lean2/library/data/int/basic.lean

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: int.basic
Authors: Floris van Doorn, Jeremy Avigad

The integers, with addition, multiplication, and subtraction. The representation of the integers is
chosen to compute efficiently.

To faciliate proving things about these operations, we show that the integers are a quotient of
ℕ × ℕ with the usual equivalence relation, ≡, and functions

  abstr : ℕ × ℕ → ℤ
  repr : ℤ → ℕ × ℕ

satisfying:

  abstr_repr (a : ℤ) : abstr (repr a) = a
  repr_abstr (p : ℕ × ℕ) : repr (abstr p) ≡ p
  abstr_eq (p q : ℕ × ℕ) : p ≡ q → abstr p = abstr q

For example, to "lift" statements about add to statements about padd, we need to prove the
following:

  repr_add (a b : ℤ) : repr (a + b) = padd (repr a) (repr b)
  padd_congr (p p' q q' : ℕ × ℕ) (H1 : p ≡ p') (H2 : q ≡ q') : padd p q ≡ p' q'

-/
import data.nat.basic data.nat.order data.nat.sub data.prod
import algebra.relation algebra.binary algebra.ordered_ring
import tools.fake_simplifier
open eq.ops
open prod relation nat
open decidable binary fake_simplifier

/- the type of integers -/

inductive int : Type :=
| of_nat : nat → int
| neg_succ_of_nat : nat → int

notation `ℤ` := int
attribute int.of_nat [coercion]
definition int.of_num [coercion] [reducible] (n : num) : ℤ := int.of_nat (nat.of_num n)

namespace int

/- definitions of basic functions -/

definition neg_of_nat (m : ℕ) : ℤ :=
nat.cases_on m 0 (take m', neg_succ_of_nat m')

definition sub_nat_nat (m n : ℕ) : ℤ :=
nat.cases_on (n - m)
  (of_nat (m - n))                                     -- m ≥ n
  (take k, neg_succ_of_nat k)                          -- m < n, and n - m = succ k

definition neg (a : ℤ) : ℤ :=
  int.cases_on a
    (take m,                                           -- a = of_nat m
      nat.cases_on m 0 (take m', neg_succ_of_nat m'))
    (take m, of_nat (succ m))                          -- a = neg_succ_of_nat m

definition add (a b : ℤ) : ℤ :=
  int.cases_on a
    (take m,                                           -- a = of_nat m
      int.cases_on b
        (take n, of_nat (m + n))                         -- b = of_nat n
        (take n, sub_nat_nat m (succ n)))                -- b = neg_succ_of_nat n
    (take m,                                           -- a = neg_succ_of_nat m
      int.cases_on b
        (take n, sub_nat_nat n (succ m))                 -- b = of_nat n
        (take n, neg_of_nat (succ m + succ n)))          -- b = neg_succ_of_nat n

definition mul (a b : ℤ) : ℤ :=
  int.cases_on a
    (take m,                                           -- a = of_nat m
      int.cases_on b
        (take n, of_nat (m * n))                         -- b = of_nat n
        (take n, neg_of_nat (m * succ n)))               -- b = neg_succ_of_nat n
    (take m,                                           -- a = neg_succ_of_nat m
      int.cases_on b
        (take n, neg_of_nat (succ m * n))                -- b = of_nat n
        (take n, of_nat (succ m * succ n)))              -- b = neg_succ_of_nat n

/- notation -/

notation `-[` n `+1]` := int.neg_succ_of_nat n    -- for pretty-printing output
prefix - := int.neg
infix +  := int.add
infix *  := int.mul

/- some basic functions and properties -/

theorem of_nat.inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
int.no_confusion H (λe, e)

theorem neg_succ_of_nat.inj {m n : ℕ} (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
int.no_confusion H (λe, e)

theorem neg_succ_of_nat_eq (n : ℕ) : -[n +1] = -(n + 1) := rfl

definition has_decidable_eq [instance] : decidable_eq ℤ :=
take a b,
int.cases_on a
  (take m,
    int.cases_on b
      (take n,
        if H : m = n then inl (congr_arg of_nat H) else inr (take H1, H (of_nat.inj H1)))
      (take n', inr (assume H, int.no_confusion H)))
  (take m',
    int.cases_on b
      (take n, inr (assume H, int.no_confusion H))
      (take n',
        (if H : m' = n' then inl (congr_arg neg_succ_of_nat H) else
            inr (take H1, H (neg_succ_of_nat.inj H1)))))

theorem of_nat_add_of_nat (n m : nat) : of_nat n + of_nat m = #nat n + m := rfl

theorem of_nat_succ (n : ℕ) : of_nat (succ n) = of_nat n + 1 := rfl

theorem of_nat_mul_of_nat (n m : ℕ) : of_nat n * of_nat m = n * m := rfl

theorem sub_nat_nat_of_ge {m n : ℕ} (H : m ≥ n) : sub_nat_nat m n = of_nat (m - n) :=
have H1 : n - m = 0, from sub_eq_zero_of_le H,
calc
  sub_nat_nat m n = nat.cases_on 0 (of_nat (m - n)) _ : H1 ▸ rfl
    ... = of_nat (m - n) : rfl

context
attribute sub_nat_nat [reducible]
theorem sub_nat_nat_of_lt {m n : ℕ} (H : m < n) :
  sub_nat_nat m n = neg_succ_of_nat (pred (n - m)) :=
have H1 : n - m = succ (pred (n - m)), from (succ_pred_of_pos (sub_pos_of_lt H))⁻¹,
calc
  sub_nat_nat m n = nat.cases_on (succ (pred (n - m))) (of_nat (m - n))
        (take k, neg_succ_of_nat k) : H1 ▸ rfl
    ... = neg_succ_of_nat (pred (n - m)) : rfl
end

definition nat_abs (a : ℤ) : ℕ := int.cases_on a (take n, n) (take n', succ n')

theorem nat_abs_of_nat (n : ℕ) : nat_abs (of_nat n) = n := rfl

theorem nat_abs_eq_zero {a : ℤ} : nat_abs a = 0 → a = 0 :=
int.cases_on a
  (take m, assume H : nat_abs (of_nat m) = 0, congr_arg of_nat H)
  (take m', assume H : nat_abs (neg_succ_of_nat m') = 0, absurd H (succ_ne_zero _))

/- int is a quotient of ordered pairs of natural numbers -/

definition equiv (p q : ℕ × ℕ) : Prop :=  pr1 p + pr2 q = pr2 p + pr1 q

local notation p `≡` q := equiv p q

theorem equiv.refl {p : ℕ × ℕ} : p ≡ p := !add.comm

theorem equiv.symm {p q : ℕ × ℕ} (H : p ≡ q) : q ≡ p :=
calc
  pr1 q + pr2 p = pr2 p + pr1 q : !add.comm
    ... = pr1 p + pr2 q         : H⁻¹
    ... = pr2 q + pr1 p         : !add.comm

theorem equiv.trans {p q r : ℕ × ℕ} (H1 : p ≡ q) (H2 : q ≡ r) : p ≡ r :=
have H3 : pr1 p + pr2 r + pr2 q = pr2 p + pr1 r + pr2 q, from
  calc
   pr1 p + pr2 r + pr2 q = pr1 p + pr2 q + pr2 r : by simp
    ... = pr2 p + pr1 q + pr2 r                  : {H1}
    ... = pr2 p + (pr1 q + pr2 r)                : by simp
    ... = pr2 p + (pr2 q + pr1 r)                : {H2}
    ... = pr2 p + pr1 r + pr2 q                  : by simp,
show pr1 p + pr2 r = pr2 p + pr1 r, from add.cancel_right H3

theorem equiv_equiv : is_equivalence equiv :=
is_equivalence.mk @equiv.refl @equiv.symm @equiv.trans

theorem equiv_cases {p q : ℕ × ℕ} (H : equiv p q) :
    (pr1 p ≥ pr2 p ∧ pr1 q ≥ pr2 q) ∨ (pr1 p < pr2 p ∧ pr1 q < pr2 q) :=
or.elim (@le_or_gt (pr2 p) (pr1 p))
  (assume H1: pr1 p ≥ pr2 p,
    have H2 : pr2 p + pr1 q ≥ pr2 p + pr2 q, from H ▸ add_le_add_right H1 (pr2 q),
    or.inl (and.intro H1 (le_of_add_le_add_left H2)))
  (assume H1: pr1 p < pr2 p,
    have H2 : pr2 p + pr1 q < pr2 p + pr2 q, from H ▸ add_lt_add_right H1 (pr2 q),
    or.inr (and.intro H1 (lt_of_add_lt_add_left H2)))

theorem equiv_of_eq {p q : ℕ × ℕ} (H : p = q) : p ≡ q := H ▸ equiv.refl

calc_trans equiv.trans
calc_refl equiv.refl
calc_symm equiv.symm

/- the representation and abstraction functions -/

definition abstr (a : ℕ × ℕ) : ℤ := sub_nat_nat (pr1 a) (pr2 a)

theorem abstr_of_ge {p : ℕ × ℕ} (H : pr1 p ≥ pr2 p) : abstr p = of_nat (pr1 p - pr2 p) :=
sub_nat_nat_of_ge H

theorem abstr_of_lt {p : ℕ × ℕ} (H : pr1 p < pr2 p) :
  abstr p = neg_succ_of_nat (pred (pr2 p - pr1 p)) :=
sub_nat_nat_of_lt H

definition repr (a : ℤ) : ℕ × ℕ := int.cases_on a (take m, (m, 0)) (take m, (0, succ m))

theorem abstr_repr (a : ℤ) : abstr (repr a) = a :=
int.cases_on a (take m, (sub_nat_nat_of_ge (zero_le m))) (take m, rfl)

theorem repr_sub_nat_nat (m n : ℕ) : repr (sub_nat_nat m n) ≡ (m, n) :=
or.elim (@le_or_gt n m)
  (take H : m ≥ n,
    have H1 : repr (sub_nat_nat m n) = (m - n, 0), from sub_nat_nat_of_ge H ▸ rfl,
    H1⁻¹ ▸
      (calc
        m - n + n = m : sub_add_cancel H
          ... = 0 + m : zero_add))
  (take H : m < n,
    have H1 : repr (sub_nat_nat m n) = (0, succ (pred (n - m))), from sub_nat_nat_of_lt H ▸ rfl,
    H1⁻¹ ▸
      (calc
        0 + n = n : zero_add
          ... = n - m + m : sub_add_cancel (le_of_lt H)
          ... = succ (pred (n - m)) + m : (succ_pred_of_pos (sub_pos_of_lt H))⁻¹))

theorem repr_abstr (p : ℕ × ℕ) : repr (abstr p) ≡ p :=
!prod.eta ▸ !repr_sub_nat_nat

theorem abstr_eq {p q : ℕ × ℕ} (Hequiv : p ≡ q) : abstr p = abstr q :=
or.elim (equiv_cases Hequiv)
  (assume H2,
    have H3 : pr1 p ≥ pr2 p, from and.elim_left H2,
    have H4 : pr1 q ≥ pr2 q, from and.elim_right H2,
    have H5 : pr1 p = pr1 q - pr2 q + pr2 p, from
      calc
        pr1 p = pr1 p + pr2 q - pr2 q : add_sub_cancel
          ... = pr2 p + pr1 q - pr2 q : Hequiv
          ... = pr2 p + (pr1 q - pr2 q) : add_sub_assoc H4
          ... = pr1 q - pr2 q + pr2 p : add.comm,
    have H6 : pr1 p - pr2 p = pr1 q - pr2 q, from
      calc
        pr1 p - pr2 p = pr1 q - pr2 q + pr2 p - pr2 p : H5
                  ... = pr1 q - pr2 q                 : add_sub_cancel,
    abstr_of_ge H3 ⬝ congr_arg of_nat H6 ⬝ (abstr_of_ge H4)⁻¹)
  (assume H2,
    have H3 : pr1 p < pr2 p, from and.elim_left H2,
    have H4 : pr1 q < pr2 q, from and.elim_right H2,
    have H5 : pr2 p = pr2 q - pr1 q + pr1 p, from
      calc
        pr2 p = pr2 p + pr1 q - pr1 q : add_sub_cancel
          ... = pr1 p + pr2 q - pr1 q : Hequiv
          ... = pr1 p + (pr2 q - pr1 q) : add_sub_assoc (le_of_lt H4)
          ... = pr2 q - pr1 q + pr1 p : add.comm,
    have H6 : pr2 p - pr1 p = pr2 q - pr1 q, from
      calc
        pr2 p - pr1 p = pr2 q - pr1 q + pr1 p - pr1 p : H5
                  ... = pr2 q - pr1 q                 : add_sub_cancel,
    abstr_of_lt H3 ⬝ congr_arg neg_succ_of_nat (congr_arg pred H6)⬝ (abstr_of_lt H4)⁻¹)

theorem equiv_iff (p q : ℕ × ℕ) : (p ≡ q) ↔ ((p ≡ p) ∧ (q ≡ q) ∧ (abstr p = abstr q)) :=
iff.intro
  (assume H : equiv p q,
    and.intro !equiv.refl (and.intro !equiv.refl (abstr_eq H)))
  (assume H : equiv p p ∧ equiv q q ∧ abstr p = abstr q,
    have H1 : abstr p = abstr q, from and.elim_right (and.elim_right H),
    equiv.trans (H1 ▸ equiv.symm (repr_abstr p)) (repr_abstr q))

theorem eq_abstr_of_equiv_repr {a : ℤ} {p : ℕ × ℕ} (Hequiv : repr a ≡ p) : a = abstr p :=
calc
  a = abstr (repr a) : abstr_repr
   ... = abstr p : abstr_eq Hequiv

theorem eq_of_repr_equiv_repr {a b : ℤ} (H : repr a ≡ repr b) : a = b :=
calc
  a = abstr (repr a) : abstr_repr
    ... = abstr (repr b) : abstr_eq H
    ... = b : abstr_repr

context
attribute abstr [reducible]
attribute dist [reducible]
theorem nat_abs_abstr (p : ℕ × ℕ) : nat_abs (abstr p) = dist (pr1 p) (pr2 p) :=
let m := pr1 p, n := pr2 p in
or.elim (@le_or_gt n m)
  (assume H : m ≥ n,
    calc
      nat_abs (abstr (m, n)) = nat_abs (of_nat (m - n)) : int.abstr_of_ge H
                         ... = dist m n                 : dist_eq_sub_of_ge H)
  (assume H : m < n,
    calc
      nat_abs (abstr (m, n)) = nat_abs (neg_succ_of_nat (pred (n - m))) : int.abstr_of_lt H
        ... = succ (pred (n - m)) : rfl
        ... = n - m               : succ_pred_of_pos (sub_pos_of_lt H)
        ... = dist m n            : dist_eq_sub_of_le (le_of_lt H))
end

theorem cases_of_nat (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - of_nat n) :=
int.cases_on a
  (take n, or.inl (exists.intro n rfl))
  (take n', or.inr (exists.intro (succ n') rfl))

theorem cases_of_nat_succ (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - (of_nat (succ n))) :=
int.cases_on a (take m, or.inl (exists.intro _ rfl)) (take m, or.inr (exists.intro _ rfl))

theorem by_cases_of_nat {P : ℤ → Prop} (a : ℤ)
    (H1 : ∀n : ℕ, P (of_nat n)) (H2 : ∀n : ℕ, P (- of_nat n)) :
  P a :=
or.elim (cases_of_nat a)
  (assume H, obtain (n : ℕ) (H3 : a = n), from H, H3⁻¹ ▸ H1 n)
  (assume H, obtain (n : ℕ) (H3 : a = -n), from H, H3⁻¹ ▸ H2 n)

theorem by_cases_of_nat_succ {P : ℤ → Prop} (a : ℤ)
    (H1 : ∀n : ℕ, P (of_nat n)) (H2 : ∀n : ℕ, P (- of_nat (succ n))) :
  P a :=
or.elim (cases_of_nat_succ a)
  (assume H, obtain (n : ℕ) (H3 : a = n), from H, H3⁻¹ ▸ H1 n)
  (assume H, obtain (n : ℕ) (H3 : a = -(succ n)), from H, H3⁻¹ ▸ H2 n)

/-
   int is a ring
-/

/- addition -/

definition padd (p q : ℕ × ℕ) : ℕ × ℕ := (pr1 p + pr1 q, pr2 p + pr2 q)

theorem repr_add (a b : ℤ) :  repr (add a b) ≡ padd (repr a) (repr b) :=
int.cases_on a
  (take m,
    int.cases_on b
      (take n, !equiv.refl)
      (take n',
        have H1 : equiv (repr (add (of_nat m) (neg_succ_of_nat n'))) (m, succ n'),
          from !repr_sub_nat_nat,
        have H2 : padd (repr (of_nat m)) (repr (neg_succ_of_nat n')) = (m, 0 + succ n'),
          from rfl,
        (!zero_add ▸ H2)⁻¹ ▸ H1))
  (take m',
    int.cases_on b
      (take n,
        have H1 : equiv (repr (add (neg_succ_of_nat m') (of_nat n))) (n, succ m'),
          from !repr_sub_nat_nat,
        have H2 : padd (repr (neg_succ_of_nat m')) (repr (of_nat n)) = (0 + n, succ m'),
          from rfl,
        (!zero_add ▸ H2)⁻¹ ▸ H1)
       (take n',!repr_sub_nat_nat))

theorem padd_congr {p p' q q' : ℕ × ℕ} (Ha : p ≡ p') (Hb : q ≡ q') : padd p q ≡ padd p' q' :=
calc
  pr1 (padd p q) + pr2 (padd p' q') = pr1 p + pr2 p' + (pr1 q + pr2 q') : by simp
    ... = pr2 p + pr1 p' + (pr1 q + pr2 q') : {Ha}
    ... = pr2 p + pr1 p' + (pr2 q + pr1 q') : {Hb}
    ... = pr2 (padd p q) + pr1 (padd p' q') : by simp

theorem padd_comm (p q : ℕ × ℕ) : padd p q = padd q p :=
calc
  padd p q = (pr1 p + pr1 q, pr2 p + pr2 q) : rfl
    ... = (pr1 q + pr1 p, pr2 p + pr2 q) : add.comm
    ... = (pr1 q + pr1 p, pr2 q + pr2 p) : add.comm
    ... = padd q p : rfl

theorem padd_assoc (p q r : ℕ × ℕ) : padd (padd p q) r = padd p (padd q r) :=
calc
  padd (padd p q) r = (pr1 p + pr1 q + pr1 r, pr2 p + pr2 q + pr2 r) : rfl
    ... = (pr1 p + (pr1 q + pr1 r), pr2 p + pr2 q + pr2 r) : add.assoc
    ... = (pr1 p + (pr1 q + pr1 r), pr2 p + (pr2 q + pr2 r)) : add.assoc
    ... = padd p (padd q r) : rfl

theorem add.comm (a b : ℤ) : a + b = b + a :=
begin
  apply eq_of_repr_equiv_repr,
  apply equiv.trans,
  apply repr_add,
  apply equiv.symm,
  apply (eq.subst (padd_comm (repr b) (repr a))),
  apply repr_add
end

theorem add.assoc (a b c : ℤ) : a + b + c = a + (b + c) :=
assert H1 : repr (a + b + c) ≡ padd (padd (repr a) (repr b)) (repr c), from
  equiv.trans (repr_add (a + b) c) (padd_congr !repr_add !equiv.refl),
assert H2 : repr (a + (b + c)) ≡ padd (repr a) (padd (repr b) (repr c)), from
  equiv.trans (repr_add a (b + c)) (padd_congr !equiv.refl !repr_add),
begin
  apply eq_of_repr_equiv_repr,
  apply equiv.trans,
  apply H1,
  apply (eq.subst ((padd_assoc _ _ _)⁻¹)),
  apply equiv.symm,
  apply H2
end

theorem add_zero (a : ℤ) : a + 0 = a := int.cases_on a (take m, rfl) (take m', rfl)

theorem zero_add (a : ℤ) : 0 + a = a := add.comm a 0 ▸ add_zero a

/- negation -/

definition pneg (p : ℕ × ℕ) : ℕ × ℕ := (pr2 p, pr1 p)

-- note: this is =, not just ≡
theorem repr_neg (a : ℤ) : repr (- a) = pneg (repr a) :=
int.cases_on a
  (take m,
    nat.cases_on m rfl (take m', rfl))
  (take m', rfl)

theorem pneg_congr {p p' : ℕ × ℕ} (H : p ≡ p') : pneg p ≡ pneg p' := eq.symm H

theorem pneg_pneg (p : ℕ × ℕ) : pneg (pneg p) = p := !prod.eta

theorem nat_abs_neg (a : ℤ) : nat_abs (-a) = nat_abs a :=
calc
  nat_abs (-a) = nat_abs (abstr (repr (-a))) : abstr_repr
    ... = nat_abs (abstr (pneg (repr a))) : repr_neg
    ... = dist (pr1 (pneg (repr a))) (pr2 (pneg (repr a))) : nat_abs_abstr
    ... = dist (pr2 (pneg (repr a))) (pr1 (pneg (repr a))) : dist.comm
    ... = nat_abs (abstr (repr a)) : nat_abs_abstr
    ... = nat_abs a : abstr_repr

theorem padd_pneg (p : ℕ × ℕ) : padd p (pneg p) ≡ (0, 0) :=
show pr1 p + pr2 p + 0 = pr2 p + pr1 p + 0, from !nat.add.comm ▸ rfl

theorem padd_padd_pneg (p q : ℕ × ℕ) : padd (padd p q) (pneg q) ≡ p :=
show pr1 p + pr1 q + pr2 q + pr2 p = pr2 p + pr2 q + pr1 q + pr1 p, from by simp

theorem add.left_inv (a : ℤ) : -a + a = 0 :=
have H : repr (-a + a) ≡ repr 0, from
  calc
    repr (-a + a) ≡ padd (repr (neg a)) (repr a) : repr_add
      ... = padd (pneg (repr a)) (repr a) : repr_neg
      ... ≡ repr 0 : padd_pneg,
eq_of_repr_equiv_repr H

/- nat abs -/

definition pabs (p : ℕ × ℕ) : ℕ := dist (pr1 p) (pr2 p)

theorem pabs_congr {p q : ℕ × ℕ} (H : p ≡ q) : pabs p = pabs q :=
calc
  pabs p = nat_abs (abstr p) : nat_abs_abstr
    ... = nat_abs (abstr q) : abstr_eq H
    ... = pabs q : nat_abs_abstr

theorem nat_abs_eq_pabs_repr (a : ℤ) : nat_abs a = pabs (repr a) :=
calc
  nat_abs a = nat_abs (abstr (repr a)) : abstr_repr
    ... = pabs (repr a) : nat_abs_abstr

theorem nat_abs_add_le (a b : ℤ) : nat_abs (a + b) ≤ nat_abs a + nat_abs b :=
have H : nat_abs (a + b) = pabs (padd (repr a) (repr b)), from
  calc
    nat_abs (a + b) = pabs (repr (a + b)) : nat_abs_eq_pabs_repr
      ... = pabs (padd (repr a) (repr b)) : pabs_congr !repr_add,
have H1 : nat_abs a = pabs (repr a), from !nat_abs_eq_pabs_repr,
have H2 : nat_abs b = pabs (repr b), from !nat_abs_eq_pabs_repr,
have H3 : pabs (padd (repr a) (repr b)) ≤ pabs (repr a) + pabs (repr b),
  from !dist_add_add_le_add_dist_dist,
H⁻¹ ▸ H1⁻¹ ▸ H2⁻¹ ▸ H3

context
attribute nat_abs [reducible]
theorem mul_nat_abs (a b : ℤ) : nat_abs (a * b) = #nat (nat_abs a) * (nat_abs b) :=
int.cases_on a
  (take m,
    int.cases_on b
      (take n, rfl)
      (take n', !nat_abs_neg ▸ rfl))
  (take m',
    int.cases_on b
      (take n, !nat_abs_neg ▸ rfl)
      (take n', rfl))
end

/- multiplication -/

definition pmul (p q : ℕ × ℕ) : ℕ × ℕ :=
  (pr1 p * pr1 q + pr2 p * pr2 q, pr1 p * pr2 q + pr2 p * pr1 q)

theorem repr_neg_of_nat (m : ℕ) : repr (neg_of_nat m) = (0, m) :=
nat.cases_on m rfl (take m', rfl)

-- note: we have =, not just ≡
theorem repr_mul (a b : ℤ) :  repr (mul a b) = pmul (repr a) (repr b) :=
int.cases_on a
  (take m,
    int.cases_on b
      (take n,
        (calc
          pmul (repr m) (repr n) = (m * n + 0 * 0, m * 0 + 0 * n) : rfl
            ... = (m * n + 0 * 0, m * 0 + 0) : zero_mul)⁻¹)
      (take n',
        (calc
          pmul (repr m) (repr (neg_succ_of_nat n')) =
              (m * 0 + 0 * succ n', m * succ n' + 0 * 0) : rfl
            ... = (m * 0 + 0, m * succ n' + 0 * 0) : zero_mul
            ... = repr (mul m (neg_succ_of_nat n')) : repr_neg_of_nat)⁻¹))
  (take m',
    int.cases_on b
      (take n,
        (calc
          pmul (repr (neg_succ_of_nat m')) (repr n) =
              (0 * n + succ m' * 0, 0 * 0 + succ m' * n) : rfl
            ... = (0 + succ m' * 0, 0 * 0 + succ m' * n) : zero_mul
            ... = (0 + succ m' * 0, succ m' * n) : {!nat.zero_add}
            ... = repr (mul (neg_succ_of_nat m') n) : repr_neg_of_nat)⁻¹)
      (take n',
        (calc
          pmul (repr (neg_succ_of_nat m')) (repr (neg_succ_of_nat n')) =
              (0 + succ m' * succ n', 0 * succ n') : rfl
            ... = (succ m' * succ n', 0 * succ n') : nat.zero_add
            ... = (succ m' * succ n', 0) : zero_mul
            ... = repr (mul (neg_succ_of_nat m') (neg_succ_of_nat n')) : rfl)⁻¹))

theorem equiv_mul_prep {xa ya xb yb xn yn xm ym : ℕ}
  (H1 : xa + yb = ya + xb) (H2 : xn + ym = yn + xm)
: xa * xn + ya * yn + (xb * ym + yb * xm) = xa * yn + ya * xn + (xb * xm + yb * ym) :=
have H3 : xa * xn + ya * yn + (xb * ym + yb * xm) + (yb * xn + xb * yn + (xb * xn + yb * yn))
    = xa * yn + ya * xn + (xb * xm + yb * ym) + (yb * xn + xb * yn + (xb * xn + yb * yn)), from
  calc
    xa * xn + ya * yn + (xb * ym + yb * xm) + (yb * xn + xb * yn + (xb * xn + yb * yn))
          = xa * xn + yb * xn + (ya * yn + xb * yn) + (xb * xn + xb * ym + (yb * yn + yb * xm))
            : by simp
      ... = (xa + yb) * xn + (ya + xb) * yn + (xb * (xn + ym) + yb * (yn + xm)) : by simp
      ... = (ya + xb) * xn + (xa + yb) * yn + (xb * (yn + xm) + yb * (xn + ym)) : by simp
      ... = ya * xn + xb * xn + (xa * yn + yb * yn) + (xb * yn + xb * xm + (yb*xn + yb*ym))
            : by simp
      ... = xa * yn + ya * xn + (xb * xm + yb * ym) + (yb * xn + xb * yn + (xb * xn + yb * yn))
            : by simp,
nat.add.cancel_right H3

theorem pmul_congr {p p' q q' : ℕ × ℕ} (H1 : p ≡ p') (H2 : q ≡ q') : pmul p q ≡ pmul p' q' :=
equiv_mul_prep H1 H2

theorem pmul_comm (p q : ℕ × ℕ) : pmul p q = pmul q p :=
calc
  (pr1 p * pr1 q + pr2 p * pr2 q, pr1 p * pr2 q + pr2 p * pr1 q) =
      (pr1 q * pr1 p + pr2 p * pr2 q, pr1 p * pr2 q + pr2 p * pr1 q) : mul.comm
    ... = (pr1 q * pr1 p + pr2 q * pr2 p, pr1 p * pr2 q + pr2 p * pr1 q) : mul.comm
    ... = (pr1 q * pr1 p + pr2 q * pr2 p, pr2 q * pr1 p + pr2 p * pr1 q) : mul.comm
    ... = (pr1 q * pr1 p + pr2 q * pr2 p, pr2 q * pr1 p + pr1 q * pr2 p) : mul.comm
    ... = (pr1 q * pr1 p + pr2 q * pr2 p, pr1 q * pr2 p + pr2 q * pr1 p) : nat.add.comm

theorem mul.comm (a b : ℤ) : a * b = b * a :=
eq_of_repr_equiv_repr
  ((calc
    repr (a * b) = pmul (repr a) (repr b) : repr_mul
      ... = pmul (repr b) (repr a) : pmul_comm
      ... = repr (b * a) : repr_mul) ▸ !equiv.refl)

theorem pmul_assoc (p q r: ℕ × ℕ) : pmul (pmul p q) r = pmul p (pmul q r) :=
by simp

theorem mul.assoc (a b c : ℤ) : (a * b) * c = a * (b * c) :=
eq_of_repr_equiv_repr
  ((calc
    repr (a * b * c) = pmul (repr (a * b)) (repr c) : repr_mul
      ... = pmul (pmul (repr a) (repr b)) (repr c) : repr_mul
      ... = pmul (repr a) (pmul (repr b) (repr c)) : pmul_assoc
      ... = pmul (repr a) (repr (b * c)) : repr_mul
      ... = repr (a * (b * c)) : repr_mul) ▸ !equiv.refl)

theorem mul_one (a : ℤ) : a * 1 = a :=
eq_of_repr_equiv_repr (equiv_of_eq
  ((calc
    repr (a * 1) = pmul (repr a) (repr 1) : repr_mul
      ... = (pr1 (repr a), pr2 (repr a)) : by simp
      ... = repr a : prod.eta)))

theorem one_mul (a : ℤ) : 1 * a = a :=
mul.comm a 1 ▸ mul_one a

theorem mul.right_distrib (a b c : ℤ) : (a + b) * c = a * c + b * c :=
eq_of_repr_equiv_repr
  (calc
    repr ((a + b) * c) = pmul (repr (a + b)) (repr c) : repr_mul
      ... ≡ pmul (padd (repr a) (repr b)) (repr c) : pmul_congr !repr_add equiv.refl
      ... = padd (pmul (repr a) (repr c)) (pmul (repr b) (repr c)) : by simp
      ... = padd (repr (a * c)) (pmul (repr b) (repr c)) : {(repr_mul a c)⁻¹}
      ... = padd (repr (a * c)) (repr (b * c)) : repr_mul
      ... ≡ repr (a * c + b * c) : equiv.symm !repr_add)

theorem mul.left_distrib (a b c : ℤ) : a * (b + c) = a * b + a * c :=
calc
  a * (b + c) = (b + c) * a : mul.comm a (b + c)
    ... = b * a + c * a : mul.right_distrib b c a
    ... = a * b + c * a : {mul.comm b a}
    ... = a * b + a * c : {mul.comm c a}

theorem zero_ne_one : (typeof 0 : int) ≠ 1 :=
assume H : 0 = 1,
show false, from succ_ne_zero 0 ((of_nat.inj H)⁻¹)

theorem eq_zero_or_eq_zero_of_mul_eq_zero {a b : ℤ} (H : a * b = 0) : a = 0 ∨ b = 0 :=
have H2 : (nat_abs a) * (nat_abs b) = nat.zero, from
  calc
    (nat_abs a) * (nat_abs b) = (nat_abs (a * b)) : (mul_nat_abs a b)⁻¹
      ... = (nat_abs 0) : {H}
      ... = nat.zero : nat_abs_of_nat nat.zero,
have H3 : (nat_abs a) = nat.zero ∨ (nat_abs b) = nat.zero,
  from eq_zero_or_eq_zero_of_mul_eq_zero H2,
or_of_or_of_imp_of_imp H3
  (assume H : (nat_abs a) = nat.zero, nat_abs_eq_zero H)
  (assume H : (nat_abs b) = nat.zero, nat_abs_eq_zero H)

section
  open [classes] algebra

  protected definition integral_domain [instance] [reducible] : algebra.integral_domain int :=
  ⦃algebra.integral_domain,
    add            := add,
    add_assoc      := add.assoc,
    zero           := zero,
    zero_add       := zero_add,
    add_zero       := add_zero,
    neg            := neg,
    add_left_inv   := add.left_inv,
    add_comm       := add.comm,
    mul            := mul,
    mul_assoc      := mul.assoc,
    one            := (of_num 1),
    one_mul        := one_mul,
    mul_one        := mul_one,
    left_distrib   := mul.left_distrib,
    right_distrib  := mul.right_distrib,
    mul_comm       := mul.comm,
    eq_zero_or_eq_zero_of_mul_eq_zero := @eq_zero_or_eq_zero_of_mul_eq_zero⦄
end

/- instantiate ring theorems to int -/

section port_algebra
  theorem mul.left_comm : ∀a b c : ℤ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
  theorem mul.right_comm : ∀a b c : ℤ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
  theorem add.left_comm : ∀a b c : ℤ, a + (b + c) = b + (a + c) := algebra.add.left_comm
  theorem add.right_comm : ∀a b c : ℤ, (a + b) + c = (a + c) + b := algebra.add.right_comm
  theorem add.left_cancel : ∀{a b c : ℤ}, a + b = a + c → b = c := @algebra.add.left_cancel _ _
  theorem add.right_cancel : ∀{a b c : ℤ}, a + b = c + b → a = c := @algebra.add.right_cancel _ _
  theorem neg_add_cancel_left : ∀a b : ℤ, -a + (a + b) = b := algebra.neg_add_cancel_left
  theorem neg_add_cancel_right : ∀a b : ℤ, a + -b + b = a := algebra.neg_add_cancel_right
  theorem neg_eq_of_add_eq_zero : ∀{a b : ℤ}, a + b = 0 → -a = b :=
    @algebra.neg_eq_of_add_eq_zero _ _
  theorem neg_zero : -0 = 0 := algebra.neg_zero
  theorem neg_neg : ∀a : ℤ, -(-a) = a := algebra.neg_neg
  theorem neg.inj : ∀{a b : ℤ}, -a = -b → a = b := @algebra.neg.inj _ _
  theorem neg_eq_neg_iff_eq : ∀a b : ℤ, -a = -b ↔ a = b := algebra.neg_eq_neg_iff_eq
  theorem neg_eq_zero_iff_eq_zero : ∀a : ℤ, -a = 0 ↔ a = 0 := algebra.neg_eq_zero_iff_eq_zero
  theorem eq_neg_of_eq_neg : ∀{a b : ℤ}, a = -b → b = -a := @algebra.eq_neg_of_eq_neg _ _
  theorem eq_neg_iff_eq_neg : ∀{a b : ℤ}, a = -b ↔ b = -a := @algebra.eq_neg_iff_eq_neg _ _
  theorem add.right_inv : ∀a : ℤ, a + -a = 0 := algebra.add.right_inv
  theorem add_neg_cancel_left : ∀a b : ℤ, a + (-a + b) = b := algebra.add_neg_cancel_left
  theorem add_neg_cancel_right : ∀a b : ℤ, a + b + -b = a := algebra.add_neg_cancel_right
  theorem neg_add_rev : ∀a b : ℤ, -(a + b) = -b + -a := algebra.neg_add_rev
  theorem eq_add_neg_of_add_eq : ∀{a b c : ℤ}, a + c = b → a = b + -c :=
    @algebra.eq_add_neg_of_add_eq _ _
  theorem eq_neg_add_of_add_eq : ∀{a b c : ℤ}, b + a = c → a = -b + c :=
    @algebra.eq_neg_add_of_add_eq _ _
  theorem neg_add_eq_of_eq_add : ∀{a b c : ℤ}, b = a + c → -a + b = c :=
    @algebra.neg_add_eq_of_eq_add _ _
  theorem add_neg_eq_of_eq_add : ∀{a b c : ℤ}, a = c + b → a + -b = c :=
    @algebra.add_neg_eq_of_eq_add _ _
  theorem eq_add_of_add_neg_eq : ∀{a b c : ℤ}, a + -c = b → a = b + c :=
    @algebra.eq_add_of_add_neg_eq _ _
  theorem eq_add_of_neg_add_eq : ∀{a b c : ℤ}, -b + a = c → a = b + c :=
    @algebra.eq_add_of_neg_add_eq _ _
  theorem add_eq_of_eq_neg_add : ∀{a b c : ℤ}, b = -a + c → a + b = c :=
    @algebra.add_eq_of_eq_neg_add _ _
  theorem add_eq_of_eq_add_neg : ∀{a b c : ℤ}, a = c + -b → a + b = c :=
    @algebra.add_eq_of_eq_add_neg _ _
  theorem add_eq_iff_eq_neg_add : ∀a b c : ℤ, a + b = c ↔ b = -a + c :=
    @algebra.add_eq_iff_eq_neg_add _ _
  theorem add_eq_iff_eq_add_neg : ∀a b c : ℤ, a + b = c ↔ a = c + -b :=
    @algebra.add_eq_iff_eq_add_neg _ _
  definition sub (a b : ℤ) : ℤ := algebra.sub a b
  infix - := int.sub
  theorem sub_eq_add_neg : ∀a b : ℤ, a - b = a + -b := algebra.sub_eq_add_neg
  theorem sub_self : ∀a : ℤ, a - a = 0 := algebra.sub_self
  theorem sub_add_cancel : ∀a b : ℤ, a - b + b = a := algebra.sub_add_cancel
  theorem add_sub_cancel : ∀a b : ℤ, a + b - b = a := algebra.add_sub_cancel
  theorem eq_of_sub_eq_zero : ∀{a b : ℤ}, a - b = 0 → a = b := @algebra.eq_of_sub_eq_zero _ _
  theorem eq_iff_sub_eq_zero : ∀a b : ℤ, a = b ↔ a - b = 0 := algebra.eq_iff_sub_eq_zero
  theorem zero_sub : ∀a : ℤ, 0 - a = -a := algebra.zero_sub
  theorem sub_zero : ∀a : ℤ, a - 0 = a := algebra.sub_zero
  theorem sub_neg_eq_add : ∀a b : ℤ, a - (-b) = a + b := algebra.sub_neg_eq_add
  theorem neg_sub : ∀a b : ℤ, -(a - b) = b - a := algebra.neg_sub
  theorem add_sub : ∀a b c : ℤ, a + (b - c) = a + b - c := algebra.add_sub
  theorem sub_add_eq_sub_sub_swap : ∀a b c : ℤ, a - (b + c) = a - c - b :=
    algebra.sub_add_eq_sub_sub_swap
  theorem sub_eq_iff_eq_add : ∀a b c : ℤ, a - b = c ↔ a = c + b := algebra.sub_eq_iff_eq_add
  theorem eq_sub_iff_add_eq : ∀a b c : ℤ, a = b - c ↔ a + c = b := algebra.eq_sub_iff_add_eq
  theorem eq_iff_eq_of_sub_eq_sub : ∀{a b c d : ℤ}, a - b = c - d → a = b ↔ c = d :=
    @algebra.eq_iff_eq_of_sub_eq_sub _ _
  theorem eq_sub_of_add_eq : ∀{a b c : ℤ}, a + c = b → a = b - c := @algebra.eq_sub_of_add_eq _ _
  theorem sub_eq_of_eq_add : ∀{a b c : ℤ}, a = c + b → a - b = c := @algebra.sub_eq_of_eq_add _ _
  theorem eq_add_of_sub_eq : ∀{a b c : ℤ}, a - c = b → a = b + c := @algebra.eq_add_of_sub_eq _ _
  theorem add_eq_of_eq_sub : ∀{a b c : ℤ}, a = c - b → a + b = c := @algebra.add_eq_of_eq_sub _ _
  theorem sub_add_eq_sub_sub : ∀a b c : ℤ, a - (b + c) = a - b - c := algebra.sub_add_eq_sub_sub
  theorem neg_add_eq_sub : ∀a b : ℤ, -a + b = b - a := algebra.neg_add_eq_sub
  theorem neg_add : ∀a b : ℤ, -(a + b) = -a + -b := algebra.neg_add
  theorem sub_add_eq_add_sub : ∀a b c : ℤ, a - b + c = a + c - b := algebra.sub_add_eq_add_sub
  theorem sub_sub_ : ∀a b c : ℤ, a - b - c = a - (b + c) := algebra.sub_sub
  theorem add_sub_add_left_eq_sub : ∀a b c : ℤ, (c + a) - (c + b) = a - b :=
    algebra.add_sub_add_left_eq_sub
  theorem eq_sub_of_add_eq' : ∀{a b c : ℤ}, c + a = b → a = b - c := @algebra.eq_sub_of_add_eq' _ _
  theorem sub_eq_of_eq_add' : ∀{a b c : ℤ}, a = b + c → a - b = c := @algebra.sub_eq_of_eq_add' _ _
  theorem eq_add_of_sub_eq' : ∀{a b c : ℤ}, a - b = c → a = b + c := @algebra.eq_add_of_sub_eq' _ _
  theorem add_eq_of_eq_sub' : ∀{a b c : ℤ}, b = c - a → a + b = c := @algebra.add_eq_of_eq_sub' _ _
  theorem ne_zero_of_mul_ne_zero_right : ∀{a b : ℤ}, a * b ≠ 0 → a ≠ 0 :=
    @algebra.ne_zero_of_mul_ne_zero_right _ _
  theorem ne_zero_of_mul_ne_zero_left : ∀{a b : ℤ}, a * b ≠ 0 → b ≠ 0 :=
    @algebra.ne_zero_of_mul_ne_zero_left _ _
  definition dvd (a b : ℤ) : Prop := algebra.dvd a b
  notation (a | b) := dvd a b
  theorem dvd.intro : ∀{a b c : ℤ} (H : a * c = b), (a | b) := @algebra.dvd.intro _ _
  theorem dvd.intro_left : ∀{a b c : ℤ} (H : c * a = b), (a | b) := @algebra.dvd.intro_left _ _
  theorem exists_eq_mul_right_of_dvd : ∀{a b : ℤ} (H : (a | b)), ∃c, b = a * c :=
    @algebra.exists_eq_mul_right_of_dvd _ _
  theorem dvd.elim : ∀{P : Prop} {a b : ℤ} (H₁ : (a | b)) (H₂ : ∀c, b = a * c → P), P :=
    @algebra.dvd.elim _ _
  theorem exists_eq_mul_left_of_dvd : ∀{a b : ℤ} (H : (a | b)), ∃c, b = c * a :=
    @algebra.exists_eq_mul_left_of_dvd _ _
  theorem dvd.elim_left : ∀{P : Prop} {a b : ℤ} (H₁ : (a | b)) (H₂ : ∀c, b = c * a → P), P :=
    @algebra.dvd.elim_left _ _
  theorem dvd.refl : ∀a : ℤ, (a | a) := algebra.dvd.refl
  theorem dvd.trans : ∀{a b c : ℤ} (H₁ : (a | b)) (H₂ : (b | c)), (a | c) := @algebra.dvd.trans _ _
  theorem eq_zero_of_zero_dvd : ∀{a : ℤ} (H : (0 | a)), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
  theorem dvd_zero : ∀a : ℤ, (a | 0) := algebra.dvd_zero
  theorem one_dvd : ∀a : ℤ, (1 | a) := algebra.one_dvd
  theorem dvd_mul_right : ∀a b : ℤ, (a | a * b) := algebra.dvd_mul_right
  theorem dvd_mul_left : ∀a b : ℤ, (a | b * a) := algebra.dvd_mul_left
  theorem dvd_mul_of_dvd_left : ∀{a b : ℤ} (H : (a | b)) (c : ℤ), (a | b * c) :=
    @algebra.dvd_mul_of_dvd_left _ _
  theorem dvd_mul_of_dvd_right : ∀{a b : ℤ} (H : (a | b)) (c : ℤ), (a | c * b) :=
    @algebra.dvd_mul_of_dvd_right _ _
  theorem mul_dvd_mul : ∀{a b c d : ℤ}, (a | b) → (c | d) → (a * c | b * d) :=
    @algebra.mul_dvd_mul _ _
  theorem dvd_of_mul_right_dvd : ∀{a b c : ℤ}, (a * b | c) → (a | c) :=
    @algebra.dvd_of_mul_right_dvd _ _
  theorem dvd_of_mul_left_dvd : ∀{a b c : ℤ}, (a * b | c) → (b | c) :=
    @algebra.dvd_of_mul_left_dvd _ _
  theorem dvd_add : ∀{a b c : ℤ}, (a | b) → (a | c) → (a | b + c) := @algebra.dvd_add _ _
  theorem zero_mul : ∀a : ℤ, 0 * a = 0 := algebra.zero_mul
  theorem mul_zero : ∀a : ℤ, a * 0 = 0 := algebra.mul_zero
  theorem neg_mul_eq_neg_mul : ∀a b : ℤ, -(a * b) = -a * b := algebra.neg_mul_eq_neg_mul
  theorem neg_mul_eq_mul_neg : ∀a b : ℤ, -(a * b) = a * -b := algebra.neg_mul_eq_mul_neg
  theorem neg_mul_neg : ∀a b : ℤ, -a * -b = a * b := algebra.neg_mul_neg
  theorem neg_mul_comm : ∀a b : ℤ, -a * b = a * -b := algebra.neg_mul_comm
  theorem neg_eq_neg_one_mul : ∀a : ℤ, -a = -1 * a := algebra.neg_eq_neg_one_mul
  theorem mul_sub_left_distrib : ∀a b c : ℤ, a * (b - c) = a * b - a * c :=
    algebra.mul_sub_left_distrib
  theorem mul_sub_right_distrib : ∀a b c : ℤ, (a - b) * c = a * c - b * c :=
    algebra.mul_sub_right_distrib
  theorem mul_add_eq_mul_add_iff_sub_mul_add_eq :
      ∀a b c d e : ℤ, a * e + c = b * e + d ↔ (a - b) * e + c = d :=
    algebra.mul_add_eq_mul_add_iff_sub_mul_add_eq
  theorem mul_self_sub_mul_self_eq : ∀a b : ℤ, a * a - b * b = (a + b) * (a - b) :=
    algebra.mul_self_sub_mul_self_eq
  theorem mul_self_sub_one_eq : ∀a : ℤ, a * a - 1 = (a + 1) * (a - 1) :=
    algebra.mul_self_sub_one_eq
  theorem dvd_neg_iff_dvd : ∀a b : ℤ, (a | -b) ↔ (a | b) := algebra.dvd_neg_iff_dvd
  theorem neg_dvd_iff_dvd : ∀a b : ℤ, (-a | b) ↔ (a | b) := algebra.neg_dvd_iff_dvd
  theorem dvd_sub : ∀a b c : ℤ, (a | b) → (a | c) → (a | b - c) := algebra.dvd_sub
  theorem mul_ne_zero : ∀{a b : ℤ}, a ≠ 0 → b ≠ 0 → a * b ≠ 0 := @algebra.mul_ne_zero _ _
  theorem mul.cancel_right : ∀{a b c : ℤ}, a ≠ 0 → b * a = c * a → b = c :=
    @algebra.mul.cancel_right _ _
  theorem mul.cancel_left : ∀{a b c : ℤ}, a ≠ 0 → a * b = a * c → b = c :=
    @algebra.mul.cancel_left _ _
  theorem mul_self_eq_mul_self_iff : ∀a b : ℤ, a * a = b * b ↔ a = b ∨ a = -b :=
    algebra.mul_self_eq_mul_self_iff
  theorem mul_self_eq_one_iff : ∀a : ℤ, a * a = 1 ↔ a = 1 ∨ a = -1 :=
    algebra.mul_self_eq_one_iff
  theorem dvd_of_mul_dvd_mul_left : ∀{a b c : ℤ}, a ≠ 0 → (a * b | a * c) → (b | c) :=
    @algebra.dvd_of_mul_dvd_mul_left _ _
  theorem dvd_of_mul_dvd_mul_right : ∀{a b c : ℤ}, a ≠ 0 → (b * a | c * a) → (b | c) :=
    @algebra.dvd_of_mul_dvd_mul_right _ _
end port_algebra

/- additional properties -/

theorem of_nat_sub_of_nat {m n : ℕ} (H : #nat m ≥ n) : of_nat m - of_nat n = of_nat (#nat m - n) :=
have H1 : m = (#nat m - n + n), from (nat.sub_add_cancel H)⁻¹,
have H2 : m = (#nat m - n) + n, from congr_arg of_nat H1,
sub_eq_of_eq_add H2

theorem neg_succ_of_nat_eq' (m : ℕ) : -[m +1] = -m - 1 :=
by rewrite [neg_succ_of_nat_eq, -of_nat_add_of_nat, neg_add]

end int