feat(hott/init/nat): add basic facts about natural numbers
This commit is contained in:
Leonardo de Moura
parent
d75ef7a07a
commit
0627ad2f56
1 changed files with 347 additions and 0 deletions
347
hott/init/nat.hlean
Normal file
347
hott/init/nat.hlean
Normal file
|
|
@ -0,0 +1,347 @@
|
||||||
|
/-
|
||||||
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||||||
|
Released under Apache 2.0 license as described in the file LICENSE.
|
||||||
|
Authors: Floris van Doorn, Leonardo de Moura
|
||||||
|
-/
|
||||||
|
prelude
|
||||||
|
import init.wf init.tactic init.hedberg init.util init.types.sum
|
||||||
|
|
||||||
|
open eq.ops decidable sum
|
||||||
|
|
||||||
|
namespace nat
|
||||||
|
open lift
|
||||||
|
notation `ℕ` := nat
|
||||||
|
|
||||||
|
inductive lt (a : nat) : nat → Type :=
|
||||||
|
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) : Type₀ := 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] : decidable_eq nat :=
|
||||||
|
λn m : nat,
|
||||||
|
have general : ∀n, decidable (n = m), from
|
||||||
|
rec_on m
|
||||||
|
(λ n, cases_on n
|
||||||
|
(inl rfl)
|
||||||
|
(λ m, inr (λ (e : succ m = zero), down (no_confusion e))))
|
||||||
|
(λ (m' : nat) (ih : ∀n, decidable (n = m')) (n : nat), cases_on n
|
||||||
|
(inr (λ h, down (no_confusion h)))
|
||||||
|
(λ (n' : nat),
|
||||||
|
decidable.rec_on (ih n')
|
||||||
|
(assume Heq : n' = m', inl (eq.rec_on Heq rfl))
|
||||||
|
(assume Hne : n' ≠ m',
|
||||||
|
have H1 : succ n' ≠ succ m', from
|
||||||
|
assume Heq, down (no_confusion Heq (λ e : n' = m', Hne e)),
|
||||||
|
inr H1))),
|
||||||
|
general n
|
||||||
|
|
||||||
|
-- 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, down (no_confusion heq))
|
||||||
|
(λ b hlt heq, down (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),
|
||||||
|
down (no_confusion heq (λ (e : n = m),
|
||||||
|
eq.rec_on e ih)))
|
||||||
|
(λ b (hlt : m < b) (heq : succ n = succ b),
|
||||||
|
down (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 → Type₀ :=
|
||||||
|
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 → empty, from
|
||||||
|
λ b H, lt.cases_on H
|
||||||
|
(λ heq, down (no_confusion heq))
|
||||||
|
(λ b h₁ heq, down (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 lt.succ_of_lt {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 (lt.succ_of_lt 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) : sum (a = b) (a < b) :=
|
||||||
|
have aux : Π (a₁ b₁ : nat) (hlt : a₁ < b₁), a₁ = a → b₁ = (succ b) → sum (a = b) (a < b), from
|
||||||
|
λ a₁ b₁ hlt, lt.rec_on hlt
|
||||||
|
(λ h₁, eq.rec_on h₁ (λ h₂, down (no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inl rfl)))))
|
||||||
|
(λ b₁ hlt ih h₁, eq.rec_on h₁ (λ h₂, down (no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inr hlt))))),
|
||||||
|
aux a (succ b) H rfl rfl
|
||||||
|
|
||||||
|
definition le.of_eq_or_lt {a b : nat} (H : sum (a = b) (a < b)) : a ≤ b :=
|
||||||
|
sum.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_iff_equiv _ (iff.intro le.of_eq_or_lt eq_or_lt_of_le)
|
||||||
|
|
||||||
|
definition le.rec_on {a : nat} {P : nat → Type} {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
|
||||||
|
(sum.inr (sum.inl rfl))
|
||||||
|
(λ a₁, sum.inr (sum.inr !zero_lt_succ)))
|
||||||
|
(λ b₁ (ih : ∀a, a < b₁ ⊎ a = b₁ ⊎ b₁ < a) (a : nat), cases_on a
|
||||||
|
(sum.inl !zero_lt_succ)
|
||||||
|
(λ a, sum.rec_on (ih a)
|
||||||
|
(λ h : a < b₁, sum.inl (lt.succ_of_lt h))
|
||||||
|
(λ h, sum.rec_on h
|
||||||
|
(λ h : a = b₁, sum.inr (sum.inl (eq.rec_on h rfl)))
|
||||||
|
(λ h : b₁ < a, sum.inr (sum.inr (lt.succ_of_lt h))))))
|
||||||
|
a
|
||||||
|
|
||||||
|
definition eq_or_lt_of_not_lt {a b : nat} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
|
||||||
|
sum.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.step {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 :=
|
||||||
|
lt.succ_of_lt 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, sum.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 :=
|
||||||
|
rec_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 :=
|
||||||
|
rec_on a
|
||||||
|
rfl
|
||||||
|
(λ a₁ (ih : zero - a₁ = zero), calc
|
||||||
|
zero - succ a₁ = pred (zero - a₁) : rfl
|
||||||
|
... = pred zero : ih
|
||||||
|
... = zero : rfl)
|
||||||
|
|
||||||
|
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 :=
|
||||||
|
rec_on b
|
||||||
|
(le.refl a)
|
||||||
|
(λ b₁ ih, le.trans !pred_le ih)
|
||||||
|
|
||||||
|
definition of_num [coercion] [reducible] (n : num) : ℕ :=
|
||||||
|
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
|
||||||
Loading…
Reference in a new issue