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.
--- Author: Floris van Doorn
-- data.nat.basic
-- ==============
--
-- Basic operations on the natural numbers.
import logic data.num tools.tactic struc.binary tools.helper_tactics
using num tactic binary eq_ops
using decidable (hiding induction_on rec_on)
using relation -- for subst_iff
using helper_tactics
-- Definition of the type
-- ----------------------
namespace nat
inductive nat : Type :=
zero : nat,
succ : nat → nat
notation `` := nat
theorem nat_rec_zero {P : → Type} (x : P zero) (f : ∀m, P m → P (succ m)) : nat_rec x f zero = x
theorem nat_rec_succ {P : → Type} (x : P zero) (f : ∀m, P m → P (succ m)) (n : ) :
nat_rec x f (succ n) = f n (nat_rec x f n)
theorem induction_on {P : → Prop} (a : ) (H1 : P zero) (H2 : ∀ (n : ) (IH : P n), P (succ n)) :
P a :=
nat_rec H1 H2 a
definition rec_on {P : → Type} (n : ) (H1 : P zero) (H2 : ∀m, P m → P (succ m)) : P n :=
nat_rec H1 H2 n
-- Coercion from num
-- -----------------
abbreviation plus (x y : ) : :=
nat_rec x (λ n r, succ r) y
definition to_nat [coercion] [inline] (n : num) : :=
num_rec zero
(λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
-- Successor and predecessor
-- -------------------------
theorem succ_ne_zero (n : ) : succ n ≠ 0 :=
assume H : succ n = 0,
have H2 : true = false, from
let f := (nat_rec false (fun a b, true)) in
calc
true = f (succ n) : rfl
... = f 0 : {H}
... = false : rfl,
absurd H2 true_ne_false
-- add_rewrite succ_ne_zero
definition pred (n : ) := nat_rec 0 (fun m x, m) n
theorem pred_zero : pred 0 = 0
theorem pred_succ (n : ) : pred (succ n) = n
opaque_hint (hiding pred)
theorem zero_or_succ_pred (n : ) : n = 0 n = succ (pred n) :=
induction_on n
(or_inl (refl 0))
(take m IH, or_inr
(show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
theorem zero_or_exists_succ (n : ) : n = 0 ∃k, n = succ k :=
or_imp_or (zero_or_succ_pred n) (assume H, H)
(assume H : n = succ (pred n), exists_intro (pred n) H)
theorem case {P : → Prop} (n : ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
induction_on n H1 (take m IH, H2 m)
theorem discriminate {B : Prop} {n : } (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
or_elim (zero_or_succ_pred n)
(take H3 : n = 0, H1 H3)
(take H3 : n = succ (pred n), H2 (pred n) H3)
theorem succ_inj {n m : } (H : succ n = succ m) : n = m :=
calc
n = pred (succ n) : pred_succ n⁻¹
... = pred (succ m) : {H}
... = m : pred_succ m
theorem succ_ne_self (n : ) : succ n ≠ n :=
induction_on n
(take H : 1 = 0,
have ne : 1 ≠ 0, from succ_ne_zero 0,
absurd H ne)
(take k IH H, IH (succ_inj H))
theorem decidable_eq [instance] (n m : ) : decidable (n = m) :=
have general : ∀n, decidable (n = m), from
rec_on m
(take n,
rec_on n
(inl (refl 0))
(λ m iH, inr (succ_ne_zero m)))
(λ (m' : ) (iH1 : ∀n, decidable (n = m')),
take n, rec_on n
(inr (ne_symm (succ_ne_zero m')))
(λ (n' : ) (iH2 : decidable (n' = succ m')),
have d1 : decidable (n' = m'), from iH1 n',
decidable.rec_on d1
(assume Heq : n' = m', inl (congr_arg succ Heq))
(assume Hne : n' ≠ m',
have H1 : succ n' ≠ succ m', from
assume Heq, absurd (succ_inj Heq) Hne,
inr H1))),
general n
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
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 induction_on n
(take m : , H1 m)
(take k : ,
assume IH : ∀m, P k m,
take m : ,
discriminate
(assume Hm : m = 0,
Hm⁻¹ ▸ (H2 k))
(take l : ,
assume Hm : m = succ l,
Hm⁻¹ ▸ (H3 k l (IH l)))),
general m
-- Addition
-- --------
definition add (x y : ) : := plus x y
infixl `+` := add
theorem add_zero_right (n : ) : n + 0 = n
theorem add_succ_right (n m : ) : n + succ m = succ (n + m)
opaque_hint (hiding add)
theorem add_zero_left (n : ) : 0 + n = n :=
induction_on n
(add_zero_right 0)
(take m IH, show 0 + succ m = succ m, from
calc
0 + succ m = succ (0 + m) : add_succ_right _ _
... = succ m : {IH})
theorem add_succ_left (n m : ) : (succ n) + m = succ (n + m) :=
induction_on m
(calc
succ n + 0 = succ n : add_zero_right (succ n)
... = succ (n + 0) : {symm (add_zero_right n)})
(take k IH,
calc
succ n + succ k = succ (succ n + k) : add_succ_right _ _
... = succ (succ (n + k)) : {IH}
... = succ (n + succ k) : {symm (add_succ_right _ _)})
theorem add_comm (n m : ) : n + m = m + n :=
induction_on m
(trans (add_zero_right _) (symm (add_zero_left _)))
(take k IH,
calc
n + succ k = succ (n+k) : add_succ_right _ _
... = succ (k + n) : {IH}
... = succ k + n : symm (add_succ_left _ _))
theorem add_move_succ (n m : ) : succ n + m = n + succ m :=
calc
succ n + m = succ (n + m) : add_succ_left n m
... = n +succ m : symm (add_succ_right n m)
theorem add_comm_succ (n m : ) : n + succ m = m + succ n :=
calc
n + succ m = succ n + m : symm (add_move_succ n m)
... = m + succ n : add_comm (succ n) m
theorem add_assoc (n m k : ) : (n + m) + k = n + (m + k) :=
induction_on k
(calc
(n + m) + 0 = n + m : add_zero_right _
... = n + (m + 0) : {symm (add_zero_right m)})
(take l IH,
calc
(n + m) + succ l = succ ((n + m) + l) : add_succ_right _ _
... = succ (n + (m + l)) : {IH}
... = n + succ (m + l) : symm (add_succ_right _ _)
... = n + (m + succ l) : {symm (add_succ_right _ _)})
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
-- add_rewrite add_zero_left add_zero_right
-- add_rewrite add_succ_left add_succ_right
-- add_rewrite add_comm add_assoc add_left_comm
-- ### cancelation
theorem add_cancel_left {n m k : } : n + m = n + k → m = k :=
induction_on n
(take H : 0 + m = 0 + k,
calc
m = 0 + m : symm (add_zero_left m)
... = 0 + k : H
... = k : add_zero_left k)
(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 : symm (add_succ_left n m)
... = succ n + k : H
... = succ (n + k) : add_succ_left n k,
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 calc
m + n = n + m : add_comm m n
... = k + m : H
... = m + k : add_comm k m,
add_cancel_left H2
theorem add_eq_zero_left {n m : } : n + m = 0 → n = 0 :=
induction_on n
(take (H : 0 + m = 0), refl 0)
(take k IH,
assume (H : succ k + m = 0),
absurd_elim (succ k = 0)
(show succ (k + m) = 0, from
calc
succ (k + m) = succ k + m : symm (add_succ_left k m)
... = 0 : H)
(succ_ne_zero (k + m)))
theorem add_eq_zero_right {n m : } (H : n + m = 0) : m = 0 :=
add_eq_zero_left (trans (add_comm m n) H)
theorem add_eq_zero {n m : } (H : n + m = 0) : n = 0 ∧ m = 0 :=
and_intro (add_eq_zero_left H) (add_eq_zero_right H)
-- ### misc
theorem add_one (n : ) : n + 1 = succ n :=
calc
n + 1 = succ (n + 0) : add_succ_right _ _
... = succ n : {add_zero_right _}
theorem add_one_left (n : ) : 1 + n = succ n :=
calc
1 + n = succ (0 + n) : add_succ_left _ _
... = succ n : {add_zero_left _}
-- TODO: rename? remove?
theorem induction_plus_one {P : nat → Prop} (a : ) (H1 : P 0)
(H2 : ∀ (n : ) (IH : P n), P (n + 1)) : P a :=
nat_rec H1 (take n IH, (add_one n) ▸ (H2 n IH)) a
-- Multiplication
-- --------------
definition mul (n m : ) := nat_rec 0 (fun m x, x + n) m
infixl `*` := mul
theorem mul_zero_right (n:) : n * 0 = 0
theorem mul_succ_right (n m:) : n * succ m = n * m + n
opaque_hint (hiding mul)
-- ### commutativity, distributivity, associativity, identity
theorem mul_zero_left (n:) : 0 * n = 0 :=
induction_on n
(mul_zero_right 0)
(take m IH,
calc
0 * succ m = 0 * m + 0 : mul_succ_right _ _
... = 0 * m : add_zero_right _
... = 0 : IH)
theorem mul_succ_left (n m:) : (succ n) * m = (n * m) + m :=
induction_on m
(calc
succ n * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * 0 + 0 : symm (add_zero_right _))
(take k IH,
calc
succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
... = (n * k) + k + succ n : { IH }
... = (n * k) + (k + succ n) : add_assoc _ _ _
... = (n * k) + (n + succ k) : {add_comm_succ _ _}
... = (n * k) + n + succ k : symm (add_assoc _ _ _)
... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
theorem mul_comm (n m:) : n * m = m * n :=
induction_on m
(trans (mul_zero_right _) (symm (mul_zero_left _)))
(take k IH,
calc
n * succ k = n * k + n : mul_succ_right _ _
... = k * n + n : {IH}
... = (succ k) * n : symm (mul_succ_left _ _))
theorem mul_distr_right (n m k : ) : (n + m) * k = n * k + m * k :=
induction_on k
(calc
(n + m) * 0 = 0 : mul_zero_right _
... = 0 + 0 : symm (add_zero_right _)
... = n * 0 + 0 : {symm (mul_zero_right _)}
... = n * 0 + m * 0 : {symm (mul_zero_right _)})
(take l IH, calc
(n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
... = n * l + m * l + (n + m) : {IH}
... = n * l + m * l + n + m : symm (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) : {symm (mul_succ_right _ _)}
... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})
theorem mul_distr_left (n m k : ) : n * (m + k) = n * m + n * k :=
calc
n * (m + k) = (m + k) * n : mul_comm _ _
... = m * n + k * n : mul_distr_right _ _ _
... = n * m + k * n : {mul_comm _ _}
... = n * m + n * k : {mul_comm _ _}
theorem mul_assoc (n m k:) : (n * m) * k = n * (m * k) :=
induction_on k
(calc
(n * m) * 0 = 0 : mul_zero_right _
... = n * 0 : symm (mul_zero_right _)
... = n * (m * 0) : {symm (mul_zero_right _)})
(take l IH,
calc
(n * m) * succ l = (n * m) * l + n * m : mul_succ_right _ _
... = n * (m * l) + n * m : {IH}
... = n * (m * l + m) : symm (mul_distr_left _ _ _)
... = n * (m * succ l) : {symm (mul_succ_right _ _)})
theorem mul_left_comm (n m k : ) : n * (m * k) = m * (n * k) :=
left_comm mul_comm mul_assoc n m k
theorem mul_right_comm (n m k : ) : n * m * k = n * k * m :=
right_comm mul_comm mul_assoc n m k
theorem mul_one_right (n : ) : n * 1 = n :=
calc
n * 1 = n * 0 + n : mul_succ_right n 0
... = 0 + n : {mul_zero_right n}
... = n : add_zero_left n
theorem mul_one_left (n : ) : 1 * n = n :=
calc
1 * n = n * 1 : mul_comm _ _
... = n : mul_one_right n
theorem mul_eq_zero {n m : } (H : n * m = 0) : n = 0 m = 0 :=
discriminate
(take Hn : n = 0, or_intro_left _ Hn)
(take (k : ),
assume (Hk : n = succ k),
discriminate
(take (Hm : m = 0), or_intro_right _ Hm)
(take (l : ),
assume (Hl : m = succ l),
have Heq : succ (k * succ l + l) = n * m, from
symm (calc
n * m = n * succ l : {Hl}
... = succ k * succ l : {Hk}
... = k * succ l + succ l : mul_succ_left _ _
... = succ (k * succ l + l) : add_succ_right _ _),
absurd_elim _ (trans Heq H) (succ_ne_zero _)))
---other inversion theorems appear below
-- add_rewrite mul_zero_left mul_zero_right mul_one_right mul_one_left
-- add_rewrite mul_succ_left mul_succ_right
-- add_rewrite mul_comm mul_assoc mul_left_comm
-- add_rewrite mul_distr_right mul_distr_left
end nat