350 lines
12 KiB
Text
350 lines
12 KiB
Text
/-
|
||
Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: data.nat.basic
|
||
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
|
||
|
||
Basic operations on the natural numbers.
|
||
-/
|
||
import logic.connectives data.num algebra.binary algebra.ring
|
||
open binary eq.ops
|
||
|
||
namespace nat
|
||
|
||
/- a variant of add, defined by recursion on the first argument -/
|
||
|
||
definition addl (x y : ℕ) : ℕ :=
|
||
nat.rec y (λ n r, succ r) x
|
||
infix `⊕`:65 := addl
|
||
|
||
theorem addl.succ_right (n m : ℕ) : n ⊕ succ m = succ (n ⊕ m) :=
|
||
nat.induction_on n
|
||
rfl
|
||
(λ n₁ ih, calc
|
||
succ n₁ ⊕ succ m = succ (n₁ ⊕ succ m) : rfl
|
||
... = succ (succ (n₁ ⊕ m)) : ih
|
||
... = succ (succ n₁ ⊕ m) : rfl)
|
||
|
||
theorem add_eq_addl (x : ℕ) : ∀y, x + y = x ⊕ y :=
|
||
nat.induction_on x
|
||
(λ y, nat.induction_on y
|
||
rfl
|
||
(λ y₁ ih, calc
|
||
zero + succ y₁ = succ (zero + y₁) : rfl
|
||
... = succ (zero ⊕ y₁) : {ih}
|
||
... = zero ⊕ (succ y₁) : rfl))
|
||
(λ x₁ ih₁ y, nat.induction_on y
|
||
(calc
|
||
succ x₁ + zero = succ (x₁ + zero) : rfl
|
||
... = succ (x₁ ⊕ zero) : {ih₁ zero}
|
||
... = succ x₁ ⊕ zero : rfl)
|
||
(λ y₁ ih₂, calc
|
||
succ x₁ + succ y₁ = succ (succ x₁ + y₁) : rfl
|
||
... = succ (succ x₁ ⊕ y₁) : {ih₂}
|
||
... = succ x₁ ⊕ succ y₁ : addl.succ_right))
|
||
|
||
/- successor and predecessor -/
|
||
|
||
theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
|
||
assume H, no_confusion H
|
||
|
||
-- add_rewrite succ_ne_zero
|
||
|
||
theorem pred_zero : pred 0 = 0 :=
|
||
rfl
|
||
|
||
theorem pred_succ (n : ℕ) : pred (succ n) = n :=
|
||
rfl
|
||
|
||
theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
|
||
induction_on n
|
||
(or.inl rfl)
|
||
(take m IH, or.inr
|
||
(show succ m = succ (pred (succ m)), from congr_arg succ !pred_succ⁻¹))
|
||
|
||
theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃k : ℕ, n = succ k :=
|
||
exists.intro _ (or_resolve_right !eq_zero_or_eq_succ_pred H)
|
||
|
||
theorem succ.inj {n m : ℕ} (H : succ n = succ m) : n = m :=
|
||
no_confusion H (λe, e)
|
||
|
||
theorem succ.ne_self {n : ℕ} : succ n ≠ n :=
|
||
induction_on n
|
||
(take H : 1 = 0,
|
||
have ne : 1 ≠ 0, from !succ_ne_zero,
|
||
absurd H ne)
|
||
(take k IH H, IH (succ.inj H))
|
||
|
||
theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
|
||
have H : n = n → B, from cases_on n H1 H2,
|
||
H rfl
|
||
|
||
theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
||
(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
|
||
have stronger : P a ∧ P (succ a), from
|
||
induction_on a
|
||
(and.intro H1 H2)
|
||
(take k IH,
|
||
have IH1 : P k, from and.elim_left IH,
|
||
have IH2 : P (succ k), from and.elim_right IH,
|
||
and.intro IH2 (H3 k IH1 IH2)),
|
||
and.elim_left stronger
|
||
|
||
theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
|
||
(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m :=
|
||
have general : ∀m, P n m, from induction_on n
|
||
(take m : ℕ, H1 m)
|
||
(take k : ℕ,
|
||
assume IH : ∀m, P k m,
|
||
take m : ℕ,
|
||
cases_on m (H2 k) (take l, (H3 k l (IH l)))),
|
||
general m
|
||
|
||
/- addition -/
|
||
|
||
theorem add_zero (n : ℕ) : n + 0 = n :=
|
||
rfl
|
||
|
||
theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
|
||
rfl
|
||
|
||
theorem zero_add (n : ℕ) : 0 + n = n :=
|
||
induction_on n
|
||
!add_zero
|
||
(take m IH, show 0 + succ m = succ m, from
|
||
calc
|
||
0 + succ m = succ (0 + m) : add_succ
|
||
... = succ m : IH)
|
||
|
||
theorem add.succ_left (n m : ℕ) : (succ n) + m = succ (n + m) :=
|
||
induction_on m
|
||
(!add_zero ▸ !add_zero)
|
||
(take k IH, calc
|
||
succ n + succ k = succ (succ n + k) : add_succ
|
||
... = succ (succ (n + k)) : IH
|
||
... = succ (n + succ k) : add_succ)
|
||
|
||
theorem add.comm (n m : ℕ) : n + m = m + n :=
|
||
induction_on m
|
||
(!add_zero ⬝ !zero_add⁻¹)
|
||
(take k IH, calc
|
||
n + succ k = succ (n+k) : add_succ
|
||
... = succ (k + n) : IH
|
||
... = succ k + n : add.succ_left)
|
||
|
||
theorem succ_add_eq_add_succ (n m : ℕ) : succ n + m = n + succ m :=
|
||
!add.succ_left ⬝ !add_succ⁻¹
|
||
|
||
theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
|
||
induction_on k
|
||
(!add_zero ▸ !add_zero)
|
||
(take l IH,
|
||
calc
|
||
(n + m) + succ l = succ ((n + m) + l) : add_succ
|
||
... = succ (n + (m + l)) : IH
|
||
... = n + succ (m + l) : add_succ
|
||
... = n + (m + succ l) : add_succ)
|
||
|
||
theorem add.left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) :=
|
||
left_comm add.comm add.assoc n m k
|
||
|
||
theorem add.right_comm (n m k : ℕ) : n + m + k = n + k + m :=
|
||
right_comm add.comm add.assoc n m k
|
||
|
||
theorem add.cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
|
||
induction_on n
|
||
(take H : 0 + m = 0 + k,
|
||
!zero_add⁻¹ ⬝ H ⬝ !zero_add)
|
||
(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
|
||
have H2 : succ (n + m) = succ (n + k),
|
||
from calc
|
||
succ (n + m) = succ n + m : add.succ_left
|
||
... = succ n + k : H
|
||
... = succ (n + k) : add.succ_left,
|
||
have H3 : n + m = n + k, from succ.inj H2,
|
||
IH H3)
|
||
|
||
theorem add.cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k :=
|
||
have H2 : m + n = m + k, from !add.comm ⬝ H ⬝ !add.comm,
|
||
add.cancel_left H2
|
||
|
||
theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0 :=
|
||
induction_on n
|
||
(take (H : 0 + m = 0), rfl)
|
||
(take k IH,
|
||
assume H : succ k + m = 0,
|
||
absurd
|
||
(show succ (k + m) = 0, from calc
|
||
succ (k + m) = succ k + m : add.succ_left
|
||
... = 0 : H)
|
||
!succ_ne_zero)
|
||
|
||
theorem eq_zero_of_add_eq_zero_left {n m : ℕ} (H : n + m = 0) : m = 0 :=
|
||
eq_zero_of_add_eq_zero_right (!add.comm ⬝ H)
|
||
|
||
theorem add.eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
|
||
and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)
|
||
|
||
theorem add_one (n : ℕ) : n + 1 = succ n :=
|
||
!add_zero ▸ !add_succ
|
||
|
||
theorem one_add (n : ℕ) : 1 + n = succ n :=
|
||
!zero_add ▸ !add.succ_left
|
||
|
||
/- multiplication -/
|
||
|
||
theorem mul_zero (n : ℕ) : n * 0 = 0 :=
|
||
rfl
|
||
|
||
theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
|
||
rfl
|
||
|
||
-- commutativity, distributivity, associativity, identity
|
||
|
||
theorem zero_mul (n : ℕ) : 0 * n = 0 :=
|
||
induction_on n
|
||
!mul_zero
|
||
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
|
||
|
||
theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
|
||
induction_on m
|
||
(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
|
||
(take k IH, calc
|
||
succ n * succ k = succ n * k + succ n : mul_succ
|
||
... = n * k + k + succ n : IH
|
||
... = n * k + (k + succ n) : add.assoc
|
||
... = n * k + (succ n + k) : add.comm
|
||
... = n * k + (n + succ k) : succ_add_eq_add_succ
|
||
... = n * k + n + succ k : add.assoc
|
||
... = n * succ k + succ k : mul_succ)
|
||
|
||
theorem mul.comm (n m : ℕ) : n * m = m * n :=
|
||
induction_on m
|
||
(!mul_zero ⬝ !zero_mul⁻¹)
|
||
(take k IH, calc
|
||
n * succ k = n * k + n : mul_succ
|
||
... = k * n + n : IH
|
||
... = (succ k) * n : succ_mul)
|
||
|
||
theorem mul.right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
|
||
induction_on k
|
||
(calc
|
||
(n + m) * 0 = 0 : mul_zero
|
||
... = 0 + 0 : add_zero
|
||
... = n * 0 + 0 : mul_zero
|
||
... = n * 0 + m * 0 : mul_zero)
|
||
(take l IH, calc
|
||
(n + m) * succ l = (n + m) * l + (n + m) : mul_succ
|
||
... = n * l + m * l + (n + m) : IH
|
||
... = n * l + m * l + n + m : add.assoc
|
||
... = n * l + n + m * l + m : add.right_comm
|
||
... = n * l + n + (m * l + m) : add.assoc
|
||
... = n * succ l + (m * l + m) : mul_succ
|
||
... = n * succ l + m * succ l : mul_succ)
|
||
|
||
theorem mul.left_distrib (n m k : ℕ) : n * (m + k) = n * m + n * k :=
|
||
calc
|
||
n * (m + k) = (m + k) * n : mul.comm
|
||
... = m * n + k * n : mul.right_distrib
|
||
... = n * m + k * n : mul.comm
|
||
... = n * m + n * k : mul.comm
|
||
|
||
theorem mul.assoc (n m k : ℕ) : (n * m) * k = n * (m * k) :=
|
||
induction_on k
|
||
(calc
|
||
(n * m) * 0 = n * (m * 0) : mul_zero)
|
||
(take l IH,
|
||
calc
|
||
(n * m) * succ l = (n * m) * l + n * m : mul_succ
|
||
... = n * (m * l) + n * m : IH
|
||
... = n * (m * l + m) : mul.left_distrib
|
||
... = n * (m * succ l) : mul_succ)
|
||
|
||
theorem mul_one (n : ℕ) : n * 1 = n :=
|
||
calc
|
||
n * 1 = n * 0 + n : mul_succ
|
||
... = 0 + n : mul_zero
|
||
... = n : zero_add
|
||
|
||
theorem one_mul (n : ℕ) : 1 * n = n :=
|
||
calc
|
||
1 * n = n * 1 : mul.comm
|
||
... = n : mul_one
|
||
|
||
theorem eq_zero_or_eq_zero_of_mul_eq_zero {n m : ℕ} : n * m = 0 → n = 0 ∨ m = 0 :=
|
||
cases_on n
|
||
(assume H, or.inl rfl)
|
||
(take n',
|
||
cases_on m
|
||
(assume H, or.inr rfl)
|
||
(take m',
|
||
assume H : succ n' * succ m' = 0,
|
||
absurd
|
||
((calc
|
||
0 = succ n' * succ m' : H
|
||
... = succ n' * m' + succ n' : mul_succ
|
||
... = succ (succ n' * m' + n') : add_succ)⁻¹)
|
||
!succ_ne_zero))
|
||
|
||
section
|
||
open [classes] algebra
|
||
|
||
protected definition comm_semiring [instance] [reducible] : algebra.comm_semiring nat :=
|
||
⦃algebra.comm_semiring,
|
||
add := add,
|
||
add_assoc := add.assoc,
|
||
zero := zero,
|
||
zero_add := zero_add,
|
||
add_zero := add_zero,
|
||
add_comm := add.comm,
|
||
mul := mul,
|
||
mul_assoc := mul.assoc,
|
||
one := succ zero,
|
||
one_mul := one_mul,
|
||
mul_one := mul_one,
|
||
left_distrib := mul.left_distrib,
|
||
right_distrib := mul.right_distrib,
|
||
zero_mul := zero_mul,
|
||
mul_zero := mul_zero,
|
||
zero_ne_one := ne.symm (succ_ne_zero zero),
|
||
mul_comm := mul.comm⦄
|
||
end
|
||
|
||
section port_algebra
|
||
theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
|
||
theorem mul.right_comm : ∀a b c : ℕ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
|
||
|
||
definition dvd (a b : ℕ) : Prop := algebra.dvd a b
|
||
infix `|` := dvd
|
||
theorem dvd.intro : ∀{a b c : ℕ} (H : a * c = b), a | b := @algebra.dvd.intro _ _
|
||
theorem dvd.intro_left : ∀{a b c : ℕ} (H : c * a = b), a | b := @algebra.dvd.intro_left _ _
|
||
theorem exists_eq_mul_right_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = a * c :=
|
||
@algebra.exists_eq_mul_right_of_dvd _ _
|
||
theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = a * c → P), P :=
|
||
@algebra.dvd.elim _ _
|
||
theorem exists_eq_mul_left_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = c * a :=
|
||
@algebra.exists_eq_mul_left_of_dvd _ _
|
||
theorem dvd.elim_left : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = c * a → P), P :=
|
||
@algebra.dvd.elim_left _ _
|
||
theorem dvd.refl : ∀a : ℕ, a | a := algebra.dvd.refl
|
||
theorem dvd.trans : ∀{a b c : ℕ} (H₁ : a | b) (H₂ : b | c), a | c := @algebra.dvd.trans _ _
|
||
theorem eq_zero_of_zero_dvd : ∀{a : ℕ} (H : 0 | a), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
|
||
theorem dvd_zero : ∀a : ℕ, a | 0 := algebra.dvd_zero
|
||
theorem one_dvd : ∀a : ℕ, 1 | a := algebra.one_dvd
|
||
theorem dvd_mul_right : ∀a b : ℕ, a | a * b := algebra.dvd_mul_right
|
||
theorem dvd_mul_left : ∀a b : ℕ, a | b * a := algebra.dvd_mul_left
|
||
theorem dvd_mul_of_dvd_left : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | b * c :=
|
||
@algebra.dvd_mul_of_dvd_left _ _
|
||
theorem dvd_mul_of_dvd_right : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | c * b :=
|
||
@algebra.dvd_mul_of_dvd_right _ _
|
||
theorem mul_dvd_mul : ∀{a b c d : ℕ}, a | b → c | d → a * c | b * d :=
|
||
@algebra.mul_dvd_mul _ _
|
||
theorem dvd_of_mul_right_dvd : ∀{a b c : ℕ}, a * b | c → a | c :=
|
||
@algebra.dvd_of_mul_right_dvd _ _
|
||
theorem dvd_of_mul_left_dvd : ∀{a b c : ℕ}, a * b | c → b | c :=
|
||
@algebra.dvd_of_mul_left_dvd _ _
|
||
theorem dvd_add : ∀{a b c : ℕ}, a | b → a | c → a | b + c := @algebra.dvd_add _ _
|
||
end port_algebra
|
||
|
||
end nat
|