lean2/library/init/nat.lean

337 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 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: init.nat
Authors: Floris van Doorn, Leonardo de Moura
-/
prelude
import init.wf init.tactic
open eq.ops decidable
namespace nat
notation `` := nat
inductive lt (a : nat) : nat → Prop :=
base : lt a (succ a),
step : Π {b}, lt a b → lt a (succ b)
notation a < b := lt a b
definition le (a b : nat) : Prop := a < succ b
notation a ≤ b := le a b
definition pred (a : nat) : nat :=
cases_on a zero (λ a₁, a₁)
protected definition is_inhabited [instance] : inhabited nat :=
inhabited.mk zero
protected definition has_decidable_eq [instance] : ∀ x y : nat, decidable (x = y),
has_decidable_eq zero zero := inl rfl,
has_decidable_eq (succ x) zero := inr (λ h, nat.no_confusion h),
has_decidable_eq zero (succ y) := inr (λ h, nat.no_confusion h),
has_decidable_eq (succ x) (succ y) :=
if H : x = y
then inl (eq.rec_on H rfl)
else inr (λ h : succ x = succ y, nat.no_confusion h (λ heq : x = y, absurd heq H))
-- less-than is well-founded
definition lt.wf [instance] : well_founded lt :=
well_founded.intro (λn, rec_on n
(acc.intro zero (λ (y : nat) (hlt : y < zero),
have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
λ n₁ hlt, lt.cases_on hlt
(λ heq, no_confusion heq)
(λ b hlt heq, no_confusion heq),
aux hlt rfl))
(λ (n : nat) (ih : acc lt n),
acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
λ n₁ hlt, lt.cases_on hlt
(λ (heq : succ n = succ m),
nat.no_confusion heq (λ (e : n = m),
eq.rec_on e ih))
(λ b (hlt : m < b) (heq : succ n = succ b),
nat.no_confusion heq (λ (e : n = b),
acc.inv (eq.rec_on e ih) hlt)),
aux hlt rfl)))
definition measure {A : Type} (f : A → nat) : A → A → Prop :=
inv_image lt f
definition measure.wf {A : Type} (f : A → nat) : well_founded (measure f) :=
inv_image.wf f lt.wf
definition not_lt_zero (a : nat) : ¬ a < zero :=
have aux : ∀ {b}, a < b → b = zero → false, from
λ b H, lt.cases_on H
(λ heq, nat.no_confusion heq)
(λ b h₁ heq, nat.no_confusion heq),
λ H, aux H rfl
definition zero_lt_succ (a : nat) : zero < succ a :=
rec_on a
(lt.base zero)
(λ a (hlt : zero < succ a), lt.step hlt)
definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
have aux : ∀ {d}, d < c → b = d → a < b → a < c, from
(λ d H, lt.rec_on H
(λ h₁ h₂, lt.step (eq.rec_on h₁ h₂))
(λ b hl ih h₁ h₂, lt.step (ih h₁ h₂))),
aux H₂ rfl H₁
definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
lt.rec_on H
(lt.base (succ a))
(λ b hlt ih, lt.trans ih (lt.base (succ b)))
definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
have aux : ∀ {a₁}, a₁ < b → succ a = a₁ → a < b, from
λ a₁ H, lt.rec_on H
(λ e₁, eq.rec_on e₁ (lt.step (lt.base a)))
(λ d hlt ih e₁, lt.step (ih e₁)),
aux H rfl
definition lt_of_succ_lt_succ {a b : nat} (H : succ a < succ b) : a < b :=
have aux : pred (succ a) < pred (succ b), from
lt.rec_on H
(lt.base a)
(λ (b : nat) (hlt : succ a < b) ih,
show pred (succ a) < pred (succ b), from
lt_of_succ_lt hlt),
aux
definition lt.is_decidable_rel [instance] : decidable_rel lt :=
λ a b, rec_on b
(λ (a : nat), inr (not_lt_zero a))
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), cases_on a
(inl !zero_lt_succ)
(λ a, decidable.rec_on (ih a)
(λ h_pos : a < b₁, inl (succ_lt_succ h_pos))
(λ h_neg : ¬ a < b₁,
have aux : ¬ succ a < succ b₁, from
λ h : succ a < succ b₁, h_neg (lt_of_succ_lt_succ h),
inr aux)))
a
definition le.refl (a : nat) : a ≤ a :=
lt.base a
definition le_of_lt {a b : nat} (H : a < b) : a ≤ b :=
lt.step H
definition eq_or_lt_of_le {a b : nat} (H : a ≤ b) : a = b a < b :=
begin
cases H with (b, hlt),
apply (or.inl rfl),
apply (or.inr hlt)
end
definition le_of_eq_or_lt {a b : nat} (H : a = b a < b) : a ≤ b :=
or.rec_on H
(λ hl, eq.rec_on hl !le.refl)
(λ hr, le_of_lt hr)
definition le.is_decidable_rel [instance] : decidable_rel le :=
λ a b, decidable_of_decidable_of_iff _ (iff.intro le_of_eq_or_lt eq_or_lt_of_le)
definition le.rec_on {a : nat} {P : nat → Prop} {b : nat} (H : a ≤ b) (H₁ : P a) (H₂ : ∀ b, a < b → P b) : P b :=
begin
cases H with (b', hlt),
apply H₁,
apply (H₂ b' hlt)
end
definition lt.irrefl (a : nat) : ¬ a < a :=
rec_on a
!not_lt_zero
(λ (a : nat) (ih : ¬ a < a) (h : succ a < succ a),
ih (lt_of_succ_lt_succ h))
definition lt.asymm {a b : nat} (H : a < b) : ¬ b < a :=
lt.rec_on H
(λ h : succ a < a, !lt.irrefl (lt_of_succ_lt h))
(λ b hlt (ih : ¬ b < a) (h : succ b < a), ih (lt_of_succ_lt h))
definition lt.trichotomy (a b : nat) : a < b a = b b < a :=
rec_on b
(λa, cases_on a
(or.inr (or.inl rfl))
(λ a₁, or.inr (or.inr !zero_lt_succ)))
(λ b₁ (ih : ∀a, a < b₁ a = b₁ b₁ < a) (a : nat), cases_on a
(or.inl !zero_lt_succ)
(λ a, or.rec_on (ih a)
(λ h : a < b₁, or.inl (succ_lt_succ h))
(λ h, or.rec_on h
(λ h : a = b₁, or.inr (or.inl (eq.rec_on h rfl)))
(λ h : b₁ < a, or.inr (or.inr (succ_lt_succ h))))))
a
definition eq_or_lt_of_not_lt {a b : nat} (hnlt : ¬ a < b) : a = b b < a :=
or.rec_on (lt.trichotomy a b)
(λ hlt, absurd hlt hnlt)
(λ h, h)
definition lt_succ_of_le {a b : nat} (h : a ≤ b) : a < succ b :=
h
definition lt_of_succ_le {a b : nat} (h : succ a ≤ b) : a < b :=
lt_of_succ_lt_succ h
definition le_succ_of_le {a b : nat} (h : a ≤ b) : a ≤ succ b :=
lt.step h
definition succ_le_of_lt {a b : nat} (h : a < b) : succ a ≤ b :=
succ_lt_succ h
definition le.trans {a b c : nat} (h₁ : a ≤ b) (h₂ : b ≤ c) : a ≤ c :=
begin
cases h₁ with (b', hlt),
apply h₂,
apply (lt.trans hlt h₂)
end
definition lt_of_le_of_lt {a b c : nat} (h₁ : a ≤ b) (h₂ : b < c) : a < c :=
begin
cases h₁ with (b', hlt),
apply h₂,
apply (lt.trans hlt h₂)
end
definition lt_of_lt_of_le {a b c : nat} (h₁ : a < b) (h₂ : b ≤ c) : a < c :=
begin
cases h₁ with (b', hlt),
apply (lt_of_succ_lt_succ h₂),
apply (lt.trans hlt (lt_of_succ_lt_succ h₂))
end
definition lt_of_lt_of_eq {a b c : nat} (h₁ : a < b) (h₂ : b = c) : a < c :=
eq.rec_on h₂ h₁
definition le_of_le_of_eq {a b c : nat} (h₁ : a ≤ b) (h₂ : b = c) : a ≤ c :=
eq.rec_on h₂ h₁
definition lt_of_eq_of_lt {a b c : nat} (h₁ : a = b) (h₂ : b < c) : a < c :=
eq.rec_on (eq.rec_on h₁ rfl) h₂
definition le_of_eq_of_le {a b c : nat} (h₁ : a = b) (h₂ : b ≤ c) : a ≤ c :=
eq.rec_on (eq.rec_on h₁ rfl) h₂
calc_trans lt.trans
calc_trans lt_of_le_of_lt
calc_trans lt_of_lt_of_le
calc_trans lt_of_lt_of_eq
calc_trans lt_of_eq_of_lt
calc_trans le.trans
calc_trans le_of_le_of_eq
calc_trans le_of_eq_of_le
definition max (a b : nat) : nat :=
if a < b then b else a
definition min (a b : nat) : nat :=
if a < b then a else b
definition max_a_a (a : nat) : a = max a a :=
eq.rec_on !if_t_t rfl
definition max.eq_right {a b : nat} (H : a < b) : max a b = b :=
if_pos H
definition max.eq_left {a b : nat} (H : ¬ a < b) : max a b = a :=
if_neg H
definition max.right_eq {a b : nat} (H : a < b) : b = max a b :=
eq.rec_on (max.eq_right H) rfl
definition max.left_eq {a b : nat} (H : ¬ a < b) : a = max a b :=
eq.rec_on (max.eq_left H) rfl
definition max.left (a b : nat) : a ≤ max a b :=
by_cases
(λ h : a < b, le_of_lt (eq.rec_on (max.right_eq h) h))
(λ h : ¬ a < b, eq.rec_on (max.eq_left h) !le.refl)
definition max.right (a b : nat) : b ≤ max a b :=
by_cases
(λ h : a < b, eq.rec_on (max.eq_right h) !le.refl)
(λ h : ¬ a < b, or.rec_on (eq_or_lt_of_not_lt h)
(λ heq, eq.rec_on heq (eq.rec_on (max_a_a a) !le.refl))
(λ h : b < a,
have aux : a = max a b, from max.left_eq (lt.asymm h),
eq.rec_on aux (le_of_lt h)))
definition gt a b := lt b a
notation a > b := gt a b
definition ge a b := le b a
notation a ≥ b := ge a b
definition add (a b : nat) : nat :=
rec_on b a (λ b₁ r, succ r)
notation a + b := add a b
definition sub (a b : nat) : nat :=
rec_on b a (λ b₁ r, pred r)
notation a - b := sub a b
definition mul (a b : nat) : nat :=
rec_on b zero (λ b₁ r, r + a)
notation a * b := mul a b
definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b :=
induction_on b
rfl
(λ b₁ (ih : succ a - succ b₁ = a - b₁),
eq.rec_on ih (eq.refl (pred (succ a - succ b₁))))
definition sub_eq_succ_sub_succ (a b : nat) : a - b = succ a - succ b :=
eq.rec_on (succ_sub_succ_eq_sub a b) rfl
definition zero_sub_eq_zero (a : nat) : zero - a = zero :=
induction_on a
rfl
(λ a₁ (ih : zero - a₁ = zero),
eq.rec_on ih (eq.refl (pred (zero - a₁))))
definition zero_eq_zero_sub (a : nat) : zero = zero - a :=
eq.rec_on (zero_sub_eq_zero a) rfl
definition sub_lt {a b : nat} : zero < a → zero < b → a - b < a :=
have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
λa h₁, lt.rec_on h₁
(λb h₂, lt.cases_on h₂
(lt.base zero)
(λ b₁ bpos,
eq.rec_on (sub_eq_succ_sub_succ zero b₁)
(eq.rec_on (zero_eq_zero_sub b₁) (lt.base zero))))
(λa₁ apos ih b h₂, lt.cases_on h₂
(lt.base a₁)
(λ b₁ bpos,
eq.rec_on (sub_eq_succ_sub_succ a₁ b₁)
(lt.trans (@ih b₁ bpos) (lt.base a₁)))),
λ h₁ h₂, aux h₁ h₂
definition pred_le (a : nat) : pred a ≤ a :=
cases_on a
(le.refl zero)
(λ a₁, le_of_lt (lt.base a₁))
definition sub_le (a b : nat) : a - b ≤ a :=
induction_on b
(le.refl a)
(λ b₁ ih, le.trans !pred_le ih)
definition of_num [coercion] [reducible] (n : num) : :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
end nat