701 lines
24 KiB
Text
701 lines
24 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
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open rat
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open nat
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open eq.ops pnat classical
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namespace rat_seq
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local postfix ⁻¹ := pnat.inv
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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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intros,
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apply 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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cases bdd_away_of_pos (reg_neg_reg Hs) H' with [N, HN],
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existsi N,
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intro m Hm,
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apply 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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cases H'' with [N, HN],
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existsi N,
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intro m Hm,
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apply 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 abs_well_defined {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
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s_abs s ≡ s_abs t :=
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begin
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rewrite [↑equiv at *],
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intro n,
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rewrite ↑s_abs,
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apply le.trans,
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apply abs_abs_sub_abs_le_abs_sub,
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apply Heq
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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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apply or.inr,
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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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private noncomputable 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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private noncomputable 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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private 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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private 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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noncomputable 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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private 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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private 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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private 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_ge Hn)
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end
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private 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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private 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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private theorem le_ps {s : seq} (Hs : regular s) (Hsep : sep s zero) (n : ℕ+) :
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abs (s_inv Hs n) ≤ (rat_of_pnat (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 (pnat.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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private 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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private 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 pnat.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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cases em (m < ps Hs Hsep) with [Hmlt, Hmlt],
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cases em (n < ps Hs Hsep) with [Hnlt, 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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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_gt 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 le.trans,
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apply mul_le_mul,
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apply Hs,
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rewrite [-(mul_one 1), -(!field.div_mul_div Hsp Hspn), abs_mul],
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apply 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_gt 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', add.comm],
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apply 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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cases em (n < ps Hs Hsep) with [Hnlt, 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_gt 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 le.trans,
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apply mul_le_mul,
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apply Hs,
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rewrite [-(mul_one 1), -(!field.div_mul_div Hspm Hsp), abs_mul],
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apply mul_le_mul,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_gt 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', 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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rewrite [(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_gt Hnlt)),
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(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_gt 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 le.trans,
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apply mul_le_mul,
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apply Hs,
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rewrite [-(mul_one 1), -(!field.div_mul_div Hspm Hspn), abs_mul],
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apply mul_le_mul,
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rewrite -(s_inv_of_sep_gt_p Hs Hsep (pnat.le_of_not_gt 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_gt 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', add.comm],
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apply 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 pnat.le_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 (pnat.lt_of_not_le 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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protected theorem mul_inv {s : seq} (Hs : regular s) (Hsep : sep s zero) :
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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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rewrite [↑smul, ↑one, mul.comm, -(mul_one_div_cancel Hnz),
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-mul_sub_left_distrib, abs_mul],
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apply le.trans,
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apply 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 pnat.le_trans,
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apply pnat.max_left,
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rotate 1,
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apply pnat.le_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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rewrite [(s_inv_of_sep_gt_p Hs Hsep Hp), (division_ring.one_div_one_div Hnz')],
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apply rat.le_trans,
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apply 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 [left_distrib, pnat.mul_comm ((ps Hs Hsep) * (ps Hs Hsep)), *pnat.mul_assoc,
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*(@pnat.inv_mul_eq_mul_inv (K₂ s (s_inv Hs))), -*mul.assoc, *pnat.inv_cancel_left,
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*one_mul, -(pnat.add_halves j)],
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apply 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 pnat.le_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 pnat.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 pnat.le_trans,
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apply pnat.max_right,
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rotate 1,
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apply pnat.le_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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protected theorem inv_mul {s : seq} (Hs : regular s) (Hsep : sep s zero) :
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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 rat_seq.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 *,
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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,
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repeat (assumption | apply reg_neg_reg | apply reg_add_reg | apply zero_is_reg)
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end
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theorem inv_unique {s t : seq} (Hs : regular s) (Ht : regular t) (Hsep : sep s zero)
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(Heq : smul s t ≡ one) : s_inv Hs ≡ t :=
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begin
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apply equiv.trans,
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rotate 3,
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apply equiv.symm,
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apply s_mul_one,
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rotate 1,
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apply equiv.trans,
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rotate 3,
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apply mul_well_defined,
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rotate 4,
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apply equiv.refl,
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apply equiv.symm,
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apply Heq,
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apply equiv.trans,
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rotate 3,
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apply equiv.symm,
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apply s_mul_assoc,
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rotate 3,
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apply equiv.trans,
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rotate 3,
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apply mul_well_defined,
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rotate 4,
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apply rat_seq.inv_mul,
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rotate 1,
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apply equiv.refl,
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apply s_one_mul,
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repeat (assumption | apply reg_inv_reg | apply reg_mul_reg | apply one_is_reg)
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end
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theorem inv_well_defined {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
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s_inv Hs ≡ s_inv Ht :=
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if Hsep : sep s zero then
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(begin
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let Hsept := sep_of_equiv_sep Hs Ht Heq Hsep,
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have Hm : smul t (s_inv Hs) ≡ smul s (s_inv Hs), begin
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apply mul_well_defined,
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repeat (assumption | apply reg_inv_reg),
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apply equiv.symm s t Heq,
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apply equiv.refl
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end,
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apply equiv.symm,
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apply inv_unique,
|
||
rotate 2,
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply Hm,
|
||
apply rat_seq.mul_inv,
|
||
repeat (assumption | apply reg_inv_reg | apply reg_mul_reg),
|
||
apply one_is_reg
|
||
end)
|
||
else
|
||
(assert 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'),
|
||
assert H' : s_inv Ht = zero, from funext (λ n, dif_neg Hsept),
|
||
by rewrite [H', H]; apply equiv.refl)
|
||
|
||
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' := 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' := iff.mp forall_iff_not_exists Hle,
|
||
intro n,
|
||
let Hn := neg_le_neg (le_of_not_gt (Hle' n)),
|
||
rewrite [↑sadd, ↑sneg, add_neg_eq_neg_add_rev],
|
||
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 := iff.mp !not_or_iff_not_and_not 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
|
||
|
||
-----------------------------
|
||
|
||
noncomputable 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
|
||
assert Hz : s_inv (reg_seq.is_reg s) = zero, from funext (λ n, dif_neg H),
|
||
by rewrite Hz; apply 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 :=
|
||
rat_seq.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
|
||
|
||
definition r_abs (s : reg_seq) : reg_seq :=
|
||
reg_seq.mk (s_abs (reg_seq.sq s)) (abs_reg_of_reg (reg_seq.is_reg s))
|
||
|
||
theorem r_abs_well_defined {s t : reg_seq} (H : requiv s t) : requiv (r_abs s) (r_abs t) :=
|
||
abs_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) H
|
||
|
||
end rat_seq
|
||
|
||
namespace real
|
||
open [classes] rat_seq
|
||
|
||
noncomputable protected definition inv (x : ℝ) : ℝ :=
|
||
quot.lift_on x (λ a, quot.mk (rat_seq.r_inv a))
|
||
(λ a b H, quot.sound (rat_seq.r_inv_well_defined H))
|
||
|
||
noncomputable definition real_has_inv [instance] [reducible] [priority real.prio] : has_inv real :=
|
||
has_inv.mk real.inv
|
||
|
||
noncomputable protected definition div (x y : ℝ) : ℝ :=
|
||
x * y⁻¹
|
||
|
||
noncomputable definition real_has_div [instance] [reducible] [priority real.prio] : has_div real :=
|
||
has_div.mk real.div
|
||
|
||
protected theorem le_total (x y : ℝ) : x ≤ y ∨ y ≤ x :=
|
||
quot.induction_on₂ x y (λ s t, rat_seq.r_le_total s t)
|
||
|
||
protected theorem mul_inv_cancel' (x : ℝ) : x ≢ 0 → x * x⁻¹ = 1 :=
|
||
quot.induction_on x (λ s H, quot.sound (rat_seq.r_mul_inv s H))
|
||
|
||
protected theorem inv_mul_cancel' (x : ℝ) : x ≢ 0 → x⁻¹ * x = 1 :=
|
||
by rewrite real.mul_comm; apply real.mul_inv_cancel'
|
||
|
||
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, rat_seq.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)
|
||
|
||
protected theorem mul_inv_cancel (x : ℝ) : x ≠ 0 → x * x⁻¹ = 1 :=
|
||
!sep_is_neq ▸ !real.mul_inv_cancel'
|
||
|
||
protected theorem inv_mul_cancel (x : ℝ) : x ≠ 0 → x⁻¹ * x = 1 :=
|
||
!sep_is_neq ▸ !real.inv_mul_cancel'
|
||
|
||
protected theorem inv_zero : (0 : ℝ)⁻¹ = 0 := quot.sound (rat_seq.r_inv_zero)
|
||
|
||
protected 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 (rat_seq.r_lt_or_equiv_of_le s t H)
|
||
(assume H1, or.inl H1)
|
||
(assume H2, or.inr (quot.sound H2)))
|
||
|
||
protected theorem le_iff_lt_or_eq (x y : ℝ) : x ≤ y ↔ x < y ∨ x = y :=
|
||
iff.intro (real.lt_or_eq_of_le x y) (real.le_of_lt_or_eq x y)
|
||
|
||
noncomputable definition dec_lt : decidable_rel real.lt :=
|
||
begin
|
||
rewrite ↑decidable_rel,
|
||
intros,
|
||
apply prop_decidable
|
||
end
|
||
|
||
protected noncomputable definition discrete_linear_ordered_field [reducible] [trans_instance]:
|
||
discrete_linear_ordered_field ℝ :=
|
||
⦃ discrete_linear_ordered_field, real.comm_ring, real.ordered_ring,
|
||
le_total := real.le_total,
|
||
mul_inv_cancel := real.mul_inv_cancel,
|
||
inv_mul_cancel := real.inv_mul_cancel,
|
||
zero_lt_one := real.zero_lt_one,
|
||
inv_zero := real.inv_zero,
|
||
le_iff_lt_or_eq := real.le_iff_lt_or_eq,
|
||
decidable_lt := dec_lt
|
||
⦄
|
||
|
||
theorem of_rat_zero : of_rat (0:rat) = (0:real) := rfl
|
||
|
||
theorem of_rat_one : of_rat (1:rat) = (1:real) := rfl
|
||
|
||
theorem of_rat_divide (x y : ℚ) : of_rat (x / y) = of_rat x / of_rat y :=
|
||
by_cases
|
||
(assume yz : y = 0, by krewrite [yz, div_zero, +of_rat_zero, div_zero])
|
||
(assume ynz : y ≠ 0,
|
||
have ynz' : of_rat y ≠ 0, from assume yz', ynz (of_rat.inj yz'),
|
||
!eq_div_of_mul_eq ynz' (by krewrite [-of_rat_mul, !div_mul_cancel ynz]))
|
||
|
||
open int
|
||
|
||
theorem of_int_div (x y : ℤ) (H : y ∣ x) : of_int (x / y) = of_int x / of_int y :=
|
||
by rewrite [of_int_eq, rat.of_int_div H, of_rat_divide]
|
||
|
||
theorem of_nat_div (x y : ℕ) (H : y ∣ x) : of_nat (x / y) = of_nat x / of_nat y :=
|
||
by rewrite [of_nat_eq, rat.of_nat_div H, of_rat_divide]
|
||
|
||
/- useful for proving equalities -/
|
||
|
||
theorem eq_zero_of_nonneg_of_forall_lt {x : ℝ} (xnonneg : x ≥ 0) (H : ∀ ε : ℝ, ε > 0 → x < ε) :
|
||
x = 0 :=
|
||
decidable.by_contradiction
|
||
(suppose x ≠ 0,
|
||
have x > 0, from lt_of_le_of_ne xnonneg (ne.symm this),
|
||
have x < x, from H x this,
|
||
show false, from !lt.irrefl this)
|
||
|
||
theorem eq_zero_of_nonneg_of_forall_le {x : ℝ} (xnonneg : x ≥ 0) (H : ∀ ε : ℝ, ε > 0 → x ≤ ε) :
|
||
x = 0 :=
|
||
have ∀ ε : ℝ, ε > 0 → x < ε, from
|
||
take ε, suppose ε > 0,
|
||
assert e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
|
||
assert ε / 2 < ε, from div_two_lt_of_pos `ε > 0`,
|
||
begin apply lt_of_le_of_lt, apply H _ e2pos, apply this end,
|
||
eq_zero_of_nonneg_of_forall_lt xnonneg this
|
||
|
||
theorem eq_zero_of_forall_abs_le {x : ℝ} (H : ∀ ε : ℝ, ε > 0 → abs x ≤ ε) :
|
||
x = 0 :=
|
||
by_contradiction
|
||
(suppose x ≠ 0,
|
||
have abs x = 0, from eq_zero_of_nonneg_of_forall_le !abs_nonneg H,
|
||
show false, from `x ≠ 0` (eq_zero_of_abs_eq_zero this))
|
||
|
||
theorem eq_of_forall_abs_sub_le {x y : ℝ} (H : ∀ ε : ℝ, ε > 0 → abs (x - y) ≤ ε) :
|
||
x = y :=
|
||
have x - y = 0, from eq_zero_of_forall_abs_le H,
|
||
eq_of_sub_eq_zero this
|
||
end real
|