feat(library/data/real): fill in sorrys in proof that R is l.o. field
This commit is contained in:
Rob Lewis
parent
9843e61583
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1 changed files with 147 additions and 56 deletions
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@ -1,16 +1,29 @@
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import data.real data.rat data.nat logic.axioms.classical --data.encodable
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/-
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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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open eq.ops
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notation 2 := pnat.pos (nat.of_num 2) dec_trivial
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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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definition s_abs (s : seq) : seq := λ n, abs (s n)
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-----------------------------
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-- helper lemmas
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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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theorem nonneg_le_nonneg_of_squares_le {a b : ℚ} (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a ≤ b * b) :
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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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@ -24,6 +37,8 @@ theorem nonneg_le_nonneg_of_squares_le {a b : ℚ} (Ha : a ≥ 0) (Hb : b ≥ 0)
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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 nonneg_le_nonneg_of_squares_le,
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@ -37,6 +52,28 @@ theorem abs_abs_sub_abs_le_abs_sub (a b : ℚ) : abs (abs a - abs b) ≤ abs (a
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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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@ -46,7 +83,8 @@ theorem abs_reg_of_reg {s : seq} (Hs : regular s) : regular (s_abs s) :=
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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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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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@ -80,7 +118,6 @@ theorem abs_pos_of_nonzero {s : seq} (Hs : regular s) (Hnz : sep s zero) : ∃ N
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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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@ -94,11 +131,17 @@ theorem sep_zero_of_pos {s : seq} (Hs : regular s) (Hpos : pos s) : sep s zero :
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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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------------------------
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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) : ∀ m : ℕ+, m ≥ (pb Hs Hpos) → abs (s m) ≥ (pb Hs Hpos)⁻¹ :=
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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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@ -107,41 +150,57 @@ theorem ps_spec {s : seq} (Hs : regular s) (Hsep : sep s zero) :
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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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(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) : pb Hs Hpos = ps Hs Hsep := sorry
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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 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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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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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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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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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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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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@ -155,15 +214,30 @@ theorem s_inv_of_zero {s : seq} (Hs : regular s) (Hz : ¬ sep s zero) : s_inv Hs
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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 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 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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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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@ -172,7 +246,8 @@ theorem reg_inv_reg {s : seq} (Hs : regular s) (Hsep : sep s zero) : regular (s_
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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 [(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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@ -195,7 +270,8 @@ theorem reg_inv_reg {s : seq} (Hs : regular s) (Hsep : sep s zero) : regular (s_
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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 [(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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@ -236,9 +312,23 @@ theorem reg_inv_reg {s : seq} (Hs : regular s) (Hsep : sep s zero) : regular (s_
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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 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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@ -248,24 +338,34 @@ theorem mul_inv {s : seq} (Hs : regular s) (Hsep : sep s zero) : smul s (s_inv H
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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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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, 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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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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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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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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@ -294,7 +394,8 @@ theorem inv_mul {s : seq} (Hs : regular s) (Hsep : sep s zero) : smul (s_inv Hs)
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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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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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@ -405,9 +506,6 @@ theorem s_neg_sub {s t : seq} (Hs : regular s) (Ht : regular t) :
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repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
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end
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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 s_le_total {s t : seq} (Hs : regular s) (Ht : regular t) : s_le s t ∨ s_le t s :=
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if H : s_le s t then or.inl H else or.inr begin
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rewrite [↑s_le at *],
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@ -440,11 +538,6 @@ theorem s_le_total {s t : seq} (Hs : regular s) (Ht : regular t) : s_le s t ∨
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repeat (assumption | apply reg_add_reg | apply reg_neg_reg)
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end
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theorem neg_add_rewrite {a b : ℚ} : a + -b = -(b + -a) := sorry
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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 s_le_of_not_lt {s t : seq} (Hle : ¬ s_lt s t) : s_le t s :=
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begin
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rewrite [↑s_le, ↑nonneg, ↑s_lt at Hle, ↑pos at Hle],
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@ -500,8 +593,6 @@ theorem r_sep_of_nequiv (s t : reg_seq) (Hneq : ¬ requiv s t) : r_sep s t :=
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theorem r_lt_or_equiv_of_le (s t : reg_seq) (Hle : r_le s t) : r_lt s t ∨ requiv s t :=
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lt_or_equiv_of_le (reg_seq.is_reg s) (reg_seq.is_reg t) Hle
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/-theorem r_inv_mul (s : reg_seq) (Hsep : r_sep s r_zero) : requiv ((r_inv s) * s) r_one :=
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inv_mul (reg_seq.is_reg s) Hsep-/
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|
||||
end s
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||||
|
||||
|
|
|
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Reference in a new issue