lean2/library/data/nat/basic.lean

318 lines
10 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
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 : ) : :=
nat.rec y (λ n r, succ r) x
infix `⊕`:65 := addl
theorem addl_succ_right (n m : ) : 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 : ) : ∀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 : ) : succ n ≠ 0 :=
by contradiction
-- add_rewrite succ_ne_zero
theorem pred_zero : pred 0 = 0 :=
rfl
theorem pred_succ (n : ) : pred (succ n) = n :=
rfl
theorem eq_zero_or_eq_succ_pred (n : ) : n = 0 n = succ (pred n) :=
nat.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 : } (H : n ≠ 0) : ∃k : , n = succ k :=
exists.intro _ (or_resolve_right !eq_zero_or_eq_succ_pred H)
theorem succ.inj {n m : } (H : succ n = succ m) : n = m :=
nat.no_confusion H (λe, e)
abbreviation eq_of_succ_eq_succ := @succ.inj
theorem succ_ne_self {n : } : succ n ≠ n :=
nat.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 : } (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
have H : n = n → B, from nat.cases_on n H1 H2,
H rfl
theorem two_step_induction_on {P : → Prop} (a : ) (H1 : P 0) (H2 : P 1)
(H3 : ∀ (n : ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
have stronger : P a ∧ P (succ a), from
nat.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 : → Prop} (n m : ) (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 nat.induction_on n
(take m : , H1 m)
(take k : ,
assume IH : ∀m, P k m,
take m : ,
nat.cases_on m (H2 k) (take l, (H3 k l (IH l)))),
general m
/- addition -/
theorem add_zero (n : ) : n + 0 = n :=
rfl
theorem add_succ (n m : ) : n + succ m = succ (n + m) :=
rfl
theorem zero_add (n : ) : 0 + n = n :=
nat.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 succ_add (n m : ) : (succ n) + m = succ (n + m) :=
nat.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 + m = m + n :=
nat.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 : succ_add)
theorem succ_add_eq_succ_add (n m : ) : succ n + m = n + succ m :=
!succ_add ⬝ !add_succ⁻¹
theorem add.assoc (n m k : ) : (n + m) + k = n + (m + k) :=
nat.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 + (m + k) = m + (n + k) :=
left_comm add.comm add.assoc n m k
theorem add.right_comm (n m k : ) : n + m + k = n + k + m :=
right_comm add.comm add.assoc n m k
theorem add.cancel_left {n m k : } : n + m = n + k → m = k :=
nat.induction_on n
(take H : 0 + m = 0 + k,
!zero_add⁻¹ ⬝ H ⬝ !zero_add)
(take (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 : succ_add
... = succ n + k : H
... = succ (n + k) : succ_add,
have H3 : n + m = n + k, from succ.inj H2,
IH H3)
theorem add.cancel_right {n m k : } (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 + m = 0 → n = 0 :=
nat.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 : succ_add
... = 0 : H)
!succ_ne_zero)
theorem eq_zero_of_add_eq_zero_left {n m : } (H : n + m = 0) : m = 0 :=
eq_zero_of_add_eq_zero_right (!add.comm ⬝ H)
theorem eq_zero_and_eq_zero_of_add_eq_zero {n m : } (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 + 1 = succ n :=
!add_zero ▸ !add_succ
theorem one_add (n : ) : 1 + n = succ n :=
!zero_add ▸ !succ_add
/- multiplication -/
theorem mul_zero (n : ) : n * 0 = 0 :=
rfl
theorem mul_succ (n m : ) : n * succ m = n * m + n :=
rfl
-- commutativity, distributivity, associativity, identity
theorem zero_mul (n : ) : 0 * n = 0 :=
nat.induction_on n
!mul_zero
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
theorem succ_mul (n m : ) : (succ n) * m = (n * m) + m :=
nat.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_succ_add
... = n * k + n + succ k : add.assoc
... = n * succ k + succ k : mul_succ)
theorem mul.comm (n m : ) : n * m = m * n :=
nat.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 + m) * k = n * k + m * k :=
nat.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 * (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 * m) * k = n * (m * k) :=
nat.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 * 1 = n :=
calc
n * 1 = n * 0 + n : mul_succ
... = 0 + n : mul_zero
... = n : zero_add
theorem one_mul (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 * m = 0 → n = 0 m = 0 :=
nat.cases_on n
(assume H, or.inl rfl)
(take n',
nat.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 migrate_algebra
open [classes] algebra
protected definition comm_semiring [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,
mul_comm := mul.comm⦄
local attribute nat.comm_semiring [instance]
definition dvd (a b : ) : Prop := algebra.dvd a b
notation a b := dvd a b
migrate from algebra with nat replacing dvd → dvd
end migrate_algebra
end nat