feat(nat/div): port to HoTT library

hott/port.md

@@ -8,6 +8,7 @@ Port instructions:
 - fix all remaining errors. Typical errors include
   - Replacing "and" by "prod" in comments
   - and.intro is replaced by prod.intro, which should be prod.mk.
+  - the usage of the simp tactic
 
 Currently, the following differences exist between the two libraries, relevant to porting:
 - All of the algebraic hierarchy is in the algebra namespace in the HoTT library (on top-level in the standard library).

hott/types/nat/div.hlean

@@ -0,0 +1,619 @@
+/-
+Copyright (c) 2014 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Leonardo de Moura
+
+Definitions prod properties of div prod mod. Much of the development follows Isabelle's library.
+-/
+import .sub
+open eq eq.ops well_founded decidable prod algebra
+
+set_option class.force_new true
+
+namespace nat
+
+/- div -/
+
+-- auxiliary lemma used to justify div
+private definition div_rec_lemma {x y : nat} : 0 < y × y ≤ x → x - y < x :=
+prod.rec (λ ypos ylex, sub_lt (lt_of_lt_of_le ypos ylex) ypos)
+
+private definition div.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
+if H : 0 < y × y ≤ x then f (x - y) (div_rec_lemma H) y + 1 else zero
+
+protected definition div := fix div.F
+
+definition nat_has_divide [reducible] [instance] [priority nat.prio] : has_div nat :=
+has_div.mk nat.div
+
+theorem div_def (x y : nat) : div x y = if 0 < y × y ≤ x then div (x - y) y + 1 else 0 :=
+congr_fun (fix_eq div.F x) y
+
+protected theorem div_zero [simp] (a : ℕ) : a / 0 = 0 :=
+div_def a 0 ⬝ if_neg (!not_prod_of_not_left (lt.irrefl 0))
+
+theorem div_eq_zero_of_lt {a b : ℕ} (h : a < b) : a / b = 0 :=
+div_def a b ⬝ if_neg (!not_prod_of_not_right (not_le_of_gt h))
+
+protected theorem zero_div [simp] (b : ℕ) : 0 / b = 0 :=
+div_def 0 b ⬝ if_neg (prod.rec not_le_of_gt)
+
+theorem div_eq_succ_sub_div {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a / b = succ ((a - b) / b) :=
+div_def a b ⬝ if_pos (pair h₁ h₂)
+
+theorem add_div_self (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) / z = succ (x / z) :=
+calc
+  (x + z) / z = if 0 < z × z ≤ x + z then (x + z - z) / z + 1 else 0 : !div_def
+            ... = (x + z - z) / z + 1 : if_pos (pair H (le_add_left z x))
+            ... = succ (x / z)        : {!nat.add_sub_cancel}
+
+theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) / x = succ (z / x) :=
+!add.comm ▸ !add_div_self H
+
+local attribute succ_mul [simp]
+
+theorem add_mul_div_self {x y z : ℕ} (H : z > 0) : (x + y * z) / z = x / z + y :=
+nat.rec_on y
+  (by rewrite [zero_mul])
+  (take y,
+    assume IH : (x + y * z) / z = x / z + y, calc
+      (x + succ y * z) / z = (x + y * z + z) / z    : by rewrite [succ_mul, add.assoc]
+                       ... = succ ((x + y * z) / z) : !add_div_self H
+                       ... = succ (x / z + y)       : IH)
+
+theorem add_mul_div_self_left (x z : ℕ) {y : ℕ} (H : y > 0) : (x + y * z) / y = x / y + z :=
+!mul.comm ▸ add_mul_div_self H
+
+protected theorem mul_div_cancel (m : ℕ) {n : ℕ} (H : n > 0) : m * n / n = m :=
+calc
+  m * n / n = (0 + m * n) / n : by rewrite [zero_add]
+          ... = 0 / n + m     : add_mul_div_self H
+          ... = m             : by rewrite [nat.zero_div, zero_add]
+
+protected theorem mul_div_cancel_left {m : ℕ} (n : ℕ) (H : m > 0) : m * n / m = n :=
+!mul.comm ▸ !nat.mul_div_cancel H
+
+/- mod -/
+
+private definition mod.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
+if H : 0 < y × y ≤ x then f (x - y) (div_rec_lemma H) y else x
+
+protected definition mod := fix mod.F
+
+definition nat_has_mod [reducible] [instance] [priority nat.prio] : has_mod nat :=
+has_mod.mk nat.mod
+
+notation [priority nat.prio] a ≡ b `[mod `:0 c:0 `]` := a % c = b % c
+
+theorem mod_def (x y : nat) : mod x y = if 0 < y × y ≤ x then mod (x - y) y else x :=
+congr_fun (fix_eq mod.F x) y
+
+theorem mod_zero [simp] (a : ℕ) : a % 0 = a :=
+mod_def a 0 ⬝ if_neg (!not_prod_of_not_left (lt.irrefl 0))
+
+theorem mod_eq_of_lt {a b : ℕ} (h : a < b) : a % b = a :=
+mod_def a b ⬝ if_neg (!not_prod_of_not_right (not_le_of_gt h))
+
+theorem zero_mod [simp] (b : ℕ) : 0 % b = 0 :=
+mod_def 0 b ⬝ if_neg (λ h, prod.rec_on h (λ l r, absurd (lt_of_lt_of_le l r) (lt.irrefl 0)))
+
+theorem mod_eq_sub_mod {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a % b = (a - b) % b :=
+mod_def a b ⬝ if_pos (pair h₁ h₂)
+
+theorem add_mod_self [simp] (x z : ℕ) : (x + z) % z = x % z :=
+by_cases_zero_pos z
+  (by rewrite add_zero)
+  (take z, assume H : z > 0,
+    calc
+      (x + z) % z = if 0 < z × z ≤ x + z then (x + z - z) % z else _ : mod_def
+                ... = (x + z - z) % z   : if_pos (pair H (le_add_left z x))
+                ... = x % z             : nat.add_sub_cancel)
+
+theorem add_mod_self_left [simp] (x z : ℕ) : (x + z) % x = z % x :=
+!add.comm ▸ !add_mod_self
+
+local attribute succ_mul [simp]
+
+theorem add_mul_mod_self [simp] (x y z : ℕ) : (x + y * z) % z = x % z :=
+nat.rec_on y (by rewrite [zero_mul, add_zero])
+             (by intro y IH; rewrite [succ_mul, -add.assoc, add_mod_self, IH])
+
+theorem add_mul_mod_self_left [simp] (x y z : ℕ) : (x + y * z) % y = x % y :=
+by rewrite [mul.comm, add_mul_mod_self]
+
+theorem mul_mod_left [simp] (m n : ℕ) : (m * n) % n = 0 :=
+calc (m * n) % n = (0 + m * n) % n : by rewrite [zero_add]
+            ...  = 0               : by rewrite [add_mul_mod_self, zero_mod]
+
+theorem mul_mod_right [simp] (m n : ℕ) : (m * n) % m = 0 :=
+by rewrite [mul.comm, mul_mod_left]
+
+theorem mod_lt (x : ℕ) {y : ℕ} (H : y > 0) : x % y < y :=
+nat.case_strong_rec_on x
+  (show 0 % y < y, from !zero_mod⁻¹ ▸ H)
+  (take x,
+    assume IH : Πx', x' ≤ x → x' % y < y,
+    show succ x % y < y, from
+      by_cases -- (succ x < y)
+        (assume H1 : succ x < y,
+          have succ x % y = succ x, from mod_eq_of_lt H1,
+          show succ x % y < y, from this⁻¹ ▸ H1)
+        (assume H1 : ¬ succ x < y,
+          have y ≤ succ x, from le_of_not_gt H1,
+          have h : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H this,
+          have succ x - y < succ x, from sub_lt !succ_pos H,
+          have succ x - y ≤ x, from le_of_lt_succ this,
+          show succ x % y < y, from h⁻¹ ▸ IH _ this))
+
+theorem mod_one (n : ℕ) : n % 1 = 0 :=
+have H1 : n % 1 < 1, from !mod_lt !succ_pos,
+eq_zero_of_le_zero (le_of_lt_succ H1)
+
+/- properties of div prod mod -/
+
+-- the quotient - remainder theorem
+theorem eq_div_mul_add_mod (x y : ℕ) : x = x / y * y + x % y :=
+begin
+  eapply by_cases_zero_pos y,
+  show x = x / 0 * 0 + x % 0, from
+    (calc
+      x / 0 * 0 + x % 0 = 0 + x % 0 : mul_zero
+        ... = x % 0                 : zero_add
+        ... = x                     : mod_zero)⁻¹,
+  intro y H,
+  show x = x / y * y + x % y,
+  begin
+    eapply nat.case_strong_rec_on x,
+    show 0 = (0 / y) * y + 0 % y, by rewrite [zero_mod, add_zero, nat.zero_div, zero_mul],
+    intro x IH,
+    show succ x = succ x / y * y + succ x % y, from
+      if H1 : succ x < y then
+         assert H2 : succ x / y = 0, from div_eq_zero_of_lt H1,
+         assert H3 : succ x % y = succ x, from mod_eq_of_lt H1,
+         begin rewrite [H2, H3, zero_mul, zero_add] end
+      else
+         have H2 : y ≤ succ x, from le_of_not_gt H1,
+         assert H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2,
+         assert H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2,
+         have H5 : succ x - y < succ x, from sub_lt !succ_pos H,
+         assert H6 : succ x - y ≤ x, from le_of_lt_succ H5,
+         (calc
+             succ x / y * y + succ x % y =
+                  succ ((succ x - y) / y) * y + succ x % y      : by rewrite H3
+            ... = ((succ x - y) / y) * y + y + succ x % y       : by rewrite succ_mul
+            ... = ((succ x - y) / y) * y + y + (succ x - y) % y : by rewrite H4
+            ... = ((succ x - y) / y) * y + (succ x - y) % y + y : add.right_comm
+            ... = succ x - y + y                                : by rewrite -(IH _ H6)
+            ... = succ x                                        : nat.sub_add_cancel H2)⁻¹
+  end
+end
+
+theorem mod_eq_sub_div_mul (x y : ℕ) : x % y = x - x / y * y :=
+nat.eq_sub_of_add_eq (!add.comm ▸ !eq_div_mul_add_mod)⁻¹
+
+theorem mod_add_mod (m n k : ℕ) : (m % n + k) % n = (m + k) % n :=
+by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m / n * n), add_mul_mod_self]
+
+theorem add_mod_mod (m n k : ℕ) : (m + n % k) % k = (m + n) % k :=
+by rewrite [add.comm, mod_add_mod, add.comm]
+
+theorem add_mod_eq_add_mod_right {m n k : ℕ} (i : ℕ) (H : m % n = k % n) :
+  (m + i) % n = (k + i) % n :=
+by rewrite [-mod_add_mod, -mod_add_mod k, H]
+
+theorem add_mod_eq_add_mod_left {m n k : ℕ} (i : ℕ) (H : m % n = k % n) :
+  (i + m) % n = (i + k) % n :=
+by rewrite [add.comm, add_mod_eq_add_mod_right _ H, add.comm]
+
+theorem mod_eq_mod_of_add_mod_eq_add_mod_right {m n k i : ℕ} :
+  (m + i) % n = (k + i) % n → m % n = k % n :=
+by_cases_zero_pos n
+  (by rewrite [*mod_zero]; apply eq_of_add_eq_add_right)
+  (take n,
+    assume npos : n > 0,
+    assume H1 : (m + i) % n = (k + i) % n,
+    have H2 : (m + i % n) % n = (k + i % n) % n, by rewrite [*add_mod_mod, H1],
+    assert H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n,
+      from add_mod_eq_add_mod_right _ H2,
+    begin
+      revert H3,
+      rewrite [*add.assoc, add_sub_of_le (le_of_lt (!mod_lt npos)), *add_mod_self],
+      intros, assumption
+    end)
+
+theorem mod_eq_mod_of_add_mod_eq_add_mod_left {m n k i : ℕ} :
+  (i + m) % n = (i + k) % n → m % n = k % n :=
+by rewrite [add.comm i m, add.comm i k]; apply mod_eq_mod_of_add_mod_eq_add_mod_right
+
+theorem mod_le {x y : ℕ} : x % y ≤ x :=
+!eq_div_mul_add_mod⁻¹ ▸ !le_add_left
+
+theorem eq_remainder {q1 r1 q2 r2 y : ℕ} (H1 : r1 < y) (H2 : r2 < y)
+  (H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
+calc
+  r1 = r1 % y               : mod_eq_of_lt H1
+    ... = (r1 + q1 * y) % y : !add_mul_mod_self⁻¹
+    ... = (q1 * y + r1) % y : add.comm
+    ... = (r2 + q2 * y) % y : by rewrite [H3, add.comm]
+    ... = r2 % y            : !add_mul_mod_self
+    ... = r2                : mod_eq_of_lt H2
+
+theorem eq_quotient {q1 r1 q2 r2 y : ℕ} (H1 : r1 < y) (H2 : r2 < y)
+  (H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 :=
+have H4 : q1 * y + r2 = q2 * y + r2, from (eq_remainder H1 H2 H3) ▸ H3,
+have H5 : q1 * y = q2 * y, from add.right_cancel H4,
+have H6 : y > 0, from lt_of_le_of_lt !zero_le H1,
+show q1 = q2, from eq_of_mul_eq_mul_right H6 H5
+
+protected theorem mul_div_mul_left {z : ℕ} (x y : ℕ) (zpos : z > 0) :
+  (z * x) / (z * y) = x / y :=
+if H : y = 0 then
+  by rewrite [H, mul_zero, *nat.div_zero]
+else
+  have ypos : y > 0, from pos_of_ne_zero H,
+  have zypos : z * y > 0, from mul_pos zpos ypos,
+  have H1 : (z * x) % (z * y) < z * y, from !mod_lt zypos,
+  have H2 : z * (x % y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
+  eq_quotient H1 H2
+    (calc
+      ((z * x) / (z * y)) * (z * y) + (z * x) % (z * y) = z * x : eq_div_mul_add_mod
+        ... = z * (x / y * y + x % y)                           : eq_div_mul_add_mod
+        ... = z * (x / y * y) + z * (x % y)                     : left_distrib
+        ... = (x / y) * (z * y) + z * (x % y)                   : mul.left_comm)
+
+protected theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) / (y * z) = x / y :=
+!mul.comm ▸ !mul.comm ▸ !nat.mul_div_mul_left zpos
+
+theorem mul_mod_mul_left (z x y : ℕ) : (z * x) % (z * y) = z * (x % y) :=
+sum.elim (eq_zero_sum_pos z)
+  (assume H : z = 0, H⁻¹ ▸ calc
+      (0 * x) % (z * y) = 0 % (z * y)   : zero_mul
+                      ... = 0           : zero_mod
+                      ... = 0 * (x % y) : zero_mul)
+  (assume zpos : z > 0,
+    sum.elim (eq_zero_sum_pos y)
+      (assume H : y = 0, by rewrite [H, mul_zero, *mod_zero])
+      (assume ypos : y > 0,
+        have zypos : z * y > 0, from mul_pos zpos ypos,
+        have H1 : (z * x) % (z * y) < z * y, from !mod_lt zypos,
+        have H2 : z * (x % y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
+        eq_remainder H1 H2
+          (calc
+            ((z * x) / (z * y)) * (z * y) + (z * x) % (z * y) = z * x : eq_div_mul_add_mod
+              ... = z * (x / y * y + x % y)                           : eq_div_mul_add_mod
+              ... = z * (x / y * y) + z * (x % y)                     : left_distrib
+              ... = (x / y) * (z * y) + z * (x % y)                   : mul.left_comm)))
+
+theorem mul_mod_mul_right (x z y : ℕ) : (x * z) % (y * z) = (x % y) * z :=
+mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ !mul_mod_mul_left
+
+theorem mod_self (n : ℕ) : n % n = 0 :=
+nat.cases_on n (by rewrite zero_mod)
+  (take n, by rewrite [-zero_add (succ n) at {1}, add_mod_self])
+
+theorem mul_mod_eq_mod_mul_mod (m n k : nat) : (m * n) % k = ((m % k) * n) % k :=
+calc
+  (m * n) % k = (((m / k) * k + m % k) * n) % k : eq_div_mul_add_mod
+            ... = ((m % k) * n) % k             :
+                    by rewrite [right_distrib, mul.right_comm, add.comm, add_mul_mod_self]
+
+theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) % k = (m * (n % k)) % k :=
+!mul.comm ▸ !mul.comm ▸ !mul_mod_eq_mod_mul_mod
+
+protected theorem div_one (n : ℕ) : n / 1 = n :=
+assert n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹,
+begin rewrite [-this at {2}, mul_one, mod_one] end
+
+protected theorem div_self {n : ℕ} (H : n > 0) : n / n = 1 :=
+assert (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H,
+by rewrite [nat.div_one at this, -this, *mul_one]
+
+theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : m / n * n = m :=
+by rewrite [eq_div_mul_add_mod m n at {2}, H, add_zero]
+
+theorem mul_div_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : n * (m / n) = m :=
+!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
+
+/- dvd -/
+
+theorem dvd_of_mod_eq_zero {m n : ℕ} (H : n % m = 0) : m ∣ n :=
+dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
+
+theorem mod_eq_zero_of_dvd {m n : ℕ} (H : m ∣ n) : n % m = 0 :=
+dvd.elim H (take z, assume H1 : n = m * z, H1⁻¹ ▸ !mul_mod_right)
+
+theorem dvd_iff_mod_eq_zero (m n : ℕ) : m ∣ n ↔ n % m = 0 :=
+iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
+
+definition dvd.decidable_rel [instance] : decidable_rel dvd :=
+take m n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
+
+protected theorem div_mul_cancel {m n : ℕ} (H : n ∣ m) : m / n * n = m :=
+div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
+
+protected theorem mul_div_cancel' {m n : ℕ} (H : n ∣ m) : n * (m / n) = m :=
+!mul.comm ▸ nat.div_mul_cancel H
+
+theorem dvd_of_dvd_add_left {m n₁ n₂ : ℕ} (H₁ : m ∣ n₁ + n₂) (H₂ : m ∣ n₁) : m ∣ n₂ :=
+obtain (c₁ : nat) (Hc₁ : n₁ + n₂ = m * c₁), from H₁,
+obtain (c₂ : nat) (Hc₂ : n₁ = m * c₂), from H₂,
+have   aux : m * (c₁ - c₂) = n₂, from calc
+  m * (c₁ - c₂) = m * c₁  - m * c₂ : nat.mul_sub_left_distrib
+           ...  = n₁ + n₂ - m * c₂ : Hc₁
+           ...  = n₁ + n₂ - n₁     : Hc₂
+           ...  = n₂               : nat.add_sub_cancel_left,
+dvd.intro aux
+
+theorem dvd_of_dvd_add_right {m n₁ n₂ : ℕ} (H : m ∣ n₁ + n₂) : m ∣ n₂ → m ∣ n₁ :=
+nat.dvd_of_dvd_add_left (!add.comm ▸ H)
+
+theorem dvd_sub {m n₁ n₂ : ℕ} (H1 : m ∣ n₁) (H2 : m ∣ n₂) : m ∣ n₁ - n₂ :=
+by_cases
+  (assume H3 : n₁ ≥ n₂,
+    have H4 : n₁ = n₁ - n₂ + n₂, from (nat.sub_add_cancel H3)⁻¹,
+    show m ∣ n₁ - n₂, from nat.dvd_of_dvd_add_right (H4 ▸ H1) H2)
+  (assume H3 : ¬ (n₁ ≥ n₂),
+    have H4 : n₁ - n₂ = 0, from sub_eq_zero_of_le (le_of_lt (lt_of_not_ge H3)),
+    show m ∣ n₁ - n₂, from H4⁻¹ ▸ dvd_zero _)
+
+theorem dvd.antisymm {m n : ℕ} : m ∣ n → n ∣ m → m = n :=
+by_cases_zero_pos n
+  (assume H1, assume H2 : 0 ∣ m, eq_zero_of_zero_dvd H2)
+  (take n,
+    assume Hpos : n > 0,
+    assume H1 : m ∣ n,
+    assume H2 : n ∣ m,
+    obtain k (Hk : n = m * k), from exists_eq_mul_right_of_dvd H1,
+    obtain l (Hl : m = n * l), from exists_eq_mul_right_of_dvd H2,
+    have n * (l * k) = n, from !mul.assoc ▸ Hl ▸ Hk⁻¹,
+    have l * k = 1,       from eq_one_of_mul_eq_self_right Hpos this,
+    have k = 1,           from eq_one_of_mul_eq_one_left this,
+    show m = n,           from (mul_one m)⁻¹ ⬝ (this ▸ Hk⁻¹))
+
+protected theorem mul_div_assoc (m : ℕ) {n k : ℕ} (H : k ∣ n) : m * n / k = m * (n / k) :=
+sum.elim (eq_zero_sum_pos k)
+  (assume H1 : k = 0,
+    calc
+      m * n / k = m * n / 0     : H1
+              ... = 0           : nat.div_zero
+              ... = m * 0       : mul_zero m
+              ... = m * (n / 0) : nat.div_zero
+              ... = m * (n / k) : H1)
+  (assume H1 : k > 0,
+    have H2 : n = n / k * k, from (nat.div_mul_cancel H)⁻¹,
+      calc
+        m * n / k = m * (n / k * k) / k   : H2
+                ... = m * (n / k) * k / k : mul.assoc
+                ... = m * (n / k)         : nat.mul_div_cancel _ H1)
+
+theorem dvd_of_mul_dvd_mul_left {m n k : ℕ} (kpos : k > 0) (H : k * m ∣ k * n) : m ∣ n :=
+dvd.elim H
+  (take l,
+    assume H1 : k * n = k * m * l,
+    have H2 : n = m * l, from eq_of_mul_eq_mul_left kpos (H1 ⬝ !mul.assoc),
+    dvd.intro H2⁻¹)
+
+theorem dvd_of_mul_dvd_mul_right {m n k : ℕ} (kpos : k > 0) (H : m * k ∣ n * k) : m ∣ n :=
+nat.dvd_of_mul_dvd_mul_left kpos (!mul.comm ▸ !mul.comm ▸ H)
+
+lemma dvd_of_eq_mul (i j n : nat) : n = j*i → j ∣ n :=
+begin intros, subst n, apply dvd_mul_right end
+
+theorem div_dvd_div {k m n : ℕ} (H1 : k ∣ m) (H2 : m ∣ n) : m / k ∣ n / k :=
+have H3 : m = m / k * k, from (nat.div_mul_cancel H1)⁻¹,
+have H4 : n = n / k * k, from (nat.div_mul_cancel (dvd.trans H1 H2))⁻¹,
+sum.elim (eq_zero_sum_pos k)
+  (assume H5 : k = 0,
+    have H6: n / k = 0, from (ap _ H5 ⬝ !nat.div_zero),
+      H6⁻¹ ▸ !dvd_zero)
+  (assume H5 : k > 0,
+    nat.dvd_of_mul_dvd_mul_right H5 (H3 ▸ H4 ▸ H2))
+
+protected theorem div_eq_iff_eq_mul_right {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
+  m / n = k ↔ m = n * k :=
+iff.intro
+  (assume H1, by rewrite [-H1, nat.mul_div_cancel' H'])
+  (assume H1, by rewrite [H1, !nat.mul_div_cancel_left H])
+
+protected theorem div_eq_iff_eq_mul_left {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
+  m / n = k ↔ m = k * n :=
+!mul.comm ▸ !nat.div_eq_iff_eq_mul_right H H'
+
+protected theorem eq_mul_of_div_eq_right {m n k : ℕ} (H1 : n ∣ m) (H2 : m / n = k) :
+  m = n * k :=
+calc
+  m     = n * (m / n) : nat.mul_div_cancel' H1
+    ... = n * k       : H2
+
+protected theorem div_eq_of_eq_mul_right {m n k : ℕ} (H1 : n > 0) (H2 : m = n * k) :
+  m / n = k :=
+calc
+  m / n = n * k / n : H2
+      ... = k       : !nat.mul_div_cancel_left H1
+
+protected theorem eq_mul_of_div_eq_left {m n k : ℕ} (H1 : n ∣ m) (H2 : m / n = k) :
+  m = k * n :=
+!mul.comm ▸ !nat.eq_mul_of_div_eq_right H1 H2
+
+protected theorem div_eq_of_eq_mul_left {m n k : ℕ} (H1 : n > 0) (H2 : m = k * n) :
+  m / n = k :=
+!nat.div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
+
+lemma add_mod_eq_of_dvd (i j n : nat) : n ∣ j → (i + j) % n = i % n :=
+assume h,
+obtain k (hk : j = n * k), from exists_eq_mul_right_of_dvd h,
+begin
+  subst j, rewrite mul.comm,
+  apply add_mul_mod_self
+end
+
+/- / prod ordering -/
+
+lemma le_of_dvd {m n : nat} : n > 0 → m ∣ n → m ≤ n :=
+assume (h₁ : n > 0) (h₂ : m ∣ n),
+assert h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂,
+by_contradiction
+ (λ nle : ¬ m ≤ n,
+   have   h₄ : m > n, from lt_of_not_ge nle,
+   assert h₅ : n % m = n, from mod_eq_of_lt h₄,
+   begin
+     rewrite h₃ at h₅, subst n,
+     exact absurd h₁ (lt.irrefl 0)
+   end)
+
+theorem div_mul_le (m n : ℕ) : m / n * n ≤ m :=
+calc
+  m = m / n * n + m % n : eq_div_mul_add_mod
+    ... ≥ m / n * n       : le_add_right
+
+protected theorem div_le_of_le_mul {m n k : ℕ} (H : m ≤ n * k) : m / k ≤ n :=
+sum.elim (eq_zero_sum_pos k)
+  (assume H1 : k = 0,
+    calc
+      m / k = m / 0 : H1
+          ... = 0       : nat.div_zero
+          ... ≤ n       : zero_le)
+  (assume H1 : k > 0,
+    le_of_mul_le_mul_right (calc
+      m / k * k ≤ m / k * k + m % k : le_add_right
+        ... = m                     : eq_div_mul_add_mod
+        ... ≤ n * k                 : H) H1)
+
+protected theorem div_le_self (m n : ℕ) : m / n ≤ m :=
+nat.cases_on n (!nat.div_zero⁻¹ ▸ !zero_le)
+  take n,
+  have H : m ≤ m * succ n, from calc
+        m = m * 1      : mul_one
+      ... ≤ m * succ n : !mul_le_mul_left (succ_le_succ !zero_le),
+  nat.div_le_of_le_mul H
+
+protected theorem mul_le_of_le_div {m n k : ℕ} (H : m ≤ n / k) : m * k ≤ n :=
+calc
+  m * k ≤ n / k * k : !mul_le_mul_right H
+    ... ≤ n           : div_mul_le
+
+protected theorem le_div_of_mul_le {m n k : ℕ} (H1 : k > 0) (H2 : m * k ≤ n) : m ≤ n / k :=
+have H3 : m * k < (succ (n / k)) * k, from
+  calc
+    m * k ≤ n                  : H2
+      ... = n / k * k + n % k  : eq_div_mul_add_mod
+      ... < n / k * k + k      : add_lt_add_left (!mod_lt H1)
+      ... = (succ (n / k)) * k : succ_mul,
+le_of_lt_succ (lt_of_mul_lt_mul_right H3)
+
+protected theorem le_div_iff_mul_le {m n k : ℕ} (H : k > 0) : m ≤ n / k ↔ m * k ≤ n :=
+iff.intro !nat.mul_le_of_le_div (!nat.le_div_of_mul_le H)
+
+protected theorem div_le_div {m n : ℕ} (k : ℕ) (H : m ≤ n) : m / k ≤ n / k :=
+by_cases_zero_pos k
+  (by rewrite [*nat.div_zero])
+  (take k, assume H1 : k > 0, nat.le_div_of_mul_le H1 (le.trans !div_mul_le H))
+
+protected theorem div_lt_of_lt_mul {m n k : ℕ} (H : m < n * k) : m / k < n :=
+lt_of_mul_lt_mul_right (calc
+  m / k * k ≤ m / k * k + m % k : le_add_right
+    ... = m                     : eq_div_mul_add_mod
+    ... < n * k                 : H)
+
+protected theorem lt_mul_of_div_lt {m n k : ℕ} (H1 : k > 0) (H2 : m / k < n) : m < n * k :=
+assert H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2),
+have H4 : m / k * k + k ≤ n * k, by rewrite [succ_mul at H3]; apply H3,
+calc
+  m     = m / k * k + m % k : eq_div_mul_add_mod
+    ... < m / k * k + k     : add_lt_add_left (!mod_lt H1)
+    ... ≤ n * k             : H4
+
+protected theorem div_lt_iff_lt_mul {m n k : ℕ} (H : k > 0) : m / k < n ↔ m < n * k :=
+iff.intro (!nat.lt_mul_of_div_lt H) !nat.div_lt_of_lt_mul
+
+protected theorem div_le_iff_le_mul_of_div {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
+  m / n ≤ k ↔ m ≤ k * n :=
+by refine iff.trans (!le_iff_mul_le_mul_right H) _; rewrite [!nat.div_mul_cancel H']
+
+protected theorem le_mul_of_div_le_of_div {m n k : ℕ} (H1 : n > 0) (H2 : n ∣ m) (H3 : m / n ≤ k) :
+  m ≤ k * n :=
+iff.mp (!nat.div_le_iff_le_mul_of_div H1 H2) H3
+
+-- needed for integer division
+theorem mul_sub_div_of_lt {m n k : ℕ} (H : k < m * n) :
+  (m * n - (k + 1)) / m = n - k / m - 1 :=
+begin
+  have H1 : k / m < n, from nat.div_lt_of_lt_mul (!mul.comm ▸ H),
+  have H2 : n - k / m ≥ 1, from
+    nat.le_sub_of_add_le (calc
+       1 + k / m = succ (k / m) : add.comm
+               ... ≤ n          : succ_le_of_lt H1),
+  have H3 : n - k / m = n - k / m - 1 + 1, from (nat.sub_add_cancel H2)⁻¹,
+  have H4 : m > 0, from pos_of_ne_zero (assume H': m = 0, not_lt_zero k (begin rewrite [H' at H, zero_mul at H], exact H end)),
+  have H5   : k % m + 1 ≤ m, from succ_le_of_lt (!mod_lt H4),
+  have H6 : m - (k % m + 1) < m, from nat.sub_lt_self H4 !succ_pos,
+calc
+  (m * n - (k + 1)) / m = (m * n - (k / m * m + k % m + 1)) / m : eq_div_mul_add_mod
+     ... = (m * n - k / m * m - (k % m + 1)) / m                  : by rewrite [*nat.sub_sub]
+     ... = ((n - k / m) * m - (k % m + 1)) / m                    :
+               by rewrite [mul.comm m, nat.mul_sub_right_distrib]
+     ... = ((n - k / m - 1) * m + m - (k % m + 1)) / m            :
+               by rewrite [H3 at {1}, right_distrib, nat.one_mul]
+     ... = ((n - k / m - 1) * m + (m - (k % m + 1))) / m          : {nat.add_sub_assoc H5 _}
+     ... = (m - (k % m + 1)) / m + (n - k / m - 1)                :
+               by rewrite [add.comm, (add_mul_div_self H4)]
+     ... = n - k / m - 1                                          :
+               by rewrite [div_eq_zero_of_lt H6, zero_add]
+end
+
+private lemma div_div_aux (a b c : nat) : b > 0 → c > 0 → (a / b) / c = a / (b * c) :=
+suppose b > 0, suppose c > 0,
+nat.strong_rec_on a
+(λ a ih,
+  let k₁ := a / (b*c) in
+  let k₂ := a %(b*c) in
+  assert bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`,
+  assert k₂ < b * c,       from mod_lt _ bc_pos,
+  assert k₂ ≤ a,           from !mod_le,
+  sum.elim (eq_sum_lt_of_le this)
+    (suppose k₂ = a,
+     assert i₁ : a < b * c,   by rewrite -this; assumption,
+     assert k₁ = 0,           from div_eq_zero_of_lt i₁,
+     assert a / b < c,      by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁,
+     begin
+       rewrite [`k₁ = 0`],
+       show (a / b) / c = 0, from div_eq_zero_of_lt `a / b < c`
+     end)
+    (suppose k₂ < a,
+     assert a = k₁*(b*c) + k₂,         from eq_div_mul_add_mod a (b*c),
+     assert a / b = k₁*c + k₂ / b, by
+       rewrite [this at {1}, mul.comm b c at {2}, -mul.assoc,
+                add.comm, add_mul_div_self `b > 0`, add.comm],
+     assert e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by
+       rewrite [this, add.comm, add_mul_div_self `c > 0`, add.comm],
+     assert e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`,
+     assert e₃ : k₂ / (b * c)   = 0,              from div_eq_zero_of_lt `k₂ < b * c`,
+     assert (k₂ / b) / c      = 0,              by rewrite [e₂, e₃],
+     show (a / b) / c = k₁,                     by rewrite [e₁, this]))
+
+protected lemma div_div_eq_div_mul (a b c : nat) : (a / b) / c = a / (b * c) :=
+begin
+ cases b with b,
+   rewrite [zero_mul, *nat.div_zero, nat.zero_div],
+   cases c with c,
+     rewrite [mul_zero, *nat.div_zero],
+     apply div_div_aux a (succ b) (succ c) dec_star dec_star
+end
+
+lemma div_lt_of_ne_zero : Π {n : nat}, n ≠ 0 → n / 2 < n
+| 0        h := absurd rfl h
+| (succ n) h :=
+  begin
+    apply nat.div_lt_of_lt_mul,
+    rewrite [-add_one, right_distrib],
+    change n + 1 < (n * 1 + n) + (1 + 1),
+    rewrite [mul_one, -add.assoc],
+    apply add_lt_add_right,
+    show n < n + n + 1,
+    begin
+      rewrite [add.assoc, -add_zero n at {1}],
+      apply add_lt_add_left,
+      apply zero_lt_succ
+    end
+  end
+end nat

library/data/nat/div.lean

@@ -5,7 +5,7 @@ Authors: Jeremy Avigad, Leonardo de Moura
 
 Definitions and properties of div and mod. Much of the development follows Isabelle's library.
 -/
-import data.nat.sub
+import .sub
 open eq.ops well_founded decidable prod
 
 namespace nat
@@ -525,7 +525,7 @@ iff.intro (!nat.lt_mul_of_div_lt H) !nat.div_lt_of_lt_mul
 
 protected theorem div_le_iff_le_mul_of_div {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
   m / n ≤ k ↔ m ≤ k * n :=
-by rewrite [propext (!le_iff_mul_le_mul_right H), !nat.div_mul_cancel H']
+by refine iff.trans (!le_iff_mul_le_mul_right H) _; rewrite [!nat.div_mul_cancel H']
 
 protected theorem le_mul_of_div_le_of_div {m n k : ℕ} (H1 : n > 0) (H2 : n ∣ m) (H3 : m / n ≤ k) :
   m ≤ k * n :=