679 lines
22 KiB
Text
679 lines
22 KiB
Text
/-
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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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This construction follows Bishop and Bridges (1985).
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At this point, we no longer proceed constructively: this file makes heavy use of decidability
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and excluded middle.
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-/
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import data.real.basic data.real.order data.rat data.nat logic.axioms.classical
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open -[coercions] rat
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open -[coercions] nat
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local notation 0 := rat.of_num 0
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local notation 1 := rat.of_num 1
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open eq.ops
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local notation 2 := pnat.pos (nat.of_num 2) dec_trivial
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namespace s
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-----------------------------
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-- helper lemmas
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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 neg_add_rewrite {a b : ℚ} : a + -b = -(b + -a) := sorry
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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 rat.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_one_div (q : ℚ) : abs (1 / q) = 1 / abs q := sorry
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theorem div_le_pnat (q : ℚ) (n : ℕ+) (H : q ≥ n⁻¹) : 1 / q ≤ pnat.to_rat n := sorry
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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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-- does this not exist already??
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theorem forall_of_not_exists {A : Type} {P : A → Prop} (H : ¬ ∃ a : A, P a) : ∀ a : A, ¬ P a :=
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take a, assume Ha, H (exists.intro a Ha)
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theorem and_of_not_or {a b : Prop} (H : ¬ (a ∨ b)) : ¬ a ∧ ¬ b :=
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and.intro (assume H', H (or.inl H')) (assume H', H (or.inr H'))
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theorem ne_zero_of_abs_ne_zero {a : ℚ} (H : abs a ≠ 0) : a ≠ 0 :=
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assume Ha, H (Ha⁻¹ ▸ abs_zero)
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-----------------------------
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-- Facts about absolute values of sequences, to define inverse
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definition s_abs (s : seq) : seq := λ n, abs (s n)
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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) :
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∃ 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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------------------------
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-- This section could be cleaned up.
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definition pb {s : seq} (Hs : regular s) (Hpos : pos s) :=
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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) :=
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some (abs_pos_of_nonzero Hs Hsep)
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theorem pb_spec {s : seq} (Hs : regular s) (Hpos : pos s) :
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∀ 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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(if n < (ps Hs H) then 1 / (s ((ps Hs H) * (ps Hs H) * (ps Hs H)))
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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) :
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pb Hs Hpos = ps Hs Hsep := rfl
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theorem s_inv_of_sep_lt_p {s : seq} (Hs : regular s) (Hsep : sep s zero) {n : ℕ+}
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(Hn : n < (ps Hs Hsep)) : s_inv Hs n = 1 / s ((ps Hs Hsep) * (ps Hs Hsep) * (ps Hs Hsep)) :=
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begin
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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_sep_gt_p {s : seq} (Hs : regular s) (Hsep : sep s zero) {n : ℕ+}
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(Hn : n ≥ (ps Hs Hsep)) : s_inv Hs n = 1 / s ((ps Hs Hsep) * (ps Hs Hsep) * n) :=
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begin
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apply eq.trans,
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apply dif_pos Hsep,
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apply dif_neg (pnat.not_lt_of_le Hn)
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end
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theorem s_inv_of_pos_lt_p {s : seq} (Hs : regular s) (Hpos : pos s) {n : ℕ+}
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(Hn : n < (pb Hs Hpos)) : s_inv Hs n = 1 / s ((pb Hs Hpos) * (pb Hs Hpos) * (pb Hs Hpos)) :=
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s_inv_of_sep_lt_p Hs (sep_zero_of_pos Hs Hpos) Hn
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theorem s_inv_of_pos_gt_p {s : seq} (Hs : regular s) (Hpos : pos s) {n : ℕ+}
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(Hn : n ≥ (pb Hs Hpos)) : s_inv Hs n = 1 / s ((pb Hs Hpos) * (pb Hs Hpos) * n) :=
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s_inv_of_sep_gt_p Hs (sep_zero_of_pos Hs Hpos) Hn
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theorem le_ps {s : seq} (Hs : regular s) (Hsep : sep s zero) (n : ℕ+) :
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abs (s_inv Hs n) ≤ (pnat.to_rat (ps Hs Hsep)) :=
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if Hn : n < ps Hs Hsep then
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(begin
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rewrite [(s_inv_of_sep_lt_p Hs Hsep Hn), abs_one_div],
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apply div_le_pnat,
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apply ps_spec,
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apply pnat.mul_le_mul_left
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end)
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else
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(begin
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rewrite [(s_inv_of_sep_gt_p Hs Hsep (le_of_not_gt Hn)), abs_one_div],
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apply div_le_pnat,
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apply ps_spec,
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rewrite pnat_mul_assoc,
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apply pnat.mul_le_mul_right
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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 s_ne_zero_of_ge_p {s : seq} (Hs : regular s) (Hsep : sep s zero) {n : ℕ+}
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(Hn : n ≥ (ps Hs Hsep)) : s n ≠ 0 :=
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begin
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let Hps := ps_spec Hs Hsep,
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apply ne_zero_of_abs_ne_zero,
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apply ne_of_gt,
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apply gt_of_ge_of_gt,
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apply Hps,
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apply Hn,
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apply inv_pos
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end
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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
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s_ne_zero_of_ge_p Hs Hsep !pnat.mul_le_mul_left,
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have Hspn : s ((ps Hs Hsep) * (ps Hs Hsep) * n) ≠ 0, from
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s_ne_zero_of_ge_p Hs Hsep (show (ps Hs Hsep) * (ps Hs Hsep) * n ≥ ps Hs Hsep, by
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rewrite pnat_mul_assoc; apply pnat.mul_le_mul_right),
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have Hspm : s ((ps Hs Hsep) * (ps Hs Hsep) * m) ≠ 0, from
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s_ne_zero_of_ge_p Hs Hsep (show (ps Hs Hsep) * (ps Hs Hsep) * m ≥ ps Hs Hsep, by
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rewrite pnat_mul_assoc; apply pnat.mul_le_mul_right),
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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),
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(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),
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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 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 s_inv_ne_zero {s : seq} (Hs : regular s) (Hsep : sep s zero) (n : ℕ+) : s_inv Hs n ≠ 0 :=
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if H : n ≥ ps Hs Hsep then
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(begin
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rewrite (s_inv_of_sep_gt_p Hs Hsep H),
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apply one_div_ne_zero,
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apply s_ne_zero_of_ge_p,
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apply ple.trans,
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apply H,
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apply pnat.mul_le_mul_left
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end)
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else
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(begin
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rewrite (s_inv_of_sep_lt_p Hs Hsep (lt_of_not_ge H)),
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apply one_div_ne_zero,
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apply s_ne_zero_of_ge_p,
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apply pnat.mul_le_mul_left
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end)
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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 s_inv_ne_zero Hs Hsep _,
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xrewrite [↑smul, ↑one, rat.mul.comm, -(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, begin
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apply ple.trans,
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apply max_left,
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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_left
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end,
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have Hnz' : s (((ps Hs Hsep) * (ps Hs Hsep)) * ((K₂ s (s_inv Hs)) * 2 * n)) ≠ 0, from
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s_ne_zero_of_ge_p Hs Hsep
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(show ps Hs Hsep ≤ ((ps Hs Hsep) * (ps Hs Hsep)) * ((K₂ s (s_inv Hs)) * 2 * n),
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by rewrite *pnat_mul_assoc; apply pnat.mul_le_mul_right),
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xrewrite [(s_inv_of_sep_gt_p Hs Hsep Hp), (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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xrewrite [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))), -*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)
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(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 *,
|
||
apply pos_of_pos_equiv,
|
||
rotate 2,
|
||
apply Hslt,
|
||
rotate_right 1,
|
||
apply add_well_defined,
|
||
rotate 5,
|
||
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 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 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)))
|
||
|
||
theorem s_le_of_equiv_le_left {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Heq : s ≡ t) (Hle : s_le s u) : s_le t u :=
|
||
begin
|
||
rewrite ↑s_le at *,
|
||
apply nonneg_of_nonneg_equiv,
|
||
rotate 2,
|
||
apply add_well_defined,
|
||
rotate 4,
|
||
apply equiv.refl,
|
||
apply neg_well_defined,
|
||
apply Heq,
|
||
repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
|
||
end
|
||
|
||
theorem s_le_of_equiv_le_right {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Heq : t ≡ u) (Hle : s_le s t) : s_le s u :=
|
||
begin
|
||
rewrite ↑s_le at *,
|
||
apply nonneg_of_nonneg_equiv,
|
||
rotate 2,
|
||
apply add_well_defined,
|
||
rotate 4,
|
||
apply Heq,
|
||
apply equiv.refl,
|
||
repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
|
||
end
|
||
|
||
-----------------------------
|
||
|
||
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_le_of_equiv_le_left {s t u : reg_seq} (Heq : requiv s t) (Hle : r_le s u) : r_le t u :=
|
||
s_le_of_equiv_le_left (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) Heq Hle
|
||
|
||
theorem r_le_of_equiv_le_right {s t u : reg_seq} (Heq : requiv t u) (Hle : r_le s t) : r_le s u :=
|
||
s_le_of_equiv_le_right (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) Heq Hle
|
||
|
||
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
|
||
|
||
open [classes] algebra
|
||
definition linear_ordered_field [instance] : 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
|