤ x,
    { rw [mul_succ] at h₁,
      apply nat.le_trans _ h₁,
      apply nat.le_add_right _ n },
    have h₄ : x - n * k ≥ n,
    { apply @nat.le_of_add_le_add_right (n*k),
      rw [nat.sub_add_cancel h₂],
      simp [mul_succ, nat.add_comm] at h₁, simp [h₁] },
    rw [mul_succ, ← nat.sub_sub, ← mod_eq_sub_mod h₄, k_ih h₂] }
end

/- div -/

theorem sub_mul_div (x n p : ℕ) (h₁ : n*p ≤ x) : (x - n*p) / n = x / n - p :=
begin
  cases nat.eq_zero_or_pos n with h₀ h₀,
  { rw [h₀, nat.div_zero, nat.div_zero, nat.zero_sub] },
  { induction p with p,
    { rw [nat.mul_zero, nat.sub_zero, nat.sub_zero] },
    { have h₂ : n*p ≤ x,
      { transitivity,
        { apply nat.mul_le_mul_left, apply le_succ },
        { apply h₁ } },
      have h₃ : x - n * p ≥ n,
      { apply nat.le_of_add_le_add_right,
        rw [nat.sub_add_cancel h₂, nat.add_comm],
        rw [mul_succ] at h₁,
        apply h₁ },
      rw [sub_succ, ← p_ih h₂],
      rw [@div_eq_sub_div (x - n*p) _ h₀ h₃],
      simp [add_one, pred_succ, mul_succ, nat.sub_sub] } }
end

theorem div_mul_le_self : ∀ (m n : ℕ), m / n * n ≤ m
| m 0        := by simp [m.zero_le, nat.zero_mul]
| m (succ n) := (le_div_iff_mul_le _ _ (nat.succ_pos _)).1 (le_refl _)

@[simp] theorem add_div_right (x : ℕ) {z : ℕ} (H : 0 < z) : (x + z) / z = succ (x / z) :=
by rw [div_eq_sub_div H (nat.le_add_left _ _), nat.add_sub_cancel]

@[simp] theorem add_div_left (x : ℕ) {z : ℕ} (H : 0 < z) : (z + x) / z = succ (x / z) :=
by rw [nat.add_comm, add_div_right x H]

@[simp] theorem mul_div_right (n : ℕ) {m : ℕ} (H : 0 < m) : m * n / m = n :=
by {induction n; simp [*, mul_succ, nat.mul_zero] }

@[simp] theorem mul_div_left (m : ℕ) {n : ℕ} (H : 0 < n) : m * n / n = m :=
by rw [nat.mul_comm, mul_div_right _ H]

protected theorem div_self {n : ℕ} (H : 0 < n) : n / n = 1 :=
let t := add_div_right 0 H in by rwa [nat.zero_add, nat.zero_div] at t

theorem add_mul_div_left (x z : ℕ) {y : ℕ} (H : 0 < y) : (x + y * z) / y = x / y + z :=
begin
  induction z with z ih,
  { rw [nat.mul_zero, nat.add_zero, nat.add_zero] },
  { rw [mul_succ, ← nat.add_assoc, add_div_right _ H, ih], refl }
end

theorem add_mul_div_right (x y : ℕ) {z : ℕ} (H : 0 < z) : (x + y * z) / z = x / z + y :=
by rw [nat.mul_comm, add_mul_div_left _ _ H]

protected theorem mul_div_cancel (m : ℕ) {n : ℕ} (H : 0 < n) : m * n / n = m :=
let t := add_mul_div_right 0 m H in by rwa [nat.zero_add, nat.zero_div, nat.zero_add] at t

protected theorem mul_div_cancel_left (m : ℕ) {n : ℕ} (H : 0 < n) : n * m / n = m :=
by rw [nat.mul_comm, nat.mul_div_cancel _ H]

protected theorem div_eq_of_eq_mul_left {m n k : ℕ} (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 {m n k : ℕ} (H1 : 0 < n) (H2 : m = n * k) :
  m / n = k :=
by rw [H2, nat.mul_div_cancel_left _ H1]

protected theorem div_eq_of_lt_le {m n k : ℕ}
  (lo : k * n ≤ m) (hi : m < succ k * n) :
  m / n = k :=
have npos : 0 < n, from n.eq_zero_or_pos.resolve_left $ λ hn,
  by rw [hn, nat.mul_zero] at hi lo; exact absurd lo (not_le_of_gt hi),
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 mul_sub_div (x n p : ℕ) (h₁ : x < n*p) : (n * p - succ x) / n = p - succ (x / n) :=
begin
  have npos : 0 < n := n.eq_zero_or_pos.resolve_left (λ n0,
    by rw [n0, nat.zero_mul] at h₁; exact nat.not_lt_zero _ h₁),
  apply nat.div_eq_of_lt_le,
  { rw [nat.mul_sub_right_distrib, nat.mul_comm],
    apply nat.sub_le_sub_left,
    exact (div_lt_iff_lt_mul _ _ npos).1 (lt_succ_self _) },
  { change succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n,
    rw [succ_pred_eq_of_pos (nat.sub_pos_of_lt h₁),
        succ_pred_eq_of_pos (nat.sub_pos_of_lt _)],
    { rw [nat.mul_sub_right_distrib, nat.mul_comm],
      apply nat.sub_le_sub_left, apply div_mul_le_self },
    { apply (div_lt_iff_lt_mul _ _ npos).2, rwa nat.mul_comm } }
end

protected lemma mul_pos {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : 0 < a * b :=
have h : 0 * b < a * b, from nat.mul_lt_mul_of_pos_right ha hb,
by rwa nat.zero_mul at h

protected theorem div_div_eq_div_mul (m n k : ℕ) : m / n / k = m / (n * k) :=
begin
  cases k.eq_zero_or_pos with k0 kpos, {rw [k0, nat.mul_zero, nat.div_zero, nat.div_zero]},
  cases n.eq_zero_or_pos with n0 npos, {rw [n0, nat.zero_mul, nat.div_zero, nat.zero_div]},
  apply 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,
    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,
    refl }
end

protected theorem mul_div_mul {m : ℕ} (n k : ℕ) (H : 0 < m) : m * n / (m * k) = n / k :=
by rw [← nat.div_div_eq_div_mul, nat.mul_div_cancel_left _ H]

/- dvd -/

protected theorem dvd_mul_right (a b : ℕ) : a ∣ a * b := ⟨b, rfl⟩

protected theorem dvd_trans {a b c : ℕ} (h₁ : a ∣ b) (h₂ : b ∣ c) : a ∣ c :=
match h₁, h₂ with
| ⟨d, (h₃ : b = a * d)⟩, ⟨e, (h₄ : c = b * e)⟩ :=
  ⟨d * e, show c = a * (d * e), by simp [h₃, h₄, nat.mul_assoc]⟩
end

protected theorem eq_zero_of_zero_dvd {a : ℕ} (h : 0 ∣ a) : a = 0 :=
exists.elim h (assume c, assume H' : a = 0 * c, eq.trans H' (nat.zero_mul c))

protected theorem dvd_add {a b c : ℕ} (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c :=
  exists.elim h₁ (λ d hd, exists.elim h₂ (λ e he, ⟨d + e, by simp [nat.left_distrib, hd, he]⟩))

protected theorem dvd_add_iff_right {k m n : ℕ} (h : k ∣ m) : k ∣ n ↔ k ∣ m + n :=
⟨nat.dvd_add h, exists.elim h $ λd hd, match m, hd with
| ._, rfl := λh₂, exists.elim h₂ $ λe he, ⟨e - d,
  by rw [nat.mul_sub_left_distrib, ← he, nat.add_sub_cancel_left]⟩
end⟩

protected theorem dvd_add_iff_left {k m n : ℕ} (h : k ∣ n) : k ∣ m ↔ k ∣ m + n :=
by rw nat.add_comm; exact nat.dvd_add_iff_right h

theorem dvd_sub {k m n : ℕ} (H : n ≤ m) (h₁ : k ∣ m) (h₂ : k ∣ n) : k ∣ m - n :=
(nat.dvd_add_iff_left h₂).2 $ by rw nat.sub_add_cancel H; exact h₁

theorem dvd_mod_iff {k m n : ℕ} (h : k ∣ n) : k ∣ m % n ↔ k ∣ m :=
let t := @nat.dvd_add_iff_left _ (m % n) _ (nat.dvd_trans h (nat.dvd_mul_right n (m / n))) in
by rwa mod_add_div at t

theorem le_of_dvd {m n : ℕ} (h : 0 < n) : m ∣ n → m ≤ n :=
λ⟨k, e⟩, by {
  revert h, rw e, refine k.cases_on _ _,
  exact λhn, absurd hn (lt_irrefl _),
  exact λk _, let t := m.mul_le_mul_left (succ_pos k) in by rwa nat.mul_one at t }

theorem dvd_antisymm : Π {m n : ℕ}, m ∣ n → n ∣ m → m = n
| m        0        h₁ h₂ := nat.eq_zero_of_zero_dvd h₂
| 0        n        h₁ h₂ := (nat.eq_zero_of_zero_dvd h₁).symm
| (succ m) (succ n) h₁ h₂ := le_antisymm (le_of_dvd (succ_pos _) h₁) (le_of_dvd (succ_pos _) h₂)

theorem pos_of_dvd_of_pos {m n : ℕ} (H1 : m ∣ n) (H2 : 0 < n) : 0 < m :=
nat.pos_of_ne_zero $ λm0, by rw m0 at H1; rw nat.eq_zero_of_zero_dvd H1 at H2; exact lt_irrefl _ H2

theorem eq_one_of_dvd_one {n : ℕ} (H : n ∣ 1) : n = 1 :=
le_antisymm (le_of_dvd dec_trivial H) (pos_of_dvd_of_pos H dec_trivial)

theorem dvd_of_mod_eq_zero {m n : ℕ} (H : n % m = 0) : m ∣ n :=
⟨n / m, by { have t := (mod_add_div n m).symm, rwa [H, nat.zero_add] at t }⟩

theorem mod_eq_zero_of_dvd {m n : ℕ} (H : m ∣ n) : n % m = 0 :=
exists.elim H (λ z H1, by rw [H1, mul_mod_right])

theorem dvd_iff_mod_eq_zero (m n : ℕ) : m ∣ n ↔ n % m = 0 :=
⟨mod_eq_zero_of_dvd, dvd_of_mod_eq_zero⟩

instance decidable_dvd : @decidable_rel ℕ (∣) :=
λm n, decidable_of_decidable_of_iff (by apply_instance) (dvd_iff_mod_eq_zero _ _).symm

protected theorem mul_div_cancel' {m n : ℕ} (H : n ∣ m) : n * (m / n) = m :=
let t := mod_add_div m n in by rwa [mod_eq_zero_of_dvd H, nat.zero_add] at t

protected theorem div_mul_cancel {m n : ℕ} (H : n ∣ m) : m / n * n = m :=
by rw [nat.mul_comm, nat.mul_div_cancel' H]

protected theorem mul_div_assoc (m : ℕ) {n k : ℕ} (H : k ∣ n) : m * n / k = m * (n / k) :=
or.elim k.eq_zero_or_pos
  (λh, by rw [h, nat.div_zero, nat.div_zero, nat.mul_zero])
  (λh, have m * n / k = m * (n / k * k) / k, by rw nat.div_mul_cancel H,
       by rw[this, ← nat.mul_assoc, nat.mul_div_cancel _ h])

theorem dvd_of_mul_dvd_mul_left {m n k : ℕ} (kpos : 0 < k) (H : k * m ∣ k * n) : m ∣ n :=
exists.elim H (λl H1, by rw nat.mul_assoc at H1; exact ⟨_, nat.eq_of_mul_eq_mul_left kpos H1⟩)

theorem dvd_of_mul_dvd_mul_right {m n k : ℕ} (kpos : 0 < k) (H : m * k ∣ n * k) : m ∣ n :=
by rw [nat.mul_comm m k, nat.mul_comm n k] at H; exact dvd_of_mul_dvd_mul_left kpos H

/- --- -/

protected lemma mul_le_mul_of_nonneg_left {a b c : ℕ} (h₁ : a ≤ b) : c * a ≤ c * b :=
begin
  by_cases hba : b ≤ a, { simp [le_antisymm hba h₁] },
  by_cases hc0 : c ≤ 0, { simp [le_antisymm hc0 c.zero_le, nat.zero_mul] },
  exact (le_not_le_of_lt
    (nat.mul_lt_mul_of_pos_left (lt_of_le_not_le h₁ hba) (lt_of_le_not_le c.zero_le hc0))).left,
end

protected lemma mul_le_mul_of_nonneg_right {a b c : ℕ} (h₁ : a ≤ b) : a * c ≤ b * c :=
begin
  by_cases hba : b ≤ a, { simp [le_antisymm hba h₁] },
  by_cases hc0 : c ≤ 0, { simp [le_antisymm hc0 c.zero_le, nat.mul_zero] },
  exact (le_not_le_of_lt
    (nat.mul_lt_mul_of_pos_right (lt_of_le_not_le h₁ hba) (lt_of_le_not_le c.zero_le hc0))).left,
end

protected lemma mul_lt_mul {a b c d : ℕ} (hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) :
  a * b < c * d :=
calc
  a * b < c * b : nat.mul_lt_mul_of_pos_right hac pos_b
    ... ≤ c * d : nat.mul_le_mul_of_nonneg_left hbd

protected lemma mul_lt_mul' {a b c d : ℕ} (h1 : a ≤ c) (h2 : b < d) (h3 : 0 < c) :
       a * b < c * d :=
calc
   a * b ≤ c * b : nat.mul_le_mul_of_nonneg_right h1
     ... < c * d : nat.mul_lt_mul_of_pos_left h2 h3

-- TODO: there are four variations, depending on which variables we assume to be nonneg
protected lemma mul_le_mul {a b c d : ℕ} (hac : a ≤ c) (hbd : b ≤ d) : a * b ≤ c * d :=
calc
  a * b ≤ c * b : nat.mul_le_mul_of_nonneg_right hac
    ... ≤ c * d : nat.mul_le_mul_of_nonneg_left hbd

lemma div_lt_self {n m : nat} : 0 < n → 1 < m → n / m < n :=
begin
  intros h₁ h₂,
  have m_pos : 0 < m, { apply lt_trans _ h₂, comp_val },
  suffices : 1 * n < m * n, {
    rw [nat.one_mul, nat.mul_comm] at this,
    exact iff.mpr (nat.div_lt_iff_lt_mul n n m_pos) this
  },
  exact nat.mul_lt_mul h₂ (le_refl _) h₁
end

end nat