lean2/library/data/real/basic.lean

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/-
Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Robert Y. Lewis
The real numbers, constructed as equivalence classes of Cauchy sequences of rationals.
This construction follows Bishop and Bridges (1985).
To do:
o Break positive naturals into their own file
o Rename things and possibly make theorems private
-/
import data.nat data.rat.order data.pnat
open nat eq eq.ops pnat
open -[coercions] rat
local notation 0 := rat.of_num 0
local notation 1 := rat.of_num 1
----------------------------------------------------------------------------------------------------
-------------------------------------
-- theorems to add to (ordered) field and/or rat
-- this can move to pnat
theorem find_midpoint {a b : } (H : a > b) : ∃ c : , a > b + c ∧ c > 0 :=
exists.intro ((a - b) / (1 + 1))
(and.intro (assert H2 : a + a > (b + b) + (a - b), from calc
a + a > b + a : rat.add_lt_add_right H
... = b + a + b - b : rat.add_sub_cancel
... = b + b + a - b : rat.add.right_comm
... = (b + b) + (a - b) : add_sub,
assert H3 : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
from div_lt_div_of_lt_of_pos H2 dec_trivial,
by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
(pos_div_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) dec_trivial))
theorem add_sub_comm (a b c d : ) : a + b - (c + d) = (a - c) + (b - d) :=
calc
a + b - (c + d) = a + b - c - d : sub_add_eq_sub_sub
... = a - c + b - d : sub_add_eq_add_sub
... = a - c + (b - d) : add_sub
theorem s_mul_assoc_lemma_4 {n : +} {ε q : } (Hε : ε > 0) (Hq : q > 0) (H : n ≥ pceil (q / ε)) :
q * n⁻¹ ≤ ε :=
begin
let H2 := pceil_helper H (pos_div_of_pos_of_pos Hq Hε),
let H3 := mul_le_of_le_div (pos_div_of_pos_of_pos Hq Hε) H2,
rewrite -(one_mul ε),
apply mul_le_mul_of_mul_div_le,
repeat assumption
end
-------------------------------------
-- small helper lemmas
theorem s_mul_assoc_lemma_3 (a b n : +) (p : ) :
p * ((a * n)⁻¹ + (b * n)⁻¹) = p * (a⁻¹ + b⁻¹) * n⁻¹ :=
by rewrite [rat.mul.assoc, rat.right_distrib, *inv_mul_eq_mul_inv]
theorem find_thirds (a b : ) (H : b > 0) : ∃ n : +, a + n⁻¹ + n⁻¹ + n⁻¹ < a + b :=
let n := pceil (of_nat 4 / b) in
have of_nat 3 * n⁻¹ < b, from calc
of_nat 3 * n⁻¹ < of_nat 4 * n⁻¹
: rat.mul_lt_mul_of_pos_right dec_trivial !inv_pos
... ≤ of_nat 4 * (b / of_nat 4)
: rat.mul_le_mul_of_nonneg_left (!inv_pceil_div dec_trivial H) !of_nat_nonneg
... = b / of_nat 4 * of_nat 4 : rat.mul.comm
... = b : rat.div_mul_cancel dec_trivial,
exists.intro n (calc
a + n⁻¹ + n⁻¹ + n⁻¹ = a + (1 + 1 + 1) * n⁻¹ : by rewrite[*rat.right_distrib,*rat.one_mul,-*rat.add.assoc]
... = a + of_nat 3 * n⁻¹ : {show 1+1+1=of_nat 3, from dec_trivial}
... < a + b : rat.add_lt_add_left this a)
theorem squeeze {a b : } (H : ∀ j : +, a ≤ b + j⁻¹ + j⁻¹ + j⁻¹) : a ≤ b :=
begin
apply rat.le_of_not_gt,
intro Hb,
apply (exists.elim (find_midpoint Hb)),
intro c Hc,
apply (exists.elim (find_thirds b c (and.right Hc))),
intro j Hbj,
have Ha : a > b + j⁻¹ + j⁻¹ + j⁻¹, from lt.trans Hbj (and.left Hc),
exact absurd !H (not_le_of_gt Ha)
end
theorem squeeze_2 {a b : } (H : ∀ ε : , ε > 0 → a ≥ b - ε) : a ≥ b :=
begin
apply rat.le_of_not_gt,
intro Hb,
apply (exists.elim (find_midpoint Hb)),
intro c Hc,
let Hc' := H c (and.right Hc),
apply (rat.not_le_of_gt (and.left Hc)) (iff.mpr !le_add_iff_sub_right_le Hc')
end
theorem rewrite_helper (a b c d : ) : a * b - c * d = a * (b - d) + (a - c) * d :=
by rewrite[rat.mul_sub_left_distrib, rat.mul_sub_right_distrib, add_sub, rat.sub_add_cancel]
theorem rewrite_helper3 (a b c d e f g: ) : a * (b + c) - (d * e + f * g) =
(a * b - d * e) + (a * c - f * g) :=
by rewrite[rat.mul.left_distrib, add_sub_comm]
theorem rewrite_helper4 (a b c d : ) : a * b - c * d = (a * b - a * d) + (a * d - c * d) :=
by rewrite[add_sub, rat.sub_add_cancel]
theorem rewrite_helper5 (a b x y : ) : a - b = (a - x) + (x - y) + (y - b) :=
by rewrite[*add_sub, *rat.sub_add_cancel]
theorem rewrite_helper7 (a b c d x : ) :
a * b * c - d = (b * c) * (a - x) + (x * b * c - d) :=
by rewrite[rat.mul_sub_left_distrib, add_sub]; exact (calc
a * b * c - d = a * b * c - x * b * c + x * b * c - d : rat.sub_add_cancel
... = b * c * a - b * c * x + x * b * c - d :
have ∀ {a b c : }, a * b * c = b * c * a, from
λa b c, !rat.mul.comm ▸ !rat.mul.right_comm,
this ▸ this ▸ rfl)
theorem ineq_helper (a b : ) (k m n : +) (H : a ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹)
(H2 : b ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹) :
(rat_of_pnat k) * a + b * (rat_of_pnat k) ≤ m⁻¹ + n⁻¹ :=
have (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹ = (2 * k)⁻¹ * (m⁻¹ + n⁻¹),
by rewrite[rat.mul.left_distrib,*inv_mul_eq_mul_inv]; exact !rat.mul.comm ▸ rfl,
have a + b ≤ k⁻¹ * (m⁻¹ + n⁻¹), from calc
a + b ≤ (2 * k)⁻¹ * (m⁻¹ + n⁻¹) + (2 * k)⁻¹ * (m⁻¹ + n⁻¹) : rat.add_le_add (this ▸ H) (this ▸ H2)
... = ((2 * k)⁻¹ + (2 * k)⁻¹) * (m⁻¹ + n⁻¹) : rat.mul.right_distrib
... = k⁻¹ * (m⁻¹ + n⁻¹) : (pnat.add_halves k) ▸ rfl,
calc (rat_of_pnat k) * a + b * (rat_of_pnat k)
= (rat_of_pnat k) * a + (rat_of_pnat k) * b : rat.mul.comm
... = (rat_of_pnat k) * (a + b) : rat.left_distrib
... ≤ (rat_of_pnat k) * (k⁻¹ * (m⁻¹ + n⁻¹)) :
iff.mp (!rat.le_iff_mul_le_mul_left !rat_of_pnat_is_pos) this
... = m⁻¹ + n⁻¹ : by rewrite[-rat.mul.assoc, inv_cancel_left, rat.one_mul]
theorem factor_lemma (a b c d e : ) : abs (a + b + c - (d + (b + e))) = abs ((a - d) + (c - e)) :=
!congr_arg (calc
a + b + c - (d + (b + e)) = a + b + c - (d + b + e) : rat.add.assoc
... = a + b - (d + b) + (c - e) : add_sub_comm
... = a + b - b - d + (c - e) : sub_add_eq_sub_sub_swap
... = a - d + (c - e) : rat.add_sub_cancel)
theorem factor_lemma_2 (a b c d : ) : (a + b) + (c + d) = (a + c) + (d + b) :=
!rat.add.comm ▸ (binary.comm4 rat.add.comm rat.add.assoc a b c d)
--------------------------------------
-- define cauchy sequences and equivalence. show equivalence actually is one
namespace s
notation `seq` := + →
definition regular (s : seq) := ∀ m n : +, abs (s m - s n) ≤ m⁻¹ + n⁻¹
definition equiv (s t : seq) := ∀ n : +, abs (s n - t n) ≤ n⁻¹ + n⁻¹
infix `≡` := equiv
theorem equiv.refl (s : seq) : s ≡ s :=
begin
rewrite ↑equiv,
intros,
rewrite [rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem equiv.symm (s t : seq) (H : s ≡ t) : t ≡ s :=
begin
rewrite ↑equiv at *,
intros,
rewrite [-abs_neg, neg_sub],
exact H n
end
theorem bdd_of_eq {s t : seq} (H : s ≡ t) :
∀ j : +, ∀ n : +, n ≥ 2 * j → abs (s n - t n) ≤ j⁻¹ :=
begin
rewrite ↑equiv at *,
intros [j, n, Hn],
apply rat.le.trans,
apply H n,
rewrite -(add_halves j),
apply rat.add_le_add,
apply inv_ge_of_le Hn,
apply inv_ge_of_le Hn
end
theorem eq_of_bdd {s t : seq} (Hs : regular s) (Ht : regular t)
(H : ∀ j : +, ∃ Nj : +, ∀ n : +, Nj ≤ n → abs (s n - t n) ≤ j⁻¹) : s ≡ t :=
begin
rewrite ↑equiv,
intros,
have Hj : (∀ j : +, abs (s n - t n) ≤ n⁻¹ + n⁻¹ + j⁻¹ + j⁻¹ + j⁻¹), begin
intros,
apply exists.elim (H j),
intros [Nj, HNj],
rewrite [-(rat.sub_add_cancel (s n) (s (max j Nj))), rat.add.assoc (s n + -s (max j Nj)),
↑regular at *],
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.le.trans,
apply rat.add_le_add,
apply Hs,
rewrite [-(rat.sub_add_cancel (s (max j Nj)) (t (max j Nj))), rat.add.assoc],
apply abs_add_le_abs_add_abs,
apply rat.le.trans,
apply rat.add_le_add_left,
apply rat.add_le_add,
apply HNj (max j Nj) (max_right j Nj),
apply Ht,
have hsimp : ∀ m : +, n⁻¹ + m⁻¹ + (j⁻¹ + (m⁻¹ + n⁻¹)) = n⁻¹ + n⁻¹ + j⁻¹ + (m⁻¹ + m⁻¹),
from λm, calc
n⁻¹ + m⁻¹ + (j⁻¹ + (m⁻¹ + n⁻¹)) = n⁻¹ + (j⁻¹ + (m⁻¹ + n⁻¹)) + m⁻¹ : rat.add.right_comm
... = n⁻¹ + (j⁻¹ + m⁻¹ + n⁻¹) + m⁻¹ : rat.add.assoc
... = n⁻¹ + (n⁻¹ + (j⁻¹ + m⁻¹)) + m⁻¹ : rat.add.comm
... = n⁻¹ + n⁻¹ + j⁻¹ + (m⁻¹ + m⁻¹) : by rewrite[-*rat.add.assoc],
rewrite hsimp,
have Hms : (max j Nj)⁻¹ + (max j Nj)⁻¹ ≤ j⁻¹ + j⁻¹, begin
apply rat.add_le_add,
apply inv_ge_of_le (max_left j Nj),
apply inv_ge_of_le (max_left j Nj),
end,
apply (calc
n⁻¹ + n⁻¹ + j⁻¹ + ((max j Nj)⁻¹ + (max j Nj)⁻¹) ≤ n⁻¹ + n⁻¹ + j⁻¹ + (j⁻¹ + j⁻¹) :
rat.add_le_add_left Hms
... = n⁻¹ + n⁻¹ + j⁻¹ + j⁻¹ + j⁻¹ : by rewrite *rat.add.assoc)
end,
apply (squeeze Hj)
end
theorem eq_of_bdd_var {s t : seq} (Hs : regular s) (Ht : regular t)
(H : ∀ ε : , ε > 0 → ∃ Nj : +, ∀ n : +, Nj ≤ n → abs (s n - t n) ≤ ε) : s ≡ t :=
begin
apply eq_of_bdd,
apply Hs,
apply Ht,
intros,
apply H j⁻¹,
apply inv_pos
end
set_option pp.beta false
theorem pnat_bound {ε : } (Hε : ε > 0) : ∃ p : +, p⁻¹ ≤ ε :=
begin
existsi (pceil (1 / ε)),
rewrite -(rat.div_div (rat.ne_of_gt Hε)) at {2},
apply pceil_helper,
apply le.refl,
apply div_pos_of_pos Hε
end
theorem bdd_of_eq_var {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
∀ ε : , ε > 0 → ∃ Nj : +, ∀ n : +, Nj ≤ n → abs (s n - t n) ≤ ε :=
begin
intro ε Hε,
apply (exists.elim (pnat_bound Hε)),
intro N HN,
let Bd' := bdd_of_eq Heq N,
existsi 2 * N,
intro n Hn,
apply rat.le.trans,
apply Bd' n Hn,
assumption
end
theorem equiv.trans (s t u : seq) (Hs : regular s) (Ht : regular t) (Hu : regular u)
(H : s ≡ t) (H2 : t ≡ u) : s ≡ u :=
begin
apply eq_of_bdd Hs Hu,
intros,
existsi 2 * (2 * j),
intro n Hn,
rewrite [-rat.sub_add_cancel (s n) (t n), rat.add.assoc],
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
have Hst : abs (s n - t n) ≤ (2 * j)⁻¹, from bdd_of_eq H _ _ Hn,
have Htu : abs (t n - u n) ≤ (2 * j)⁻¹, from bdd_of_eq H2 _ _ Hn,
rewrite -(add_halves j),
apply rat.add_le_add,
repeat assumption
end
-----------------------------------
-- define operations on cauchy sequences. show operations preserve regularity
definition K (s : seq) : + := pnat.pos (ubound (abs (s pone)) + 1 + 1) dec_trivial
theorem canon_bound {s : seq} (Hs : regular s) (n : +) : abs (s n) ≤ rat_of_pnat (K s) :=
calc
abs (s n) = abs (s n - s pone + s pone) : by rewrite rat.sub_add_cancel
... ≤ abs (s n - s pone) + abs (s pone) : abs_add_le_abs_add_abs
... ≤ n⁻¹ + pone⁻¹ + abs (s pone) : rat.add_le_add_right !Hs
... = n⁻¹ + (1 + abs (s pone)) : by rewrite [pone_inv, rat.add.assoc]
... ≤ 1 + (1 + abs (s pone)) : rat.add_le_add_right (inv_le_one n)
... = abs (s pone) + (1 + 1) :
by rewrite [add.comm 1 (abs (s pone)), rat.add.comm 1, rat.add.assoc]
... ≤ of_nat (ubound (abs (s pone))) + (1 + 1) : rat.add_le_add_right (!ubound_ge)
... = of_nat (ubound (abs (s pone)) + (1 + 1)) : of_nat_add
... = of_nat (ubound (abs (s pone)) + 1 + 1) : nat.add.assoc
definition K₂ (s t : seq) := max (K s) (K t)
theorem K₂_symm (s t : seq) : K₂ s t = K₂ t s :=
if H : K s < K t then
(assert H1 : K₂ s t = K t, from max_eq_right H,
assert H2 : K₂ t s = K t, from max_eq_left (not_lt_of_ge (le_of_lt H)),
by rewrite [H1, -H2])
else
(assert H1 : K₂ s t = K s, from max_eq_left H,
if J : K t < K s then
(assert H2 : K₂ t s = K s, from max_eq_right J, by rewrite [H1, -H2])
else
(assert Heq : K t = K s, from
eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt J),
by rewrite [↑K₂, Heq]))
theorem canon_2_bound_left (s t : seq) (Hs : regular s) (n : +) :
abs (s n) ≤ rat_of_pnat (K₂ s t) :=
calc
abs (s n) ≤ rat_of_pnat (K s) : canon_bound Hs n
... ≤ rat_of_pnat (K₂ s t) : rat_of_pnat_le_of_pnat_le (!max_left)
theorem canon_2_bound_right (s t : seq) (Ht : regular t) (n : +) :
abs (t n) ≤ rat_of_pnat (K₂ s t) :=
calc
abs (t n) ≤ rat_of_pnat (K t) : canon_bound Ht n
... ≤ rat_of_pnat (K₂ s t) : rat_of_pnat_le_of_pnat_le (!max_right)
definition sadd (s t : seq) : seq := λ n, (s (2 * n)) + (t (2 * n))
theorem reg_add_reg {s t : seq} (Hs : regular s) (Ht : regular t) : regular (sadd s t) :=
begin
rewrite [↑regular at *, ↑sadd],
intros,
rewrite add_sub_comm,
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
rewrite add_halves_double,
apply rat.add_le_add,
apply Hs,
apply Ht
end
definition smul (s t : seq) : seq := λ n : +, (s ((K₂ s t) * 2 * n)) * (t ((K₂ s t) * 2 * n))
theorem reg_mul_reg {s t : seq} (Hs : regular s) (Ht : regular t) : regular (smul s t) :=
begin
rewrite [↑regular at *, ↑smul],
intros,
rewrite rewrite_helper,
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.le.trans,
apply rat.add_le_add,
rewrite abs_mul,
apply rat.mul_le_mul_of_nonneg_right,
apply canon_2_bound_left s t Hs,
apply abs_nonneg,
rewrite abs_mul,
apply rat.mul_le_mul_of_nonneg_left,
apply canon_2_bound_right s t Ht,
apply abs_nonneg,
apply ineq_helper,
apply Ht,
apply Hs
end
definition sneg (s : seq) : seq := λ n : +, - (s n)
theorem reg_neg_reg {s : seq} (Hs : regular s) : regular (sneg s) :=
begin
rewrite [↑regular at *, ↑sneg],
intros,
rewrite [-abs_neg, neg_sub, sub_neg_eq_add, rat.add.comm],
apply Hs
end
-----------------------------------
-- show properties of +, *, -
definition zero : seq := λ n, 0
definition one : seq := λ n, 1
theorem s_add_comm (s t : seq) : sadd s t ≡ sadd t s :=
begin
esimp [sadd],
intro n,
rewrite [sub_add_eq_sub_sub, rat.add_sub_cancel, rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem s_add_assoc (s t u : seq) (Hs : regular s) (Hu : regular u) :
sadd (sadd s t) u ≡ sadd s (sadd t u) :=
begin
rewrite [↑sadd, ↑equiv, ↑regular at *],
intros,
rewrite factor_lemma,
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.le.trans,
rotate 1,
apply rat.add_le_add_right,
apply inv_two_mul_le_inv,
rewrite [-(add_halves (2 * n)), -(add_halves n), factor_lemma_2],
apply rat.add_le_add,
apply Hs,
apply Hu
end
theorem s_mul_comm (s t : seq) : smul s t ≡ smul t s :=
begin
rewrite ↑smul,
intros n,
rewrite [*(K₂_symm s t), rat.mul.comm, rat.sub_self, abs_zero],
apply add_invs_nonneg
end
definition DK (s t : seq) := (K₂ s t) * 2
theorem DK_rewrite (s t : seq) : (K₂ s t) * 2 = DK s t := rfl
definition TK (s t u : seq) := (DK (λ (n : +), s (mul (DK s t) n) * t (mul (DK s t) n)) u)
theorem TK_rewrite (s t u : seq) :
(DK (λ (n : +), s (mul (DK s t) n) * t (mul (DK s t) n)) u) = TK s t u := rfl
theorem s_mul_assoc_lemma (s t u : seq) (a b c d : +) :
abs (s a * t a * u b - s c * t d * u d) ≤ abs (t a) * abs (u b) * abs (s a - s c) +
abs (s c) * abs (t a) * abs (u b - u d) + abs (s c) * abs (u d) * abs (t a - t d) :=
begin
rewrite (rewrite_helper7 _ _ _ _ (s c)),
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
rewrite rat.add.assoc,
apply rat.add_le_add,
rewrite 2 abs_mul,
apply rat.le.refl,
rewrite [*rat.mul.assoc, -rat.mul_sub_left_distrib, -rat.left_distrib, abs_mul],
apply rat.mul_le_mul_of_nonneg_left,
rewrite rewrite_helper,
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.add_le_add,
rewrite abs_mul, apply rat.le.refl,
rewrite [abs_mul, rat.mul.comm], apply rat.le.refl,
apply abs_nonneg
end
definition Kq (s : seq) := rat_of_pnat (K s) + 1
theorem Kq_bound {s : seq} (H : regular s) : ∀ n, abs (s n) ≤ Kq s :=
begin
intros,
apply rat.le_of_lt,
apply rat.lt_of_le_of_lt,
apply canon_bound H,
apply rat.lt_add_of_pos_right,
apply rat.zero_lt_one
end
theorem Kq_bound_nonneg {s : seq} (H : regular s) : 0 ≤ Kq s :=
rat.le.trans !abs_nonneg (Kq_bound H 2)
theorem Kq_bound_pos {s : seq} (H : regular s) : 0 < Kq s :=
have H1 : 0 ≤ rat_of_pnat (K s), from rat.le.trans (!abs_nonneg) (canon_bound H 2),
add_pos_of_nonneg_of_pos H1 rat.zero_lt_one
theorem s_mul_assoc_lemma_5 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(a b c : +) : abs (t a) * abs (u b) * abs (s a - s c) ≤ (Kq t) * (Kq u) * (a⁻¹ + c⁻¹) :=
begin
repeat apply rat.mul_le_mul,
apply Kq_bound Ht,
apply Kq_bound Hu,
apply abs_nonneg,
apply Kq_bound_nonneg Ht,
apply Hs,
apply abs_nonneg,
apply rat.mul_nonneg,
apply Kq_bound_nonneg Ht,
apply Kq_bound_nonneg Hu,
end
theorem s_mul_assoc_lemma_2 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(a b c d : +) :
abs (t a) * abs (u b) * abs (s a - s c) + abs (s c) * abs (t a) * abs (u b - u d)
+ abs (s c) * abs (u d) * abs (t a - t d) ≤
(Kq t) * (Kq u) * (a⁻¹ + c⁻¹) + (Kq s) * (Kq t) * (b⁻¹ + d⁻¹) + (Kq s) * (Kq u) * (a⁻¹ + d⁻¹) :=
begin
apply add_le_add_three,
repeat apply rat.mul_le_mul,
apply Kq_bound Ht,
apply Kq_bound Hu,
apply abs_nonneg,
apply Kq_bound_nonneg Ht,
apply Hs,
apply abs_nonneg,
apply rat.mul_nonneg,
apply Kq_bound_nonneg Ht,
apply Kq_bound_nonneg Hu,
repeat apply rat.mul_le_mul,
apply Kq_bound Hs,
apply Kq_bound Ht,
apply abs_nonneg,
apply Kq_bound_nonneg Hs,
apply Hu,
apply abs_nonneg,
apply rat.mul_nonneg,
apply Kq_bound_nonneg Hs,
apply Kq_bound_nonneg Ht,
repeat apply rat.mul_le_mul,
apply Kq_bound Hs,
apply Kq_bound Hu,
apply abs_nonneg,
apply Kq_bound_nonneg Hs,
apply Ht,
apply abs_nonneg,
apply rat.mul_nonneg,
apply Kq_bound_nonneg Hs,
apply Kq_bound_nonneg Hu
end
theorem s_mul_assoc {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u) :
smul (smul s t) u ≡ smul s (smul t u) :=
begin
apply eq_of_bdd_var,
repeat apply reg_mul_reg,
apply Hs,
apply Ht,
apply Hu,
apply reg_mul_reg Hs,
apply reg_mul_reg Ht Hu,
intros,
fapply exists.intro,
rotate 1,
intros,
rewrite [↑smul, *DK_rewrite, *TK_rewrite, -*pnat.mul.assoc, -*rat.mul.assoc],
apply rat.le.trans,
apply s_mul_assoc_lemma,
apply rat.le.trans,
apply s_mul_assoc_lemma_2,
apply Hs,
apply Ht,
apply Hu,
rewrite [*s_mul_assoc_lemma_3, -rat.distrib_three_right],
apply s_mul_assoc_lemma_4,
apply a,
repeat apply rat.add_pos,
repeat apply rat.mul_pos,
apply Kq_bound_pos Ht,
apply Kq_bound_pos Hu,
apply rat.add_pos,
repeat apply inv_pos,
repeat apply rat.mul_pos,
apply Kq_bound_pos Hs,
apply Kq_bound_pos Ht,
apply rat.add_pos,
repeat apply inv_pos,
repeat apply rat.mul_pos,
apply Kq_bound_pos Hs,
apply Kq_bound_pos Hu,
apply rat.add_pos,
repeat apply inv_pos,
apply a_1
end
theorem zero_is_reg : regular zero :=
begin
rewrite [↑regular, ↑zero],
intros,
rewrite [rat.sub_zero, abs_zero],
apply add_invs_nonneg
end
theorem s_zero_add (s : seq) (H : regular s) : sadd zero s ≡ s :=
begin
rewrite [↑sadd, ↑zero, ↑equiv, ↑regular at H],
intros,
rewrite [rat.zero_add],
apply rat.le.trans,
apply H,
apply rat.add_le_add,
apply inv_two_mul_le_inv,
apply rat.le.refl
end
theorem s_add_zero (s : seq) (H : regular s) : sadd s zero ≡ s :=
begin
rewrite [↑sadd, ↑zero, ↑equiv, ↑regular at H],
intros,
rewrite [rat.add_zero],
apply rat.le.trans,
apply H,
apply rat.add_le_add,
apply inv_two_mul_le_inv,
apply rat.le.refl
end
theorem s_neg_cancel (s : seq) (H : regular s) : sadd (sneg s) s ≡ zero :=
begin
rewrite [↑sadd, ↑sneg, ↑regular at H, ↑zero, ↑equiv],
intros,
rewrite [neg_add_eq_sub, rat.sub_self, rat.sub_zero, abs_zero],
apply add_invs_nonneg
end
theorem neg_s_cancel (s : seq) (H : regular s) : sadd s (sneg s) ≡ zero :=
begin
apply equiv.trans,
rotate 3,
apply s_add_comm,
apply s_neg_cancel s H,
apply reg_add_reg,
apply H,
apply reg_neg_reg,
apply H,
apply reg_add_reg,
apply reg_neg_reg,
repeat apply H,
apply zero_is_reg
end
theorem add_well_defined {s t u v : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(Hv : regular v) (Esu : s ≡ u) (Etv : t ≡ v) : sadd s t ≡ sadd u v :=
begin
rewrite [↑sadd, ↑equiv at *],
intros,
rewrite [add_sub_comm, add_halves_double],
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.add_le_add,
apply Esu,
apply Etv
end
set_option tactic.goal_names false
theorem mul_bound_helper {s t : seq} (Hs : regular s) (Ht : regular t) (a b c : +) (j : +) :
∃ N : +, ∀ n : +, n ≥ N → abs (s (a * n) * t (b * n) - s (c * n) * t (c * n)) ≤ j⁻¹ :=
begin
existsi pceil (((rat_of_pnat (K s)) * (b⁻¹ + c⁻¹) + (a⁻¹ + c⁻¹) *
(rat_of_pnat (K t))) * (rat_of_pnat j)),
intros n Hn,
rewrite rewrite_helper4,
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.le.trans,
rotate 1,
show n⁻¹ * ((rat_of_pnat (K s)) * (b⁻¹ + c⁻¹)) +
n⁻¹ * ((a⁻¹ + c⁻¹) * (rat_of_pnat (K t))) ≤ j⁻¹, begin
rewrite -rat.left_distrib,
apply rat.le.trans,
apply rat.mul_le_mul_of_nonneg_right,
apply pceil_helper Hn,
repeat (apply rat.mul_pos | apply rat.add_pos | apply inv_pos | apply rat_of_pnat_is_pos),
apply rat.le_of_lt,
apply rat.add_pos,
apply rat.mul_pos,
apply rat_of_pnat_is_pos,
apply rat.add_pos,
repeat apply inv_pos,
apply rat.mul_pos,
apply rat.add_pos,
repeat apply inv_pos,
apply rat_of_pnat_is_pos,
have H : (rat_of_pnat (K s) * (b⁻¹ + c⁻¹) + (a⁻¹ + c⁻¹) * rat_of_pnat (K t)) ≠ 0, begin
apply rat.ne_of_gt,
repeat (apply rat.mul_pos | apply rat.add_pos | apply rat_of_pnat_is_pos | apply inv_pos),
end,
rewrite (rat.div_helper H),
apply rat.le.refl
end,
apply rat.add_le_add,
rewrite [-rat.mul_sub_left_distrib, abs_mul],
apply rat.le.trans,
apply rat.mul_le_mul,
apply canon_bound,
apply Hs,
apply Ht,
apply abs_nonneg,
apply rat.le_of_lt,
apply rat_of_pnat_is_pos,
rewrite [*inv_mul_eq_mul_inv, -rat.right_distrib, -rat.mul.assoc, rat.mul.comm],
apply rat.mul_le_mul_of_nonneg_left,
apply rat.le.refl,
apply rat.le_of_lt,
apply inv_pos,
rewrite [-rat.mul_sub_right_distrib, abs_mul],
apply rat.le.trans,
apply rat.mul_le_mul,
apply Hs,
apply canon_bound,
apply Ht,
apply abs_nonneg,
apply add_invs_nonneg,
rewrite [*inv_mul_eq_mul_inv, -rat.right_distrib, mul.comm _ n⁻¹, rat.mul.assoc],
apply rat.mul_le_mul,
apply rat.le.refl,
apply rat.le.refl,
apply rat.le_of_lt,
apply rat.mul_pos,
apply rat.add_pos,
repeat apply inv_pos,
apply rat_of_pnat_is_pos,
apply rat.le_of_lt,
apply inv_pos
end
theorem s_distrib {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u) :
smul s (sadd t u) ≡ sadd (smul s t) (smul s u) :=
begin
apply eq_of_bdd,
repeat (assumption | apply reg_add_reg | apply reg_mul_reg),
intros,
let exh1 := λ a b c, mul_bound_helper Hs Ht a b c (2 * j),
apply exists.elim,
apply exh1,
rotate 3,
intros N1 HN1,
let exh2 := λ d e f, mul_bound_helper Hs Hu d e f (2 * j),
apply exists.elim,
apply exh2,
rotate 3,
intros N2 HN2,
existsi max N1 N2,
intros n Hn,
rewrite [↑sadd at *, ↑smul, rewrite_helper3, -add_halves j, -*pnat.mul.assoc at *],
apply rat.le.trans,
apply abs_add_le_abs_add_abs,
apply rat.add_le_add,
apply HN1,
apply le.trans,
apply max_left N1 N2,
apply Hn,
apply HN2,
apply le.trans,
apply max_right N1 N2,
apply Hn
end
theorem mul_zero_equiv_zero {s t : seq} (Hs : regular s) (Ht : regular t) (Htz : t ≡ zero) :
smul s t ≡ zero :=
begin
apply eq_of_bdd_var,
apply reg_mul_reg Hs Ht,
apply zero_is_reg,
intro ε Hε,
let Bd := bdd_of_eq_var Ht zero_is_reg Htz (ε / (Kq s))
(pos_div_of_pos_of_pos Hε (Kq_bound_pos Hs)),
apply exists.elim Bd,
intro N HN,
existsi N,
intro n Hn,
rewrite [↑equiv at Htz, ↑zero at *, rat.sub_zero, ↑smul, abs_mul],
apply rat.le.trans,
apply rat.mul_le_mul,
apply Kq_bound Hs,
let HN' := λ n, (!rat.sub_zero ▸ HN n),
apply HN',
apply le.trans Hn,
apply pnat.mul_le_mul_left,
apply abs_nonneg,
apply rat.le_of_lt (Kq_bound_pos Hs),
rewrite (rat.mul_div_cancel' (ne.symm (rat.ne_of_lt (Kq_bound_pos Hs)))),
apply rat.le.refl
end
theorem neg_bound_eq_bound (s : seq) : K (sneg s) = K s :=
by rewrite [↑K, ↑sneg, abs_neg]
theorem neg_bound2_eq_bound2 (s t : seq) : K₂ s (sneg t) = K₂ s t :=
by rewrite [↑K₂, neg_bound_eq_bound]
theorem sneg_def (s : seq) : (λ (n : +), -(s n)) = sneg s := rfl
theorem mul_neg_equiv_neg_mul {s t : seq} : smul s (sneg t) ≡ sneg (smul s t) :=
begin
rewrite [↑equiv, ↑smul],
intros,
rewrite [↑sneg, *sub_neg_eq_add, -neg_mul_eq_mul_neg, rat.add.comm, *sneg_def,
*neg_bound2_eq_bound2, rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem equiv_of_diff_equiv_zero {s t : seq} (Hs : regular s) (Ht : regular t)
(H : sadd s (sneg t) ≡ zero) : s ≡ t :=
begin
have hsimp : ∀ a b c d e : , a + b + c + (d + e) = b + d + a + e + c, from
λ a b c d e, calc
a + b + c + (d + e) = a + b + (d + e) + c : rat.add.right_comm
... = a + (b + d) + e + c : by rewrite[-*rat.add.assoc]
... = b + d + a + e + c : rat.add.comm,
apply eq_of_bdd Hs Ht,
intros,
let He := bdd_of_eq H,
existsi 2 * (2 * (2 * j)),
intros n Hn,
rewrite (rewrite_helper5 _ _ (s (2 * n)) (t (2 * n))),
apply rat.le.trans,
apply abs_add_three,
apply rat.le.trans,
apply add_le_add_three,
apply Hs,
rewrite [↑sadd at He, ↑sneg at He, ↑zero at He],
let He' := λ a b c, !rat.sub_zero ▸ (He a b c),
apply (He' _ _ Hn),
apply Ht,
rewrite [hsimp, add_halves, -(add_halves j), -(add_halves (2 * j)), -*rat.add.assoc],
apply rat.add_le_add_right,
apply add_le_add_three,
repeat (apply rat.le.trans; apply inv_ge_of_le Hn; apply inv_two_mul_le_inv)
end
theorem s_sub_cancel (s : seq) : sadd s (sneg s) ≡ zero :=
begin
rewrite [↑equiv, ↑sadd, ↑sneg, ↑zero],
intros,
rewrite [rat.sub_zero, rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem diff_equiv_zero_of_equiv {s t : seq} (Hs : regular s) (Ht : regular t) (H : s ≡ t) :
sadd s (sneg t) ≡ zero :=
begin
let Hnt := reg_neg_reg Ht,
let Hsnt := reg_add_reg Hs Hnt,
let Htnt := reg_add_reg Ht Hnt,
apply equiv.trans,
rotate 4,
apply s_sub_cancel t,
rotate 2,
apply zero_is_reg,
apply add_well_defined,
repeat assumption,
apply equiv.refl,
repeat assumption
end
theorem mul_well_defined_half1 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(Etu : t ≡ u) : smul s t ≡ smul s u :=
begin
apply equiv_of_diff_equiv_zero,
rotate 2,
apply equiv.trans,
rotate 3,
apply equiv.symm,
apply add_well_defined,
rotate 4,
apply equiv.refl,
apply mul_neg_equiv_neg_mul,
apply equiv.trans,
rotate 3,
apply equiv.symm,
apply s_distrib,
rotate 3,
apply mul_zero_equiv_zero,
rotate 2,
apply diff_equiv_zero_of_equiv,
repeat (assumption | apply reg_mul_reg | apply reg_neg_reg | apply reg_add_reg |
apply zero_is_reg)
end
theorem mul_well_defined_half2 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(Est : s ≡ t) : smul s u ≡ smul t u :=
begin
apply equiv.trans,
rotate 3,
apply s_mul_comm,
apply equiv.trans,
rotate 3,
apply mul_well_defined_half1,
rotate 2,
apply Ht,
rotate 1,
apply s_mul_comm,
repeat (assumption | apply reg_mul_reg)
end
theorem mul_well_defined {s t u v : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
(Hv : regular v) (Esu : s ≡ u) (Etv : t ≡ v) : smul s t ≡ smul u v :=
begin
apply equiv.trans,
exact reg_mul_reg Hs Ht,
exact reg_mul_reg Hs Hv,
exact reg_mul_reg Hu Hv,
apply mul_well_defined_half1,
repeat assumption,
apply mul_well_defined_half2,
repeat assumption
end
theorem neg_well_defined {s t : seq} (Est : s ≡ t) : sneg s ≡ sneg t :=
begin
rewrite [↑sneg, ↑equiv at *],
intros,
rewrite [-abs_neg, neg_sub, sub_neg_eq_add, rat.add.comm],
apply Est
end
theorem one_is_reg : regular one :=
begin
rewrite [↑regular, ↑one],
intros,
rewrite [rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem s_one_mul {s : seq} (H : regular s) : smul one s ≡ s :=
begin
rewrite ↑equiv,
intros,
rewrite [↑smul, ↑one, rat.one_mul],
apply rat.le.trans,
apply H,
apply rat.add_le_add_right,
apply inv_mul_le_inv
end
theorem s_mul_one {s : seq} (H : regular s) : smul s one ≡ s :=
begin
apply equiv.trans,
apply reg_mul_reg H one_is_reg,
rotate 2,
apply s_mul_comm,
apply s_one_mul H,
apply reg_mul_reg one_is_reg H,
apply H
end
theorem zero_nequiv_one : ¬ zero ≡ one :=
begin
intro Hz,
rewrite [↑equiv at Hz, ↑zero at Hz, ↑one at Hz],
let H := Hz (2 * 2),
rewrite [rat.zero_sub at H, abs_neg at H, add_halves at H],
have H' : pone⁻¹ ≤ 2⁻¹, from calc
pone⁻¹ = 1 : by rewrite -pone_inv
... = abs 1 : abs_of_pos zero_lt_one
... ≤ 2⁻¹ : H,
let H'' := ge_of_inv_le H',
apply absurd (one_lt_two) (not_lt_of_ge H'')
end
---------------------------------------------
-- constant sequences
definition const (a : ) : seq := λ n, a
theorem const_reg (a : ) : regular (const a) :=
begin
intros,
rewrite [↑const, rat.sub_self, abs_zero],
apply add_invs_nonneg
end
theorem add_consts (a b : ) : sadd (const a) (const b) ≡ const (a + b) :=
begin
rewrite [↑sadd, ↑const],
apply equiv.refl
end
---------------------------------------------
-- create the type of regular sequences and lift theorems
record reg_seq : Type :=
(sq : seq) (is_reg : regular sq)
definition requiv (s t : reg_seq) := (reg_seq.sq s) ≡ (reg_seq.sq t)
definition requiv.refl (s : reg_seq) : requiv s s := equiv.refl (reg_seq.sq s)
definition requiv.symm (s t : reg_seq) (H : requiv s t) : requiv t s :=
equiv.symm (reg_seq.sq s) (reg_seq.sq t) H
definition requiv.trans (s t u : reg_seq) (H : requiv s t) (H2 : requiv t u) : requiv s u :=
equiv.trans _ _ _ (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) H H2
definition radd (s t : reg_seq) : reg_seq :=
reg_seq.mk (sadd (reg_seq.sq s) (reg_seq.sq t))
(reg_add_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
infix `+` := radd
definition rmul (s t : reg_seq) : reg_seq :=
reg_seq.mk (smul (reg_seq.sq s) (reg_seq.sq t))
(reg_mul_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
infix `*` := rmul
definition rneg (s : reg_seq) : reg_seq :=
reg_seq.mk (sneg (reg_seq.sq s)) (reg_neg_reg (reg_seq.is_reg s))
prefix `-` := rneg
definition radd_well_defined {s t u v : reg_seq} (H : requiv s u) (H2 : requiv t v) :
requiv (s + t) (u + v) :=
add_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) (reg_seq.is_reg v) H H2
definition rmul_well_defined {s t u v : reg_seq} (H : requiv s u) (H2 : requiv t v) :
requiv (s * t) (u * v) :=
mul_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) (reg_seq.is_reg v) H H2
definition rneg_well_defined {s t : reg_seq} (H : requiv s t) : requiv (-s) (-t) :=
neg_well_defined H
theorem requiv_is_equiv : equivalence requiv :=
mk_equivalence requiv requiv.refl requiv.symm requiv.trans
definition reg_seq.to_setoid [instance] : setoid reg_seq :=
⦃setoid, r := requiv, iseqv := requiv_is_equiv⦄
definition r_zero : reg_seq :=
reg_seq.mk (zero) (zero_is_reg)
definition r_one : reg_seq :=
reg_seq.mk (one) (one_is_reg)
theorem r_add_comm (s t : reg_seq) : requiv (s + t) (t + s) :=
s_add_comm (reg_seq.sq s) (reg_seq.sq t)
theorem r_add_assoc (s t u : reg_seq) : requiv (s + t + u) (s + (t + u)) :=
s_add_assoc (reg_seq.sq s) (reg_seq.sq t) (reg_seq.sq u) (reg_seq.is_reg s) (reg_seq.is_reg u)
theorem r_zero_add (s : reg_seq) : requiv (r_zero + s) s :=
s_zero_add (reg_seq.sq s) (reg_seq.is_reg s)
theorem r_add_zero (s : reg_seq) : requiv (s + r_zero) s :=
s_add_zero (reg_seq.sq s) (reg_seq.is_reg s)
theorem r_neg_cancel (s : reg_seq) : requiv (-s + s) r_zero :=
s_neg_cancel (reg_seq.sq s) (reg_seq.is_reg s)
theorem r_mul_comm (s t : reg_seq) : requiv (s * t) (t * s) :=
s_mul_comm (reg_seq.sq s) (reg_seq.sq t)
theorem r_mul_assoc (s t u : reg_seq) : requiv (s * t * u) (s * (t * u)) :=
s_mul_assoc (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u)
theorem r_mul_one (s : reg_seq) : requiv (s * r_one) s :=
s_mul_one (reg_seq.is_reg s)
theorem r_one_mul (s : reg_seq) : requiv (r_one * s) s :=
s_one_mul (reg_seq.is_reg s)
theorem r_distrib (s t u : reg_seq) : requiv (s * (t + u)) (s * t + s * u) :=
s_distrib (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u)
theorem r_zero_nequiv_one : ¬ requiv r_zero r_one :=
zero_nequiv_one
definition r_const (a : ) : reg_seq := reg_seq.mk (const a) (const_reg a)
theorem r_add_consts (a b : ) : requiv (r_const a + r_const b) (r_const (a + b)) := add_consts a b
end s
----------------------------------------------
-- take quotients to get and show it's a comm ring
open s
definition real := quot reg_seq.to_setoid
namespace real
notation `` := real
definition add (x y : ) : :=
(quot.lift_on₂ x y (λ a b, quot.mk (a + b))
(take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
quot.sound (radd_well_defined Hab Hcd)))
protected definition prio := num.pred rat.prio
infix [priority real.prio] `+` := add
definition mul (x y : ) : :=
(quot.lift_on₂ x y (λ a b, quot.mk (a * b))
(take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
quot.sound (rmul_well_defined Hab Hcd)))
infix [priority real.prio] `*` := mul
definition neg (x : ) : :=
(quot.lift_on x (λ a, quot.mk (-a)) (take a b : reg_seq, take Hab : requiv a b,
quot.sound (rneg_well_defined Hab)))
prefix [priority real.prio] `-` := neg
definition zero : := quot.mk r_zero
--notation 0 := zero
definition one : := quot.mk r_one
theorem add_comm (x y : ) : x + y = y + x :=
quot.induction_on₂ x y (λ s t, quot.sound (r_add_comm s t))
theorem add_assoc (x y z : ) : x + y + z = x + (y + z) :=
quot.induction_on₃ x y z (λ s t u, quot.sound (r_add_assoc s t u))
theorem zero_add (x : ) : zero + x = x :=
quot.induction_on x (λ s, quot.sound (r_zero_add s))
theorem add_zero (x : ) : x + zero = x :=
quot.induction_on x (λ s, quot.sound (r_add_zero s))
theorem neg_cancel (x : ) : -x + x = zero :=
quot.induction_on x (λ s, quot.sound (r_neg_cancel s))
theorem mul_assoc (x y z : ) : x * y * z = x * (y * z) :=
quot.induction_on₃ x y z (λ s t u, quot.sound (r_mul_assoc s t u))
theorem mul_comm (x y : ) : x * y = y * x :=
quot.induction_on₂ x y (λ s t, quot.sound (r_mul_comm s t))
theorem one_mul (x : ) : one * x = x :=
quot.induction_on x (λ s, quot.sound (r_one_mul s))
theorem mul_one (x : ) : x * one = x :=
quot.induction_on x (λ s, quot.sound (r_mul_one s))
theorem distrib (x y z : ) : x * (y + z) = x * y + x * z :=
quot.induction_on₃ x y z (λ s t u, quot.sound (r_distrib s t u))
theorem distrib_l (x y z : ) : (x + y) * z = x * z + y * z :=
by rewrite [mul_comm, distrib, {x * _}mul_comm, {y * _}mul_comm] -- this shouldn't be necessary
theorem zero_ne_one : ¬ zero = one :=
take H : zero = one,
absurd (quot.exact H) (r_zero_nequiv_one)
protected definition comm_ring [reducible] : algebra.comm_ring :=
begin
fapply algebra.comm_ring.mk,
exact add,
exact add_assoc,
exact zero,
exact zero_add,
exact add_zero,
exact neg,
exact neg_cancel,
exact add_comm,
exact mul,
exact mul_assoc,
apply one,
apply one_mul,
apply mul_one,
apply distrib,
apply distrib_l,
apply mul_comm
end
definition const (a : ) : := quot.mk (s.r_const a)
theorem add_consts (a b : ) : const a + const b = const (a + b) :=
quot.sound (s.r_add_consts a b)
theorem sub_consts (a b : ) : const a + -const b = const (a - b) := !add_consts
theorem add_half_const (n : +) : const (2 * n)⁻¹ + const (2 * n)⁻¹ = const (n⁻¹) :=
by rewrite [add_consts, pnat.add_halves]
end real