239 lines
10 KiB
Text
239 lines
10 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.int.div
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Author: Jeremy Avigad
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Definitions and properties of div, mod, gcd, lcm, coprime. Following the SSReflect library
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(and the SMT lib standard), we define a mod b so that 0 ≤ a mod b < |b| when b ≠ 0.
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-/
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import data.int.order data.nat.div
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open [coercions] [reduce-hints] nat
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open [declarations] nat (succ)
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open eq.ops
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notation `ℕ` := nat
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set_option pp.beta true
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namespace int
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/- definitions -/
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definition divide (a b : ℤ) : ℤ :=
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sign b *
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(match a with
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| of_nat m := #nat m div (nat_abs b)
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| -[ m +1] := -[ (#nat m div (nat_abs b)) +1]
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end)
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notation a div b := divide a b
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definition modulo (a b : ℤ) : ℤ := a - a div b * b
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notation a mod b := modulo a b
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/- div -/
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theorem of_nat_div_of_nat (m n : nat) : m div n = of_nat (#nat m div n) :=
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nat.cases_on n
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(by rewrite [↑divide, sign_zero, zero_mul, nat.div_zero])
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(take n, by rewrite [↑divide, sign_of_succ, one_mul])
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theorem neg_succ_of_nat_div (m : nat) {b : ℤ} (H : b > 0) :
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-[m +1] div b = -(m div b + 1) :=
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calc
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-[m +1] div b = sign b * _ : rfl
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... = -[(#nat m div (nat_abs b)) +1] : by rewrite [(sign_of_pos H), one_mul]
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... = -(m div b + 1) : by rewrite [↑divide, (sign_of_pos H), one_mul]
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theorem div_neg (a b : ℤ) : a div -b = -(a div b) :=
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calc
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a div -b = sign (-b) * _ : rfl
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... = -(sign b) * _ : sign_neg
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... = -(sign b * _) : neg_mul_eq_neg_mul
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... = -(sign b * _) : nat_abs_neg
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... = -(a div b) : rfl
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theorem zero_div (b : ℤ) : 0 div b = 0 :=
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calc
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0 div b = sign b * (#nat 0 div (nat_abs b)) : rfl
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... = sign b * 0 : nat.zero_div
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... = 0 : mul_zero
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theorem div_zero (a : ℤ) : a div 0 = 0 :=
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by rewrite [↑divide, sign_zero, zero_mul]
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theorem eq_div_mul_add_mod {a b : ℤ} : a = a div b * b + a mod b :=
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!add.comm ▸ eq_add_of_sub_eq rfl
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/- mod -/
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theorem of_nat_mod_of_nat (m n : nat) : m mod n = (#nat m mod n) :=
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have H : m = (#nat m mod n) + m div n * n, from calc
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m = of_nat (#nat m div n * n + m mod n) : nat.eq_div_mul_add_mod
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... = (#nat m div n) * n + (#nat m mod n) : rfl
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... = m div n * n + (#nat m mod n) : of_nat_div_of_nat
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... = (#nat m mod n) + m div n * n : add.comm,
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calc
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m mod n = m - m div n * n : rfl
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... = (#nat m mod n) : sub_eq_of_eq_add H
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theorem neg_succ_of_nat_mod (m : ℕ) {b : ℤ} (bpos : b > 0) :
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-[m +1] mod b = b - 1 - m mod b :=
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calc
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-[m +1] mod b = -(m + 1) - -[m +1] div b * b : rfl
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... = -(m + 1) - -(m div b + 1) * b : neg_succ_of_nat_div _ bpos
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... = -m + -1 + (b + m div b * b) :
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by rewrite [neg_add, -neg_mul_eq_neg_mul, sub_neg_eq_add, mul.right_distrib,
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one_mul, (add.comm b)]
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... = b + -1 + (-m + m div b * b) :
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by rewrite [-*add.assoc, (add.comm (-m)), (add.right_comm (-1)), (add.comm b)]
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... = b - 1 - m mod b :
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by rewrite [↑modulo, *sub_eq_add_neg, neg_add, neg_neg]
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theorem mod_neg (a b : ℤ) : a mod -b = a mod b :=
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calc
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a mod -b = a - (a div -b) * -b : rfl
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... = a - -(a div b) * -b : div_neg
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... = a - a div b * b : neg_mul_neg
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... = a mod b : rfl
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theorem mod_abs (a b : ℤ) : a mod (abs b) = a mod b :=
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abs.by_cases rfl !mod_neg
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theorem zero_mod (b : ℤ) : 0 mod b = 0 :=
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by rewrite [↑modulo, zero_div, zero_mul, sub_zero]
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theorem mod_zero (a : ℤ) : a mod 0 = a :=
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by rewrite [↑modulo, mul_zero, sub_zero]
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private lemma of_nat_mod_abs (m : ℕ) (b : ℤ) : m mod (abs b) = (#nat m mod (nat_abs b)) :=
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calc
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m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
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... = (#nat m mod (nat_abs b)) : of_nat_mod_of_nat
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private lemma of_nat_mod_abs_lt (m : ℕ) {b : ℤ} (H : b ≠ 0) : m mod (abs b) < (abs b) :=
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have H1 : abs b > 0, from abs_pos_of_ne_zero H,
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have H2 : (#nat nat_abs b > 0), from lt_of_of_nat_lt_of_nat (!of_nat_nat_abs⁻¹ ▸ H1),
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calc
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m mod (abs b) = (#nat m mod (nat_abs b)) : of_nat_mod_abs m b
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... < nat_abs b : of_nat_lt_of_nat (nat.mod_lt H2)
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... = abs b : of_nat_nat_abs _
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theorem mod_nonneg (a : ℤ) {b : ℤ} (H : b ≠ 0) : a mod b ≥ 0 :=
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have H1 : abs b > 0, from abs_pos_of_ne_zero H,
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have H2 : a mod (abs b) ≥ 0, from
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int.cases_on a
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(take m, (of_nat_mod_abs m b)⁻¹ ▸ !of_nat_nonneg)
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(take m,
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have H3 : 1 + m mod (abs b) ≤ (abs b), from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
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calc
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-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
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... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
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... ≥ 0 : iff.mp' !sub_nonneg_iff_le H3),
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!mod_abs ▸ H2
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theorem mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a mod b < (abs b) :=
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have H1 : abs b > 0, from abs_pos_of_ne_zero H,
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have H2 : a mod (abs b) < abs b, from
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int.cases_on a
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(take m, of_nat_mod_abs_lt m H)
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(take m,
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have H3 : abs b ≠ 0, from assume H', H (eq_zero_of_abs_eq_zero H'),
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have H4 : 1 + m mod (abs b) > 0, from add_pos_of_pos_of_nonneg dec_trivial (mod_nonneg _ H3),
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calc
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-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
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... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
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... < abs b : sub_lt_self _ H4),
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!mod_abs ▸ H2
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/- both div and mod -/
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private theorem add_mul_div_self_right_aux1 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a ≥ 0) (H2 : #nat k > 0) :
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(a + n * k) div k = a div k + n :=
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obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1,
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Hm⁻¹ ▸ (calc
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(m + n * k) div k = (#nat (m + n * k)) div k : rfl
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... = (#nat (m + n * k) div k) : of_nat_div_of_nat
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... = (#nat m div k + n) : !nat.add_mul_div_self_right H2
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... = (#nat m div k) + n : rfl
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... = m div k + n : of_nat_div_of_nat)
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private theorem add_mul_div_self_right_aux2 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a < 0) (H2 : #nat k > 0) :
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(a + n * k) div k = a div k + n :=
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obtain m (Hm : a = -[m +1]), from exists_eq_neg_succ_of_nat H1,
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or.elim (nat.lt_or_ge m (#nat n * k))
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(assume m_lt_nk : #nat m < n * k,
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have H3 : #nat (m + 1 ≤ n * k), from nat.succ_le_of_lt m_lt_nk,
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have H4 : #nat m div k + 1 ≤ n,
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from nat.succ_le_of_lt (nat.div_lt_of_lt_mul (!nat.mul.comm ▸ m_lt_nk)),
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Hm⁻¹ ▸ (calc
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(-[m +1] + n * k) div k = (n * k - (m + 1)) div k : by rewrite [add.comm, neg_succ_of_nat_eq]
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... = ((#nat n * k) - (#nat m + 1)) div k : rfl
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... = (#nat n * k - (m + 1)) div k : {of_nat_sub_of_nat H3}
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... = #nat (n * k - (m + 1)) div k : of_nat_div_of_nat
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... = #nat (k * n - (m + 1)) div k : nat.mul.comm
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... = #nat n - m div k - 1 :
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nat.mul_sub_div_of_lt (!nat.mul.comm ▸ m_lt_nk)
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... = #nat n - (m div k + 1) : nat.sub_sub
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... = n - (#nat m div k + 1) : of_nat_sub_of_nat H4
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... = -(m div k + 1) + n :
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by rewrite [add.comm, -sub_eq_add_neg, -of_nat_add_of_nat, of_nat_div_of_nat]
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... = -[m +1] div k + n :
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neg_succ_of_nat_div m (of_nat_lt_of_nat H2)))
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(assume nk_le_m : #nat n * k ≤ m,
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eq.symm (Hm⁻¹ ▸ (calc
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-[m +1] div k + n = -(m div k + 1) + n :
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neg_succ_of_nat_div m (of_nat_lt_of_nat H2)
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... = -((#nat m div k) + 1) + n : of_nat_div_of_nat
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... = -((#nat (m - n * k + n * k) div k) + 1) + n : nat.sub_add_cancel nk_le_m
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... = -((#nat (m - n * k) div k + n) + 1) + n : nat.add_mul_div_self_right H2
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... = -((#nat m - n * k) div k + 1) :
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by rewrite [-of_nat_add_of_nat, *neg_add, add.right_comm, neg_add_cancel_right,
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of_nat_div_of_nat]
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... = -[(#nat m - n * k) +1] div k :
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neg_succ_of_nat_div _ (of_nat_lt_of_nat H2)
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... = -((#nat m - n * k) + 1) div k : rfl
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... = -(m - (#nat n * k) + 1) div k : of_nat_sub_of_nat nk_le_m
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... = (-(m + 1) + n * k) div k :
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by rewrite [sub_eq_add_neg, -*add.assoc, *neg_add, neg_neg, add.right_comm]
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... = (-[m +1] + n * k) div k : rfl)))
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private theorem add_mul_div_self_right_aux3 (a : ℤ) {b c : ℤ} (H1 : b ≥ 0) (H2 : c > 0) :
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(a + b * c) div c = a div c + b :=
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obtain n (Hn : b = of_nat n), from exists_eq_of_nat H1,
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obtain k (Hk : c = of_nat k), from exists_eq_of_nat (le_of_lt H2),
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have knz : k ≠ 0, from assume kz, !lt.irrefl (kz ▸ Hk ▸ H2),
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have kgt0 : (#nat k > 0), from nat.pos_of_ne_zero knz,
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have H3 : (a + n * k) div k = a div k + n, from
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or.elim (lt_or_ge a 0)
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(assume Ha : a < 0, add_mul_div_self_right_aux2 _ Ha kgt0)
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(assume Ha : a ≥ 0, add_mul_div_self_right_aux1 _ Ha kgt0),
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Hn⁻¹ ▸ Hk⁻¹ ▸ H3
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private theorem add_mul_div_self_right_aux4 (a b : ℤ) {c : ℤ} (H : c > 0) :
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(a + b * c) div c = a div c + b :=
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or.elim (le.total 0 b)
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(assume H1 : 0 ≤ b, add_mul_div_self_right_aux3 _ H1 H)
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(assume H1 : 0 ≥ b,
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eq.symm (calc
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a div c + b = (a + b * c + -b * c) div c + b :
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by rewrite [-neg_mul_eq_neg_mul, add_neg_cancel_right]
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... = (a + b * c) div c + - b + b :
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add_mul_div_self_right_aux3 _ (neg_nonneg_of_nonpos H1) H
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... = (a + b * c) div c : neg_add_cancel_right))
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theorem add_mul_div_self_right (a b : ℤ) {c : ℤ} (H : c ≠ 0) : (a + b * c) div c = a div c + b :=
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lt.by_cases
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(assume H1 : 0 < c, !add_mul_div_self_right_aux4 H1)
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(assume H1 : 0 = c, absurd H1⁻¹ H)
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(assume H1 : 0 > c,
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have H2 : -c > 0, from neg_pos_of_neg H1,
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calc
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(a + b * c) div c = - ((a + -b * -c) div -c) : by rewrite [div_neg, neg_mul_neg, neg_neg]
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... = -(a div -c + -b) : !add_mul_div_self_right_aux4 H2
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... = a div c + b : by rewrite [div_neg, neg_add, *neg_neg])
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end int
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