feat(library/data/real): define inverses of reals, prove (classically) that R is a discrete linear ordered field
This commit is contained in:
Rob Lewis
parent
fabdae6d54
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3 changed files with 614 additions and 23 deletions
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@ -1,5 +1,5 @@
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/-
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Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
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Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Robert Y. Lewis
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The real numbers, constructed as equivalence classes of Cauchy sequences of rationals.
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@ -144,19 +144,27 @@ theorem inv_mul_le_inv (p q : ℕ+) : (p * q)⁻¹ ≤ q⁻¹ :=
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apply rat.le_of_lt rat.zero_lt_one
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end
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theorem pnat_mul_le_mul_left' (a b c : ℕ+) (H : a ≤ b) : c * a ≤ c * b := sorry
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theorem pnat_mul_assoc (a b c : ℕ+) : a * b * c = a * (b * c) := sorry
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theorem pnat_mul_comm (a b : ℕ+) : a * b = b * a := sorry
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theorem s_mul_assoc_lemma_3 (a b n : ℕ+) (p : ℚ) :
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p * ((a * n)⁻¹ + (b * n)⁻¹) = p * (a⁻¹ + b⁻¹) * n⁻¹ := sorry
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theorem pnat.mul_le_mul_left (p q : ℕ+) : q ≤ p * q := sorry
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theorem pnat.mul_le_mul_right (p q : ℕ+) : p ≤ p * q := sorry
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theorem one_lt_two : pone < 2 := sorry
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theorem pnat.lt_of_not_le {p q : ℕ+} (H : ¬ p ≤ q) : q < p := sorry
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theorem pnat.inv_cancel (p : ℕ+) : pnat.to_rat p * p⁻¹ = (1 : ℚ) := sorry
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theorem pnat.inv_cancel_right (p : ℕ+) : p⁻¹ * pnat.to_rat p = (1 : ℚ) := sorry
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-------------------------------------
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-- theorems to add to (ordered) field and/or rat
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@ -438,7 +446,6 @@ theorem canon_2_bound_right (s t : seq) (Ht : regular t) (n : ℕ+) :
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abs (t n) ≤ pnat.to_rat (K t) : canon_bound Ht n
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... ≤ pnat.to_rat (K₂ s t) : pnat_le_to_rat_le (!max_right)
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definition sadd (s t : seq) : seq := λ n, (s (2 * n)) + (t (2 * n))
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theorem reg_add_reg {s t : seq} (Hs : regular s) (Ht : regular t) : regular (sadd s t) :=
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565
library/data/real/division.lean
Normal file
565
library/data/real/division.lean
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@ -0,0 +1,565 @@
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import data.real data.rat data.nat logic.axioms.classical --data.encodable
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open -[coercions] rat
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open -[coercions] nat
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open eq.ops
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notation 2 := pnat.pos (nat.of_num 2) dec_trivial
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namespace s
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definition s_abs (s : seq) : seq := λ n, abs (s n)
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theorem nonneg_le_nonneg_of_squares_le {a b : ℚ} (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a ≤ b * b) : a ≤ b :=
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begin
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apply rat.le_of_not_gt,
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intro Hab,
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let Hposa := rat.lt_of_le_of_lt Hb Hab,
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let H' := calc
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b * b ≤ a * b : rat.mul_le_mul_of_nonneg_right (rat.le_of_lt Hab) Hb
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... < a * a : rat.mul_lt_mul_of_pos_left Hab Hposa,
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apply (rat.not_le_of_gt H') H
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end
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theorem abs_sub_square (a b : ℚ) : abs (a - b) * abs (a - b) = a * a + b * b - (1 + 1) * a * b :=
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sorry --begin rewrite [abs_mul_self, *rat.left_distrib, *rat.right_distrib, *one_mul] end
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theorem abs_abs_sub_abs_le_abs_sub (a b : ℚ) : abs (abs a - abs b) ≤ abs (a - b) :=
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begin
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apply nonneg_le_nonneg_of_squares_le,
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repeat apply abs_nonneg,
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rewrite [*(abs_sub_square _ _), *abs_abs, *abs_mul_self],
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apply sub_le_sub_left,
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rewrite *rat.mul.assoc,
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apply rat.mul_le_mul_of_nonneg_left,
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rewrite -abs_mul,
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apply le_abs_self,
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apply trivial
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end
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theorem abs_reg_of_reg {s : seq} (Hs : regular s) : regular (s_abs s) :=
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begin
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rewrite ↑regular at *,
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intros,
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apply rat.le.trans,
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apply abs_abs_sub_abs_le_abs_sub,
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apply Hs
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end
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theorem abs_pos_of_nonzero {s : seq} (Hs : regular s) (Hnz : sep s zero) : ∃ N : ℕ+, ∀ m : ℕ+, m ≥ N → abs (s m) ≥ N⁻¹ :=
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begin
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rewrite [↑sep at Hnz, ↑s_lt at Hnz],
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apply or.elim Hnz,
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intro Hnz1,
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have H' : pos (sneg s), begin
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apply pos_of_pos_equiv,
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rotate 2,
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apply Hnz1,
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rotate 1,
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apply s_zero_add,
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repeat (assumption | apply reg_add_reg | apply reg_neg_reg | apply zero_is_reg)
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end,
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let H'' := bdd_away_of_pos (reg_neg_reg Hs) H',
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apply exists.elim H'',
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intro N HN,
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existsi N,
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intro m Hm,
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apply rat.le.trans,
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apply HN m Hm,
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rewrite ↑sneg,
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apply neg_le_abs_self,
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intro Hnz2,
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let H' := pos_of_pos_equiv (reg_add_reg Hs (reg_neg_reg zero_is_reg)) (s_add_zero s Hs) Hnz2,
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let H'' := bdd_away_of_pos Hs H',
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apply exists.elim H'',
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intro N HN,
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existsi N,
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intro m Hm,
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apply rat.le.trans,
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apply HN m Hm,
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apply le_abs_self
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end
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theorem sep_zero_of_pos {s : seq} (Hs : regular s) (Hpos : pos s) : sep s zero :=
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begin
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rewrite ↑sep,
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apply or.inr,
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rewrite ↑s_lt,
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apply pos_of_pos_equiv,
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rotate 2,
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apply Hpos,
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apply Hs,
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apply equiv.symm,
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apply s_sub_zero Hs
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end
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definition pb {s : seq} (Hs : regular s) (Hpos : pos s) := some (abs_pos_of_nonzero Hs (sep_zero_of_pos Hs Hpos))
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definition ps {s : seq} (Hs : regular s) (Hsep : sep s zero) := some (abs_pos_of_nonzero Hs Hsep)
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theorem pb_spec {s : seq} (Hs : regular s) (Hpos : pos s) : ∀ m : ℕ+, m ≥ (pb Hs Hpos) → abs (s m) ≥ (pb Hs Hpos)⁻¹ :=
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some_spec (abs_pos_of_nonzero Hs (sep_zero_of_pos Hs Hpos))
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theorem ps_spec {s : seq} (Hs : regular s) (Hsep : sep s zero) :
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∀ m : ℕ+, m ≥ (ps Hs Hsep) → abs (s m) ≥ (ps Hs Hsep)⁻¹ :=
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some_spec (abs_pos_of_nonzero Hs Hsep)
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definition s_inv {s : seq} (Hs : regular s) (n : ℕ+) : ℚ :=
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if H : sep s zero then
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--let N := some (abs_pos_of_nonzero Hs H) in
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(if n < (ps Hs H) then 1 / (s ((ps Hs H) * (ps Hs H) * (ps Hs H))) else 1 / (s ((ps Hs H) * (ps Hs H) * n)))
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else 0
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theorem peq {s : seq} (Hsep : sep s zero) (Hpos : pos s) (Hs : regular s) : pb Hs Hpos = ps Hs Hsep := sorry
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theorem le_ps {s : seq} (Hs : regular s) (Hsep : sep s zero) (n : ℕ+) : abs (s_inv Hs n) ≤ (pnat.to_rat (ps Hs Hsep)) := sorry
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theorem s_inv_of_sep_lt_p {s : seq} (Hs : regular s) (Hsep : sep s zero) {n : ℕ+} (Hn : n < (ps Hs Hsep)) :
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s_inv Hs n = 1 / s ((ps Hs Hsep) * (ps Hs Hsep) * (ps Hs Hsep)) := sorry
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theorem s_inv_of_sep_gt_p {s : seq} (Hs : regular s) (Hsep : sep s zero) {n : ℕ+} (Hn : n ≥ (ps Hs Hsep)) :
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s_inv Hs n = 1 / s ((ps Hs Hsep) * (ps Hs Hsep) * n) := sorry
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theorem s_inv_of_pos_lt_p {s : seq} (Hs : regular s) (Hpos : pos s) {n : ℕ+} (Hn : n < (pb Hs Hpos)) :
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s_inv Hs n = 1 / s ((pb Hs Hpos) * (pb Hs Hpos) * (pb Hs Hpos)) :=
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begin
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let Hsep := sep_zero_of_pos Hs Hpos,
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apply eq.trans,
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apply dif_pos Hsep,
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apply dif_pos Hn
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end
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theorem s_inv_of_pos_gt_p {s : seq} (Hs : regular s) (Hpos : pos s) {n : ℕ+} (Hn : n ≥ (pb Hs Hpos)) :
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s_inv Hs n = 1 / s ((pb Hs Hpos) * (pb Hs Hpos) * n) :=
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begin
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let Hsep := sep_zero_of_pos Hs Hpos,
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apply eq.trans,
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apply dif_pos Hsep,
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apply eq.trans,
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rewrite *(peq Hsep Hpos) at *,
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apply dif_neg (pnat.not_lt_of_le Hn),
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rewrite *(peq Hsep Hpos)
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end
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theorem s_inv_zero : s_inv zero_is_reg = zero :=
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funext (λ n, dif_neg (!not_sep_self))
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theorem s_inv_of_zero' {s : seq} (Hs : regular s) (Hz : ¬ sep s zero) (n : ℕ+) : s_inv Hs n = 0 :=
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dif_neg Hz
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theorem s_inv_of_zero {s : seq} (Hs : regular s) (Hz : ¬ sep s zero) : s_inv Hs = zero :=
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begin
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apply funext,
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intro n,
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apply s_inv_of_zero' Hs Hz n
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end
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theorem pnat_cancel' (n m : ℕ+) : (n * n * m)⁻¹ * (pnat.to_rat n * pnat.to_rat n) = m⁻¹ := sorry
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theorem reg_inv_reg {s : seq} (Hs : regular s) (Hsep : sep s zero) : regular (s_inv Hs) :=
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begin
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rewrite ↑regular,
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intros,
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have Hsp : s ((ps Hs Hsep) * (ps Hs Hsep) * (ps Hs Hsep)) ≠ 0, from sorry,
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have Hspn : s ((ps Hs Hsep) * (ps Hs Hsep) * n) ≠ 0, from sorry,
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have Hspm : s ((ps Hs Hsep) * (ps Hs Hsep) * m) ≠ 0, from sorry,
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apply @decidable.cases_on (m < (ps Hs Hsep)) _ _,
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intro Hmlt,
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apply @decidable.cases_on (n < (ps Hs Hsep)) _ _,
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intro Hnlt,
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rewrite [(s_inv_of_sep_lt_p Hs Hsep Hmlt), (s_inv_of_sep_lt_p Hs Hsep Hnlt)],
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rewrite [sub_self, abs_zero],
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apply add_invs_nonneg,
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intro Hnlt,
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rewrite [(s_inv_of_sep_lt_p Hs Hsep Hmlt), (s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hnlt))],
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rewrite [(div_sub_div Hsp Hspn), div_eq_mul_one_div, *abs_mul, *mul_one, *one_mul],
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apply rat.le.trans,
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apply rat.mul_le_mul,
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apply Hs,
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xrewrite [-(mul_one 1), -(div_mul_div Hsp Hspn), abs_mul],
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apply rat.mul_le_mul,
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rewrite -(s_inv_of_sep_lt_p Hs Hsep Hmlt),
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apply le_ps Hs Hsep,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hnlt)),
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apply le_ps Hs Hsep,
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apply abs_nonneg,
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apply le_of_lt !rat_of_pnat_is_pos,
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apply abs_nonneg,
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apply add_invs_nonneg,
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rewrite [right_distrib, *pnat_cancel', rat.add.comm],
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apply rat.add_le_add_right,
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apply inv_ge_of_le,
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apply pnat.le_of_lt,
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apply Hmlt,
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intro Hmlt,
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apply @decidable.cases_on (n < (ps Hs Hsep)) _ _,
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intro Hnlt,
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rewrite [(s_inv_of_sep_lt_p Hs Hsep Hnlt), (s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hmlt))],
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rewrite [(div_sub_div Hspm Hsp), div_eq_mul_one_div, *abs_mul, *mul_one, *one_mul],
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apply rat.le.trans,
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apply rat.mul_le_mul,
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apply Hs,
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xrewrite [-(mul_one 1), -(div_mul_div Hspm Hsp), abs_mul],
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apply rat.mul_le_mul,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hmlt)),
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apply le_ps Hs Hsep,
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rewrite -(s_inv_of_sep_lt_p Hs Hsep Hnlt),
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apply le_ps Hs Hsep,
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apply abs_nonneg,
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apply le_of_lt !rat_of_pnat_is_pos,
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apply abs_nonneg,
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apply add_invs_nonneg,
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rewrite [right_distrib, *pnat_cancel', rat.add.comm],
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apply rat.add_le_add_left,
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apply inv_ge_of_le,
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apply pnat.le_of_lt,
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apply Hnlt,
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intro Hnlt,
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rewrite [(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hnlt)),
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(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hmlt))],
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rewrite [(div_sub_div Hspm Hspn), div_eq_mul_one_div, abs_mul, *one_mul, *mul_one],
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apply rat.le.trans,
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apply rat.mul_le_mul,
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apply Hs,
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xrewrite [-(mul_one 1), -(div_mul_div Hspm Hspn), abs_mul],
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apply rat.mul_le_mul,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hmlt)),
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apply le_ps Hs Hsep,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_lt Hnlt)),
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apply le_ps Hs Hsep,
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apply abs_nonneg,
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apply le_of_lt !rat_of_pnat_is_pos,
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apply abs_nonneg,
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apply add_invs_nonneg,
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rewrite [right_distrib, *pnat_cancel', rat.add.comm],
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apply rat.le.refl
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end
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theorem one_rewrite : 1 = of_num 1 := rfl
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theorem fun_rewrite {s : seq} (Hs : regular s) : (λ a : ℕ+, s_inv Hs a) = s_inv Hs := rfl
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theorem mul_inv {s : seq} (Hs : regular s) (Hsep : sep s zero) : smul s (s_inv Hs) ≡ one :=
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begin
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let Rsi := reg_inv_reg Hs Hsep,
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let Rssi := reg_mul_reg Hs Rsi,
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apply eq_of_bdd Rssi one_is_reg,
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intros,
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existsi max (ps Hs Hsep) j,
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intro n Hn,
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have Hnz : s_inv Hs ((K₂ s (s_inv Hs)) * 2 * n) ≠ 0, from sorry,
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rewrite [↑smul, ↑one, rat.mul.comm, one_rewrite, -(mul_one_div_cancel Hnz),
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-rat.mul_sub_left_distrib, abs_mul],
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apply rat.le.trans,
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apply rat.mul_le_mul_of_nonneg_right,
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apply canon_2_bound_right s,
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apply Rsi,
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apply abs_nonneg,
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have Hp : (K₂ s (s_inv Hs)) * 2 * n ≥ ps Hs Hsep, from sorry,
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have Hnz' : s (((ps Hs Hsep) * (ps Hs Hsep)) * ((K₂ s (s_inv Hs)) * 2 * n)) ≠ 0, from sorry,
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rewrite [(s_inv_of_sep_gt_p Hs Hsep Hp), *one_rewrite, (div_div Hnz')],
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apply rat.le.trans,
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apply rat.mul_le_mul_of_nonneg_left,
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apply Hs,
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apply le_of_lt,
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apply rat_of_pnat_is_pos,
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rewrite [rat.mul.left_distrib, pnat_mul_comm ((ps Hs Hsep) * (ps Hs Hsep)), *pnat_mul_assoc,
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*(@pnat_div_helper (K₂ s (s_inv Hs))), fun_rewrite, -*rat.mul.assoc, *pnat.inv_cancel,
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*one_mul, -(padd_halves j)],
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apply rat.add_le_add,
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apply inv_ge_of_le,
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apply pnat_mul_le_mul_left',
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apply ple.trans,
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rotate 1,
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apply Hn,
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rotate_right 1,
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apply max_right,
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apply inv_ge_of_le,
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apply pnat_mul_le_mul_left',
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apply ple.trans,
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apply max_right,
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rotate 1,
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apply ple.trans,
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apply Hn,
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apply pnat.mul_le_mul_right
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end
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theorem inv_mul {s : seq} (Hs : regular s) (Hsep : sep s zero) : smul (s_inv Hs) s ≡ one :=
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begin
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apply equiv.trans,
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rotate 3,
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apply s_mul_comm,
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apply mul_inv,
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repeat (assumption | apply reg_mul_reg | apply reg_inv_reg | apply zero_is_reg)
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end
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theorem sep_of_equiv_sep {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) (Hsep : sep s zero) : sep t zero :=
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begin
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apply or.elim Hsep,
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intro Hslt,
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apply or.inl,
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rewrite ↑s_lt at *,
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apply pos_of_pos_equiv,
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rotate 2,
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apply Hslt,
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rotate_right 1,
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apply add_well_defined,
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rotate 4,
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apply equiv.refl,
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apply neg_well_defined,
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apply Heq,
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intro Hslt,
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apply or.inr,
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rewrite ↑s_lt at *,
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apply pos_of_pos_equiv,
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rotate 2,
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apply Hslt,
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rotate_right 1,
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apply add_well_defined,
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rotate 5,
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apply equiv.refl,
|
||||
repeat (assumption | apply reg_neg_reg | apply reg_add_reg | apply zero_is_reg)
|
||||
end
|
||||
|
||||
theorem inv_unique {s t : seq} (Hs : regular s) (Ht : regular t) (Hsep : sep s zero)
|
||||
(Heq : smul s t ≡ one) : s_inv Hs ≡ t :=
|
||||
begin
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply equiv.symm,
|
||||
apply s_mul_one,
|
||||
rotate 1,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply mul_well_defined,
|
||||
rotate 4,
|
||||
apply equiv.refl,
|
||||
apply equiv.symm,
|
||||
apply Heq,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply equiv.symm,
|
||||
apply s_mul_assoc,
|
||||
rotate 3,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply mul_well_defined,
|
||||
rotate 4,
|
||||
apply inv_mul,
|
||||
rotate 1,
|
||||
apply equiv.refl,
|
||||
apply s_one_mul,
|
||||
repeat (assumption | apply reg_inv_reg | apply reg_mul_reg | apply one_is_reg)
|
||||
end
|
||||
|
||||
theorem inv_well_defined {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
|
||||
s_inv Hs ≡ s_inv Ht :=
|
||||
if Hsep : sep s zero then
|
||||
(begin
|
||||
let Hsept := sep_of_equiv_sep Hs Ht Heq Hsep,
|
||||
have Hm : smul t (s_inv Hs) ≡ smul s (s_inv Hs), begin
|
||||
apply mul_well_defined,
|
||||
repeat (assumption | apply reg_inv_reg),
|
||||
apply equiv.symm s t Heq,
|
||||
apply equiv.refl
|
||||
end,
|
||||
apply equiv.symm,
|
||||
apply inv_unique,
|
||||
rotate 2,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply Hm,
|
||||
apply mul_inv,
|
||||
repeat (assumption | apply reg_inv_reg | apply reg_mul_reg),
|
||||
apply one_is_reg
|
||||
end)
|
||||
else
|
||||
(have H : s_inv Hs = zero, from funext (λ n, dif_neg Hsep),
|
||||
have Hsept : ¬ sep t zero, from
|
||||
assume H', Hsep (sep_of_equiv_sep Ht Hs (equiv.symm _ _ Heq) H'),
|
||||
have H' : s_inv Ht = zero, from funext (λ n, dif_neg Hsept),
|
||||
H'⁻¹ ▸ (H⁻¹ ▸ equiv.refl zero))
|
||||
|
||||
theorem s_neg_neg {s : seq} : sneg (sneg s) ≡ s :=
|
||||
begin
|
||||
rewrite [↑equiv, ↑sneg],
|
||||
intro n,
|
||||
rewrite [neg_neg, sub_self, abs_zero],
|
||||
apply add_invs_nonneg
|
||||
end
|
||||
|
||||
theorem s_neg_sub {s t : seq} (Hs : regular s) (Ht : regular t) :
|
||||
sneg (sadd s (sneg t)) ≡ sadd t (sneg s) :=
|
||||
begin
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply s_neg_add_eq_s_add_neg,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply add_well_defined,
|
||||
rotate 4,
|
||||
apply equiv.refl,
|
||||
apply s_neg_neg,
|
||||
apply s_add_comm,
|
||||
repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
|
||||
end
|
||||
|
||||
theorem forall_of_not_exists {A : Type} {P : A → Prop} (H : ¬ ∃ a : A, P a) : ∀ a : A, ¬ P a :=
|
||||
take a, assume Ha, H (exists.intro a Ha)
|
||||
|
||||
theorem s_le_total {s t : seq} (Hs : regular s) (Ht : regular t) : s_le s t ∨ s_le t s :=
|
||||
if H : s_le s t then or.inl H else or.inr begin
|
||||
rewrite [↑s_le at *],
|
||||
have H' : ∃ n : ℕ+, -n⁻¹ > sadd t (sneg s) n, begin
|
||||
apply by_contradiction,
|
||||
intro Hex,
|
||||
have Hex' : ∀ n : ℕ+, -n⁻¹ ≤ sadd t (sneg s) n, begin
|
||||
intro m,
|
||||
apply by_contradiction,
|
||||
intro Hm,
|
||||
let Hm' := rat.lt_of_not_ge Hm,
|
||||
let Hex'' := exists.intro m Hm',
|
||||
apply Hex Hex''
|
||||
end,
|
||||
apply H Hex'
|
||||
end,
|
||||
eapply exists.elim H',
|
||||
intro m Hm,
|
||||
let Hm' := neg_lt_neg Hm,
|
||||
rewrite neg_neg at Hm',
|
||||
apply s_nonneg_of_pos,
|
||||
rotate 1,
|
||||
apply pos_of_pos_equiv,
|
||||
rotate 1,
|
||||
apply s_neg_sub,
|
||||
rotate 2,
|
||||
rewrite [↑pos, ↑sneg],
|
||||
existsi m,
|
||||
apply Hm',
|
||||
repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
|
||||
end
|
||||
|
||||
theorem neg_add_rewrite {a b : ℚ} : a + -b = -(b + -a) := sorry
|
||||
|
||||
theorem and_of_not_or {a b : Prop} (H : ¬ (a ∨ b)) : ¬ a ∧ ¬ b :=
|
||||
and.intro (assume H', H (or.inl H')) (assume H', H (or.inr H'))
|
||||
|
||||
theorem s_le_of_not_lt {s t : seq} (Hle : ¬ s_lt s t) : s_le t s :=
|
||||
begin
|
||||
rewrite [↑s_le, ↑nonneg, ↑s_lt at Hle, ↑pos at Hle],
|
||||
let Hle' := forall_of_not_exists Hle,
|
||||
intro n,
|
||||
let Hn := neg_le_neg (rat.le_of_not_gt (Hle' n)),
|
||||
rewrite [↑sadd, ↑sneg, neg_add_rewrite],
|
||||
apply Hn
|
||||
end
|
||||
|
||||
theorem sep_of_nequiv {s t : seq} (Hs : regular s) (Ht : regular t) (Hneq : ¬ equiv s t) :
|
||||
sep s t :=
|
||||
begin
|
||||
rewrite ↑sep,
|
||||
apply by_contradiction,
|
||||
intro Hnor,
|
||||
let Hand := and_of_not_or Hnor,
|
||||
let Hle1 := s_le_of_not_lt (and.left Hand),
|
||||
let Hle2 := s_le_of_not_lt (and.right Hand),
|
||||
apply Hneq (equiv_of_le_of_ge Hs Ht Hle2 Hle1)
|
||||
end
|
||||
|
||||
theorem s_zero_inv_equiv_zero : s_inv zero_is_reg ≡ zero :=
|
||||
by rewrite s_inv_zero; apply equiv.refl
|
||||
|
||||
theorem lt_or_equiv_of_le {s t : seq} (Hs : regular s) (Ht : regular t) (Hle : s_le s t) :
|
||||
s_lt s t ∨ s ≡ t :=
|
||||
if H : s ≡ t then or.inr H else
|
||||
or.inl (lt_of_le_and_sep Hs Ht (and.intro Hle (sep_of_nequiv Hs Ht H)))
|
||||
|
||||
-----------------------------
|
||||
|
||||
definition r_inv (s : reg_seq) : reg_seq := reg_seq.mk (s_inv (reg_seq.is_reg s))
|
||||
(if H : sep (reg_seq.sq s) zero then reg_inv_reg (reg_seq.is_reg s) H else
|
||||
have Hz : s_inv (reg_seq.is_reg s) = zero, from funext (λ n, dif_neg H), Hz⁻¹ ▸ zero_is_reg)
|
||||
|
||||
theorem r_inv_zero : requiv (r_inv r_zero) r_zero :=
|
||||
s_zero_inv_equiv_zero
|
||||
|
||||
|
||||
theorem r_inv_well_defined {s t : reg_seq} (H : requiv s t) : requiv (r_inv s) (r_inv t) :=
|
||||
inv_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) H
|
||||
|
||||
theorem r_le_total (s t : reg_seq) : r_le s t ∨ r_le t s :=
|
||||
s_le_total (reg_seq.is_reg s) (reg_seq.is_reg t)
|
||||
|
||||
theorem r_mul_inv (s : reg_seq) (Hsep : r_sep s r_zero) : requiv (s * (r_inv s)) r_one :=
|
||||
mul_inv (reg_seq.is_reg s) Hsep
|
||||
|
||||
theorem r_sep_of_nequiv (s t : reg_seq) (Hneq : ¬ requiv s t) : r_sep s t :=
|
||||
sep_of_nequiv (reg_seq.is_reg s) (reg_seq.is_reg t) Hneq
|
||||
|
||||
theorem r_lt_or_equiv_of_le (s t : reg_seq) (Hle : r_le s t) : r_lt s t ∨ requiv s t :=
|
||||
lt_or_equiv_of_le (reg_seq.is_reg s) (reg_seq.is_reg t) Hle
|
||||
|
||||
/-theorem r_inv_mul (s : reg_seq) (Hsep : r_sep s r_zero) : requiv ((r_inv s) * s) r_one :=
|
||||
inv_mul (reg_seq.is_reg s) Hsep-/
|
||||
|
||||
end s
|
||||
|
||||
|
||||
namespace real
|
||||
|
||||
definition inv (x : ℝ) : ℝ := quot.lift_on x (λ a, quot.mk (s.r_inv a))
|
||||
(λ a b H, quot.sound (s.r_inv_well_defined H))
|
||||
postfix `⁻¹` := inv
|
||||
|
||||
theorem le_total (x y : ℝ) : x ≤ y ∨ y ≤ x :=
|
||||
quot.induction_on₂ x y (λ s t, s.r_le_total s t)
|
||||
|
||||
theorem mul_inv' (x : ℝ) : x ≢ zero → x * x⁻¹ = one :=
|
||||
quot.induction_on x (λ s H, quot.sound (s.r_mul_inv s H))
|
||||
|
||||
theorem inv_mul' (x : ℝ) : x ≢ zero → x⁻¹ * x = one :=
|
||||
by rewrite real.mul_comm; apply mul_inv'
|
||||
|
||||
theorem neq_of_sep {x y : ℝ} (H : x ≢ y) : ¬ x = y :=
|
||||
assume Heq, !not_sep_self (Heq ▸ H)
|
||||
|
||||
theorem sep_of_neq {x y : ℝ} : ¬ x = y → x ≢ y :=
|
||||
quot.induction_on₂ x y (λ s t H, s.r_sep_of_nequiv s t (assume Heq, H (quot.sound Heq)))
|
||||
|
||||
theorem sep_is_neq (x y : ℝ) : (x ≢ y) = (¬ x = y) :=
|
||||
propext (iff.intro neq_of_sep sep_of_neq)
|
||||
|
||||
theorem mul_inv (x : ℝ) : x ≠ zero → x * x⁻¹ = one := !sep_is_neq ▸ !mul_inv'
|
||||
|
||||
theorem inv_mul (x : ℝ) : x ≠ zero → x⁻¹ * x = one := !sep_is_neq ▸ !inv_mul'
|
||||
|
||||
theorem inv_zero : zero⁻¹ = zero := quot.sound (s.r_inv_zero)
|
||||
|
||||
theorem lt_or_eq_of_le (x y : ℝ) : x ≤ y → x < y ∨ x = y :=
|
||||
quot.induction_on₂ x y (λ s t H, or.elim (s.r_lt_or_equiv_of_le s t H)
|
||||
(assume H1, or.inl H1)
|
||||
(assume H2, or.inr (quot.sound H2)))
|
||||
|
||||
theorem le_iff_lt_or_eq (x y : ℝ) : x ≤ y ↔ x < y ∨ x = y :=
|
||||
iff.intro (lt_or_eq_of_le x y) (le_of_lt_or_eq x y)
|
||||
|
||||
theorem dec_lt : decidable_rel lt :=
|
||||
begin
|
||||
rewrite ↑decidable_rel,
|
||||
intros,
|
||||
apply prop_decidable
|
||||
end
|
||||
|
||||
definition linear_ordered_field : algebra.discrete_linear_ordered_field ℝ :=
|
||||
⦃ algebra.discrete_linear_ordered_field, comm_ring, ordered_ring,
|
||||
le_total := le_total,
|
||||
mul_inv_cancel := mul_inv,
|
||||
inv_mul_cancel := inv_mul,
|
||||
zero_lt_one := zero_lt_one,
|
||||
inv_zero := inv_zero,
|
||||
le_iff_lt_or_eq := le_iff_lt_or_eq,
|
||||
decidable_lt := dec_lt
|
||||
⦄
|
||||
|
||||
end real
|
||||
|
|
@ -1,5 +1,5 @@
|
|||
/-
|
||||
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
|
||||
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.
|
||||
|
|
@ -220,6 +220,15 @@ theorem zero_nonneg : nonneg zero :=
|
|||
apply inv_pos
|
||||
end
|
||||
|
||||
theorem s_zero_lt_one : s_lt zero one :=
|
||||
begin
|
||||
rewrite [↑s_lt, ↑zero, ↑sadd, ↑sneg, ↑one, neg_zero, add_zero, ↑pos],
|
||||
fapply exists.intro,
|
||||
exact 2,
|
||||
apply inv_lt_one_of_gt,
|
||||
apply one_lt_two
|
||||
end
|
||||
|
||||
theorem le.refl {s : seq} (Hs : regular s) : s_le s s :=
|
||||
begin
|
||||
apply nonneg_of_nonneg_equiv,
|
||||
|
|
@ -898,7 +907,7 @@ theorem s_lt_of_le_of_lt {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : r
|
|||
--------
|
||||
-- These are currently needed for lin_ordered_comm_ring.
|
||||
|
||||
theorem le_or_ge {s t : seq} (Hs : regular s) (Ht : regular t) : s_le s t ∨ s_le t s :=
|
||||
/-theorem le_or_ge {s t : seq} (Hs : regular s) (Ht : regular t) : s_le s t ∨ s_le t s :=
|
||||
sorry
|
||||
|
||||
theorem lt_or_equiv_of_le {s t : seq} (Hs : regular s) (Ht : regular t) (Hle : s_le s t) :
|
||||
|
|
@ -909,7 +918,7 @@ theorem lt_or_equiv_of_le {s t : seq} (Hs : regular s) (Ht : regular t) (Hle : s
|
|||
|
||||
theorem le_iff_lt_or_equiv {s t : seq} (Hs : regular s) (Ht : regular t) :
|
||||
s_le s t ↔ s_lt s t ∨ s ≡ t :=
|
||||
iff.intro (lt_or_equiv_of_le Hs Ht) (le_of_lt_or_equiv Hs Ht)
|
||||
iff.intro (lt_or_equiv_of_le Hs Ht) (le_of_lt_or_equiv Hs Ht)-/
|
||||
|
||||
|
||||
|
||||
|
|
@ -984,13 +993,18 @@ theorem r_add_lt_add_left (s t : reg_seq) (H : r_lt s t) (u : reg_seq) : r_lt (u
|
|||
theorem r_add_lt_add_left_var (s t u : reg_seq) (H : r_lt s t) : r_lt (u + s) (u + t) :=
|
||||
r_add_lt_add_left s t H u
|
||||
|
||||
theorem r_zero_lt_one : r_lt r_zero r_one := s_zero_lt_one
|
||||
|
||||
theorem r_le_of_lt_or_eq (s t : reg_seq) (H : r_lt s t ∨ requiv s t) : r_le s t :=
|
||||
le_of_lt_or_equiv (reg_seq.is_reg s) (reg_seq.is_reg t) H
|
||||
|
||||
----------
|
||||
-- earlier versions are sorried
|
||||
theorem r_le_iff_lt_or_equiv (s t : reg_seq) : r_le s t ↔ r_lt s t ∨ requiv s t :=
|
||||
/-theorem r_le_iff_lt_or_equiv (s t : reg_seq) : r_le s t ↔ r_lt s t ∨ requiv s t :=
|
||||
le_iff_lt_or_equiv (reg_seq.is_reg s) (reg_seq.is_reg t)
|
||||
|
||||
theorem r_le_or_ge (s t : reg_seq) : r_le s t ∨ r_le t s :=
|
||||
le_or_ge (reg_seq.is_reg s) (reg_seq.is_reg t)
|
||||
le_or_ge (reg_seq.is_reg s) (reg_seq.is_reg t)-/
|
||||
-----------
|
||||
|
||||
end s
|
||||
|
|
@ -1052,9 +1066,21 @@ theorem add_lt_add_left_var (x y z : ℝ) : x < y → z + x < z + y :=
|
|||
theorem add_lt_add_left (x y : ℝ) : x < y → ∀ z : ℝ, z + x < z + y :=
|
||||
take H z, add_lt_add_left_var x y z H
|
||||
|
||||
theorem zero_lt_one : zero < one := s.r_zero_lt_one
|
||||
|
||||
theorem le_of_lt_or_eq (x y : ℝ) : x < y ∨ x = y → x ≤ y :=
|
||||
(quot.induction_on₂ x y (λ s t H, or.elim H (take H', begin
|
||||
apply s.r_le_of_lt_or_eq,
|
||||
apply or.inl H'
|
||||
end)
|
||||
(take H', begin
|
||||
apply s.r_le_of_lt_or_eq,
|
||||
apply (or.inr (quot.exact H'))
|
||||
end)))
|
||||
|
||||
----------
|
||||
-- earlier versions are sorried
|
||||
theorem le_iff_lt_or_eq (x y : ℝ) : x ≤ y ↔ x < y ∨ x = y :=
|
||||
/-theorem le_iff_lt_or_eq (x y : ℝ) : x ≤ y ↔ x < y ∨ x = y :=
|
||||
iff.intro
|
||||
(quot.induction_on₂ x y (λ s t H, or.elim (iff.mp ((s.r_le_iff_lt_or_equiv s t)) H)
|
||||
(take H1, or.inl H1)
|
||||
|
|
@ -1069,10 +1095,10 @@ theorem le_iff_lt_or_eq (x y : ℝ) : x ≤ y ↔ x < y ∨ x = y :=
|
|||
end)))
|
||||
|
||||
theorem le_or_ge (x y : ℝ) : x ≤ y ∨ y ≤ x :=
|
||||
quot.induction_on₂ x y (λ s t, s.r_le_or_ge s t)
|
||||
quot.induction_on₂ x y (λ s t, s.r_le_or_ge s t)-/
|
||||
-------------
|
||||
|
||||
theorem ordered_ring : algebra.ordered_ring ℝ :=
|
||||
definition ordered_ring : algebra.ordered_ring ℝ :=
|
||||
⦃ algebra.ordered_ring, comm_ring,
|
||||
le_refl := le.refl,
|
||||
le_trans := le.trans,
|
||||
|
|
@ -1090,7 +1116,7 @@ theorem ordered_ring : algebra.ordered_ring ℝ :=
|
|||
|
||||
-----------------------------------
|
||||
--- here is where classical logic comes in
|
||||
theorem sep_is_eq (x y : ℝ) : x ≢ y = ¬ (x = y) := sorry
|
||||
--theorem sep_is_eq (x y : ℝ) : x ≢ y = ¬ (x = y) := sorry
|
||||
|
||||
/-theorem sep_is_eq (x y : ℝ) : x ≢ y = ¬ (x = y) := begin
|
||||
apply propext,
|
||||
|
|
@ -1102,19 +1128,12 @@ theorem sep_is_eq (x y : ℝ) : x ≢ y = ¬ (x = y) := sorry
|
|||
intro Hneq,
|
||||
end-/
|
||||
|
||||
|
||||
/-theorem linear_ordered_comm_ring : algebra.linear_ordered_comm_ring ℝ :=
|
||||
⦃ algebra.linear_ordered_comm_ring, comm_ring,
|
||||
le_refl := le.refl,
|
||||
le_trans := le.trans,
|
||||
le_antisymm := eq_of_le_of_ge,
|
||||
lt_iff_le_and_ne := λ x y, (sep_is_eq x y) ▸ (lt_iff_le_and_sep x y),
|
||||
le_iff_lt_or_eq := le_iff_lt_or_eq,
|
||||
add_le_add_left := add_le_add_of_le_right,
|
||||
mul_pos := mul_gt_zero_of_gt_zero,
|
||||
mul_nonneg := mul_ge_zero_of_ge_zero,
|
||||
/-definition linear_ordered_comm_ring : algebra.linear_ordered_comm_ring ℝ :=
|
||||
⦃ algebra.linear_ordered_comm_ring, ordered_ring, comm_ring,
|
||||
zero_lt_one := zero_lt_one,
|
||||
le_total := le_or_ge,
|
||||
zero_ne_one := zero_ne_one
|
||||
le_iff_lt_or_eq := le_iff_lt_or_eq
|
||||
⦄-/
|
||||
|
||||
|
||||
end real
|
||||
|
|
|
|||
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Reference in a new issue