lean2/library/data/nat/basic.lean

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/-
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
2015-09-30 15:06:31 +00:00
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 [simp] : pred 0 = 0 :=
rfl
theorem pred_succ [simp] (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 imp.id
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 H1
(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 [simp] (n : ) : n + 0 = n :=
rfl
theorem add_succ [simp] (n m : ) : n + succ m = succ (n + m) :=
rfl
theorem zero_add [simp] (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 [simp] (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 [simp] (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 [simp] (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 [simp] : Π (n m k : ), n + (m + k) = m + (n + k) :=
left_comm add.comm add.assoc
theorem add.right_comm : Π (n m k : ), n + m + k = n + k + m :=
right_comm add.comm add.assoc
theorem add.comm4 : Π {n m k l : }, n + m + (k + l) = n + k + (m + l) :=
comm4 add.comm add.assoc
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 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 n + m = n + k, from succ.inj this,
IH this)
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 [simp] (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 [simp] (n : ) : n * 0 = 0 :=
rfl
theorem mul_succ [simp] (n m : ) : n * succ m = n * m + n :=
rfl
-- commutativity, distributivity, associativity, identity
theorem zero_mul [simp] (n : ) : 0 * n = 0 :=
nat.induction_on n
!mul_zero
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
theorem succ_mul [simp] (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 [simp] (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 [simp] (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 [simp] (n : ) : n * 1 = n :=
calc
n * 1 = n * 0 + n : mul_succ
... = 0 + n : mul_zero
... = n : zero_add
theorem one_mul [simp] (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
section
open nat
definition iterate {A : Type} (op : A → A) : → A → A
| 0 := λ a, a
| (succ k) := λ a, op (iterate k a)
notation f`^[`n`]` := iterate f n
end