1057 lines
34 KiB
Text
1057 lines
34 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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To do:
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o Break positive naturals into their own file and fill in sorry's
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o Fill in sorrys for helper lemmas that will not be handled by simplifier
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o Rename things and possibly make theorems private
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-/
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import data.nat data.rat.order data.pnat
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open nat eq eq.ops pnat
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open -[coercions] rat
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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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----------------------------------------------------------------------------------------------------
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-------------------------------------
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-- theorems to add to (ordered) field and/or rat
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-- this can move to pnat once sorry is filled in
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theorem find_midpoint {a b : ℚ} (H : a > b) : ∃ c : ℚ, a > b + c ∧ c > 0 :=
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exists.intro ((a - b) / (1 + 1))
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(and.intro (have H2 [visible] : a + a > (b + b) + (a - b), from calc
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a + a > b + a : rat.add_lt_add_right H
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... = b + a + b - b : rat.add_sub_cancel
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... = (b + b) + (a - b) : sorry, -- simp
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have H3 [visible] : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
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from div_lt_div_of_lt_of_pos H2 dec_trivial,
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by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
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(pos_div_of_pos_of_pos (iff.mp' !sub_pos_iff_lt H) dec_trivial))
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theorem add_sub_comm (a b c d : ℚ) : a + b - (c + d) = (a - c) + (b - d) := sorry
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theorem s_mul_assoc_lemma_4 {n : ℕ+} {ε q : ℚ} (Hε : ε > 0) (Hq : q > 0) (H : n ≥ pceil (q / ε)) :
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q * n⁻¹ ≤ ε :=
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begin
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let H2 := pceil_helper H (pos_div_of_pos_of_pos Hq Hε),
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let H3 := mul_le_of_le_div (pos_div_of_pos_of_pos Hq Hε) H2,
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rewrite -(one_mul ε),
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apply mul_le_mul_of_mul_div_le,
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repeat assumption
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end
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-------------------------------------
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-- small helper lemmas
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theorem s_mul_assoc_lemma_3 (a b n : ℕ+) (p : ℚ) :
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p * ((a * n)⁻¹ + (b * n)⁻¹) = p * (a⁻¹ + b⁻¹) * n⁻¹ :=
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by rewrite [rat.mul.assoc, rat.right_distrib, *inv_mul_eq_mul_inv]
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theorem find_thirds (a b : ℚ) : ∃ n : ℕ+, a + n⁻¹ + n⁻¹ + n⁻¹ < a + b := sorry
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theorem squeeze {a b : ℚ} (H : ∀ j : ℕ+, a ≤ b + j⁻¹ + j⁻¹ + j⁻¹) : a ≤ b :=
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begin
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apply rat.le_of_not_gt,
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intro Hb,
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apply (exists.elim (find_midpoint Hb)),
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intro c Hc,
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apply (exists.elim (find_thirds b c)),
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intro j Hbj,
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have Ha : a > b + j⁻¹ + j⁻¹ + j⁻¹, from lt.trans Hbj (and.left Hc),
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exact absurd !H (not_le_of_gt Ha)
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end
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theorem squeeze_2 {a b : ℚ} (H : ∀ ε : ℚ, ε > 0 → a ≥ b - ε) : a ≥ b :=
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begin
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apply rat.le_of_not_gt,
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intro Hb,
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apply (exists.elim (find_midpoint Hb)),
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intro c Hc,
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let Hc' := H c (and.right Hc),
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apply (rat.not_le_of_gt (and.left Hc)) ((iff.mp' !le_add_iff_sub_right_le) Hc')
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end
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theorem rewrite_helper (a b c d : ℚ) : a * b - c * d = a * (b - d) + (a - c) * d :=
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sorry
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theorem rewrite_helper3 (a b c d e f g: ℚ) : a * (b + c) - (d * e + f * g) =
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(a * b - d * e) + (a * c - f * g) := sorry
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theorem rewrite_helper4 (a b c d : ℚ) : a * b - c * d = (a * b - a * d) + (a * d - c * d) := sorry
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theorem rewrite_helper5 (a b x y : ℚ) : a - b = (a - x) + (x - y) + (y - b) := sorry
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theorem rewrite_helper7 (a b c d x : ℚ) :
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a * b * c - d = (b * c) * (a - x) + (x * b * c - d) := sorry
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theorem ineq_helper (a b : ℚ) (k m n : ℕ+) (H : a ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹)
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(H2 : b ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹) :
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(rat_of_pnat k) * a + b * (rat_of_pnat k) ≤ m⁻¹ + n⁻¹ := sorry
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theorem factor_lemma (a b c d e : ℚ) : abs (a + b + c - (d + (b + e))) = abs ((a - d) + (c - e)) :=
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sorry
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theorem factor_lemma_2 (a b c d : ℚ) : (a + b) + (c + d) = (a + c) + (d + b) := sorry
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-------------------------------------
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-- The only sorry's after this point are for the simplifier.
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--------------------------------------
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-- define cauchy sequences and equivalence. show equivalence actually is one
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notation `seq` := ℕ+ → ℚ
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definition regular (s : seq) := ∀ m n : ℕ+, abs (s m - s n) ≤ m⁻¹ + n⁻¹
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definition equiv (s t : seq) := ∀ n : ℕ+, abs (s n - t n) ≤ n⁻¹ + n⁻¹
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infix `≡` := equiv
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theorem equiv.refl (s : seq) : s ≡ s :=
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begin
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rewrite ↑equiv,
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intros,
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rewrite [rat.sub_self, abs_zero],
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apply add_invs_nonneg
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end
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theorem equiv.symm (s t : seq) (H : s ≡ t) : t ≡ s :=
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begin
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rewrite ↑equiv at *,
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intros,
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rewrite [-abs_neg, neg_sub],
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exact H n
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end
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theorem bdd_of_eq {s t : seq} (H : s ≡ t) :
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∀ j : ℕ+, ∀ n : ℕ+, n ≥ 2 * j → abs (s n - t n) ≤ j⁻¹ :=
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begin
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rewrite ↑equiv at *,
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intros [j, n, Hn],
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apply rat.le.trans,
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apply H n,
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rewrite -(add_halves j),
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apply rat.add_le_add,
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apply inv_ge_of_le Hn,
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apply inv_ge_of_le Hn
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end
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theorem eq_of_bdd {s t : seq} (Hs : regular s) (Ht : regular t)
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(H : ∀ j : ℕ+, ∃ Nj : ℕ+, ∀ n : ℕ+, Nj ≤ n → abs (s n - t n) ≤ j⁻¹) : s ≡ t :=
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begin
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rewrite ↑equiv,
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intros,
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have Hj : (∀ j : ℕ+, abs (s n - t n) ≤ n⁻¹ + n⁻¹ + j⁻¹ + j⁻¹ + j⁻¹), begin
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intros,
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apply exists.elim (H j),
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intros [Nj, HNj],
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rewrite [-(rat.sub_add_cancel (s n) (s (max j Nj))), rat.add.assoc (s n + -s (max j Nj)),
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↑regular at *],
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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apply rat.le.trans,
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apply rat.add_le_add,
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apply Hs,
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rewrite [-(rat.sub_add_cancel (s (max j Nj)) (t (max j Nj))), rat.add.assoc],
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apply abs_add_le_abs_add_abs,
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apply rat.le.trans,
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apply rat.add_le_add_left,
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apply rat.add_le_add,
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apply HNj (max j Nj) (max_right j Nj),
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apply Ht,
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have simp : ∀ m : ℕ+, n⁻¹ + m⁻¹ + (j⁻¹ + (m⁻¹ + n⁻¹)) = (n⁻¹ + n⁻¹ + j⁻¹) + (m⁻¹ + m⁻¹),
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from sorry, -- simplifier
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rewrite simp,
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have Hms : (max j Nj)⁻¹ + (max j Nj)⁻¹ ≤ j⁻¹ + j⁻¹, begin
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apply rat.add_le_add,
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apply inv_ge_of_le (max_left j Nj),
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apply inv_ge_of_le (max_left j Nj),
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end,
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apply (calc
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n⁻¹ + n⁻¹ + j⁻¹ + ((max j Nj)⁻¹ + (max j Nj)⁻¹) ≤ n⁻¹ + n⁻¹ + j⁻¹ + (j⁻¹ + j⁻¹) :
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rat.add_le_add_left Hms
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... = n⁻¹ + n⁻¹ + j⁻¹ + j⁻¹ + j⁻¹ : by rewrite *rat.add.assoc)
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end,
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apply (squeeze Hj)
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end
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theorem eq_of_bdd_var {s t : seq} (Hs : regular s) (Ht : regular t)
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(H : ∀ ε : ℚ, ε > 0 → ∃ Nj : ℕ+, ∀ n : ℕ+, Nj ≤ n → abs (s n - t n) ≤ ε) : s ≡ t :=
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begin
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apply eq_of_bdd,
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apply Hs,
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apply Ht,
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intros,
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apply H j⁻¹,
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apply inv_pos
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end
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set_option pp.beta false
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theorem pnat_bound {ε : ℚ} (Hε : ε > 0) : ∃ p : ℕ+, p⁻¹ ≤ ε :=
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begin
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existsi (pceil (1 / ε)),
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rewrite -(rat.div_div (rat.ne_of_gt Hε)) at {2},
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apply pceil_helper,
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apply le.refl,
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apply div_pos_of_pos Hε
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end
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theorem bdd_of_eq_var {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
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∀ ε : ℚ, ε > 0 → ∃ Nj : ℕ+, ∀ n : ℕ+, Nj ≤ n → abs (s n - t n) ≤ ε :=
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begin
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intro ε Hε,
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apply (exists.elim (pnat_bound Hε)),
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intro N HN,
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let Bd' := bdd_of_eq Heq N,
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existsi 2 * N,
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intro n Hn,
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apply rat.le.trans,
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apply Bd' n Hn,
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assumption
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end
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theorem equiv.trans (s t u : seq) (Hs : regular s) (Ht : regular t) (Hu : regular u)
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(H : s ≡ t) (H2 : t ≡ u) : s ≡ u :=
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begin
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apply eq_of_bdd Hs Hu,
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intros,
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existsi 2 * (2 * j),
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intro n Hn,
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rewrite [-rat.sub_add_cancel (s n) (t n), rat.add.assoc],
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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have Hst : abs (s n - t n) ≤ (2 * j)⁻¹, from bdd_of_eq H _ _ Hn,
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have Htu : abs (t n - u n) ≤ (2 * j)⁻¹, from bdd_of_eq H2 _ _ Hn,
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rewrite -(add_halves j),
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apply rat.add_le_add,
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repeat assumption
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end
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-----------------------------------
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-- define operations on cauchy sequences. show operations preserve regularity
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definition K (s : seq) : ℕ+ := pnat.pos (ubound (abs (s pone)) + 1 + 1) dec_trivial
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theorem canon_bound {s : seq} (Hs : regular s) (n : ℕ+) : abs (s n) ≤ rat_of_pnat (K s) :=
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calc
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abs (s n) = abs (s n - s pone + s pone) : by rewrite rat.sub_add_cancel
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... ≤ abs (s n - s pone) + abs (s pone) : abs_add_le_abs_add_abs
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... ≤ n⁻¹ + pone⁻¹ + abs (s pone) : rat.add_le_add_right !Hs
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... = n⁻¹ + (1 + abs (s pone)) : by rewrite [pone_inv, rat.add.assoc]
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... ≤ 1 + (1 + abs (s pone)) : rat.add_le_add_right (inv_le_one n)
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... = abs (s pone) + (1 + 1) :
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by rewrite [add.comm 1 (abs (s pone)), rat.add.comm 1, rat.add.assoc]
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... ≤ of_nat (ubound (abs (s pone))) + (1 + 1) : rat.add_le_add_right (!ubound_ge)
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... = of_nat (ubound (abs (s pone)) + (1 + 1)) : by rewrite of_nat_add
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... = of_nat (ubound (abs (s pone)) + 1 + 1) : by rewrite nat.add.assoc
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definition K₂ (s t : seq) := max (K s) (K t)
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theorem K₂_symm (s t : seq) : K₂ s t = K₂ t s :=
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if H : K s < K t then
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(have H1 [visible] : K₂ s t = K t, from max_eq_right H,
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have H2 [visible] : K₂ t s = K t, from max_eq_left (not_lt_of_ge (le_of_lt H)),
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by rewrite [H1, -H2])
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else
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(have H1 [visible] : K₂ s t = K s, from max_eq_left H,
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if J : K t < K s then
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(have H2 [visible] : K₂ t s = K s, from max_eq_right J, by rewrite [H1, -H2])
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else
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(have Heq [visible] : K t = K s, from
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eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt J),
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by rewrite [↑K₂, Heq]))
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theorem canon_2_bound_left (s t : seq) (Hs : regular s) (n : ℕ+) :
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abs (s n) ≤ rat_of_pnat (K₂ s t) :=
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calc
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abs (s n) ≤ rat_of_pnat (K s) : canon_bound Hs n
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... ≤ rat_of_pnat (K₂ s t) : rat_of_pnat_le_of_pnat_le (!max_left)
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theorem canon_2_bound_right (s t : seq) (Ht : regular t) (n : ℕ+) :
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abs (t n) ≤ rat_of_pnat (K₂ s t) :=
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calc
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abs (t n) ≤ rat_of_pnat (K t) : canon_bound Ht n
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... ≤ rat_of_pnat (K₂ s t) : rat_of_pnat_le_of_pnat_le (!max_right)
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definition sadd (s t : seq) : seq := λ n, (s (2 * n)) + (t (2 * n))
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theorem reg_add_reg {s t : seq} (Hs : regular s) (Ht : regular t) : regular (sadd s t) :=
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begin
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rewrite [↑regular at *, ↑sadd],
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intros,
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rewrite add_sub_comm,
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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rewrite add_halves_double,
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apply rat.add_le_add,
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apply Hs,
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apply Ht
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end
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definition smul (s t : seq) : seq := λ n : ℕ+, (s ((K₂ s t) * 2 * n)) * (t ((K₂ s t) * 2 * n))
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theorem reg_mul_reg {s t : seq} (Hs : regular s) (Ht : regular t) : regular (smul s t) :=
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begin
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rewrite [↑regular at *, ↑smul],
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intros,
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rewrite rewrite_helper,
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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apply rat.le.trans,
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apply rat.add_le_add,
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rewrite abs_mul,
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apply rat.mul_le_mul_of_nonneg_right,
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apply canon_2_bound_left s t Hs,
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apply abs_nonneg,
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rewrite abs_mul,
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apply rat.mul_le_mul_of_nonneg_left,
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apply canon_2_bound_right s t Ht,
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apply abs_nonneg,
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apply ineq_helper,
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apply Ht,
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apply Hs
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end
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definition sneg (s : seq) : seq := λ n : ℕ+, - (s n)
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theorem reg_neg_reg {s : seq} (Hs : regular s) : regular (sneg s) :=
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begin
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rewrite [↑regular at *, ↑sneg],
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intros,
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rewrite [-abs_neg, neg_sub, sub_neg_eq_add, rat.add.comm],
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apply Hs
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end
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-----------------------------------
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-- show properties of +, *, -
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definition zero : seq := λ n, 0
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definition one : seq := λ n, 1
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theorem s_add_comm (s t : seq) : sadd s t ≡ sadd t s :=
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begin
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esimp [sadd],
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intro n,
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rewrite [sub_add_eq_sub_sub, rat.add_sub_cancel, rat.sub_self, abs_zero],
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apply add_invs_nonneg
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end
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theorem s_add_assoc (s t u : seq) (Hs : regular s) (Hu : regular u) :
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sadd (sadd s t) u ≡ sadd s (sadd t u) :=
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begin
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rewrite [↑sadd, ↑equiv, ↑regular at *],
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intros,
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rewrite factor_lemma,
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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apply rat.le.trans,
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rotate 1,
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apply rat.add_le_add_right,
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apply inv_two_mul_le_inv,
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rewrite [-(add_halves (2 * n)), -(add_halves n), factor_lemma_2],
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apply rat.add_le_add,
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apply Hs,
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apply Hu
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end
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theorem s_mul_comm (s t : seq) : smul s t ≡ smul t s :=
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begin
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rewrite ↑smul,
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intros n,
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rewrite [*(K₂_symm s t), rat.mul.comm, rat.sub_self, abs_zero],
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apply add_invs_nonneg
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end
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definition DK (s t : seq) := (K₂ s t) * 2
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theorem DK_rewrite (s t : seq) : (K₂ s t) * 2 = DK s t := rfl
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definition TK (s t u : seq) := (DK (λ (n : ℕ+), s (mul (DK s t) n) * t (mul (DK s t) n)) u)
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theorem TK_rewrite (s t u : seq) :
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(DK (λ (n : ℕ+), s (mul (DK s t) n) * t (mul (DK s t) n)) u) = TK s t u := rfl
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theorem s_mul_assoc_lemma (s t u : seq) (a b c d : ℕ+) :
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abs (s a * t a * u b - s c * t d * u d) ≤ abs (t a) * abs (u b) * abs (s a - s c) +
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abs (s c) * abs (t a) * abs (u b - u d) + abs (s c) * abs (u d) * abs (t a - t d) :=
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begin
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rewrite (rewrite_helper7 _ _ _ _ (s c)),
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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rewrite rat.add.assoc,
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apply rat.add_le_add,
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rewrite 2 abs_mul,
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apply rat.le.refl,
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rewrite [*rat.mul.assoc, -rat.mul_sub_left_distrib, -rat.left_distrib, abs_mul],
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apply rat.mul_le_mul_of_nonneg_left,
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rewrite rewrite_helper,
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apply rat.le.trans,
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apply abs_add_le_abs_add_abs,
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apply rat.add_le_add,
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rewrite abs_mul, apply rat.le.refl,
|
||
rewrite [abs_mul, rat.mul.comm], apply rat.le.refl,
|
||
apply abs_nonneg
|
||
end
|
||
|
||
definition Kq (s : seq) := rat_of_pnat (K s) + 1
|
||
theorem Kq_bound {s : seq} (H : regular s) : ∀ n, abs (s n) ≤ Kq s :=
|
||
begin
|
||
intros,
|
||
apply rat.le_of_lt,
|
||
apply rat.lt_of_le_of_lt,
|
||
apply canon_bound H,
|
||
apply rat.lt_add_of_pos_right,
|
||
apply rat.zero_lt_one
|
||
end
|
||
|
||
theorem Kq_bound_nonneg {s : seq} (H : regular s) : 0 ≤ Kq s :=
|
||
rat.le.trans !abs_nonneg (Kq_bound H 2)
|
||
|
||
theorem Kq_bound_pos {s : seq} (H : regular s) : 0 < Kq s :=
|
||
have H1 : 0 ≤ rat_of_pnat (K s), from rat.le.trans (!abs_nonneg) (canon_bound H 2),
|
||
add_pos_of_nonneg_of_pos H1 rat.zero_lt_one
|
||
|
||
theorem s_mul_assoc_lemma_5 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(a b c : ℕ+) : abs (t a) * abs (u b) * abs (s a - s c) ≤ (Kq t) * (Kq u) * (a⁻¹ + c⁻¹) :=
|
||
begin
|
||
repeat apply rat.mul_le_mul,
|
||
apply Kq_bound Ht,
|
||
apply Kq_bound Hu,
|
||
apply abs_nonneg,
|
||
apply Kq_bound_nonneg Ht,
|
||
apply Hs,
|
||
apply abs_nonneg,
|
||
apply rat.mul_nonneg,
|
||
apply Kq_bound_nonneg Ht,
|
||
apply Kq_bound_nonneg Hu,
|
||
end
|
||
|
||
theorem s_mul_assoc_lemma_2 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(a b c d : ℕ+) :
|
||
abs (t a) * abs (u b) * abs (s a - s c) + abs (s c) * abs (t a) * abs (u b - u d)
|
||
+ abs (s c) * abs (u d) * abs (t a - t d) ≤
|
||
(Kq t) * (Kq u) * (a⁻¹ + c⁻¹) + (Kq s) * (Kq t) * (b⁻¹ + d⁻¹) + (Kq s) * (Kq u) * (a⁻¹ + d⁻¹) :=
|
||
begin
|
||
apply add_le_add_three,
|
||
repeat apply rat.mul_le_mul,
|
||
apply Kq_bound Ht,
|
||
apply Kq_bound Hu,
|
||
apply abs_nonneg,
|
||
apply Kq_bound_nonneg Ht,
|
||
apply Hs,
|
||
apply abs_nonneg,
|
||
apply rat.mul_nonneg,
|
||
apply Kq_bound_nonneg Ht,
|
||
apply Kq_bound_nonneg Hu,
|
||
repeat apply rat.mul_le_mul,
|
||
apply Kq_bound Hs,
|
||
apply Kq_bound Ht,
|
||
apply abs_nonneg,
|
||
apply Kq_bound_nonneg Hs,
|
||
apply Hu,
|
||
apply abs_nonneg,
|
||
apply rat.mul_nonneg,
|
||
apply Kq_bound_nonneg Hs,
|
||
apply Kq_bound_nonneg Ht,
|
||
repeat apply rat.mul_le_mul,
|
||
apply Kq_bound Hs,
|
||
apply Kq_bound Hu,
|
||
apply abs_nonneg,
|
||
apply Kq_bound_nonneg Hs,
|
||
apply Ht,
|
||
apply abs_nonneg,
|
||
apply rat.mul_nonneg,
|
||
apply Kq_bound_nonneg Hs,
|
||
apply Kq_bound_nonneg Hu
|
||
end
|
||
|
||
theorem s_mul_assoc {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u) :
|
||
smul (smul s t) u ≡ smul s (smul t u) :=
|
||
begin
|
||
apply eq_of_bdd_var,
|
||
repeat apply reg_mul_reg,
|
||
apply Hs,
|
||
apply Ht,
|
||
apply Hu,
|
||
apply reg_mul_reg Hs,
|
||
apply reg_mul_reg Ht Hu,
|
||
intros,
|
||
fapply exists.intro,
|
||
rotate 1,
|
||
intros,
|
||
rewrite [↑smul, *DK_rewrite, *TK_rewrite, -*pnat.mul.assoc, -*rat.mul.assoc],
|
||
apply rat.le.trans,
|
||
apply s_mul_assoc_lemma,
|
||
apply rat.le.trans,
|
||
apply s_mul_assoc_lemma_2,
|
||
apply Hs,
|
||
apply Ht,
|
||
apply Hu,
|
||
rewrite [*s_mul_assoc_lemma_3, -rat.distrib_three_right],
|
||
apply s_mul_assoc_lemma_4,
|
||
apply a,
|
||
repeat apply rat.add_pos,
|
||
repeat apply rat.mul_pos,
|
||
apply Kq_bound_pos Ht,
|
||
apply Kq_bound_pos Hu,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
repeat apply rat.mul_pos,
|
||
apply Kq_bound_pos Hs,
|
||
apply Kq_bound_pos Ht,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
repeat apply rat.mul_pos,
|
||
apply Kq_bound_pos Hs,
|
||
apply Kq_bound_pos Hu,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
apply a_1
|
||
end
|
||
|
||
theorem zero_is_reg : regular zero :=
|
||
begin
|
||
rewrite [↑regular, ↑zero],
|
||
intros,
|
||
rewrite [rat.sub_zero, abs_zero],
|
||
apply add_invs_nonneg
|
||
end
|
||
|
||
theorem s_zero_add (s : seq) (H : regular s) : sadd zero s ≡ s :=
|
||
begin
|
||
rewrite [↑sadd, ↑zero, ↑equiv, ↑regular at H],
|
||
intros,
|
||
rewrite [rat.zero_add],
|
||
apply rat.le.trans,
|
||
apply H,
|
||
apply rat.add_le_add,
|
||
apply inv_two_mul_le_inv,
|
||
apply rat.le.refl
|
||
end
|
||
|
||
theorem s_add_zero (s : seq) (H : regular s) : sadd s zero ≡ s :=
|
||
begin
|
||
rewrite [↑sadd, ↑zero, ↑equiv, ↑regular at H],
|
||
intros,
|
||
rewrite [rat.add_zero],
|
||
apply rat.le.trans,
|
||
apply H,
|
||
apply rat.add_le_add,
|
||
apply inv_two_mul_le_inv,
|
||
apply rat.le.refl
|
||
end
|
||
|
||
theorem s_neg_cancel (s : seq) (H : regular s) : sadd (sneg s) s ≡ zero :=
|
||
begin
|
||
rewrite [↑sadd, ↑sneg, ↑regular at H, ↑zero, ↑equiv],
|
||
intros,
|
||
rewrite [neg_add_eq_sub, rat.sub_self, rat.sub_zero, abs_zero],
|
||
apply add_invs_nonneg
|
||
end
|
||
|
||
theorem neg_s_cancel (s : seq) (H : regular s) : sadd s (sneg s) ≡ zero :=
|
||
begin
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply s_add_comm,
|
||
apply s_neg_cancel s H,
|
||
apply reg_add_reg,
|
||
apply H,
|
||
apply reg_neg_reg,
|
||
apply H,
|
||
apply reg_add_reg,
|
||
apply reg_neg_reg,
|
||
repeat apply H,
|
||
apply zero_is_reg
|
||
end
|
||
|
||
theorem add_well_defined {s t u v : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Hv : regular v) (Esu : s ≡ u) (Etv : t ≡ v) : sadd s t ≡ sadd u v :=
|
||
begin
|
||
rewrite [↑sadd, ↑equiv at *],
|
||
intros,
|
||
rewrite [add_sub_comm, add_halves_double],
|
||
apply rat.le.trans,
|
||
apply abs_add_le_abs_add_abs,
|
||
apply rat.add_le_add,
|
||
apply Esu,
|
||
apply Etv
|
||
end
|
||
|
||
theorem mul_bound_helper {s t : seq} (Hs : regular s) (Ht : regular t) (a b c : ℕ+) (j : ℕ+) :
|
||
∃ N : ℕ+, ∀ n : ℕ+, n ≥ N → abs (s (a * n) * t (b * n) - s (c * n) * t (c * n)) ≤ j⁻¹ :=
|
||
begin
|
||
existsi pceil (((rat_of_pnat (K s)) * (b⁻¹ + c⁻¹) + (a⁻¹ + c⁻¹) *
|
||
(rat_of_pnat (K t))) * (rat_of_pnat j)),
|
||
intros n Hn,
|
||
rewrite rewrite_helper4,
|
||
apply rat.le.trans,
|
||
apply abs_add_le_abs_add_abs,
|
||
apply rat.le.trans,
|
||
rotate 1,
|
||
show n⁻¹ * ((rat_of_pnat (K s)) * (b⁻¹ + c⁻¹)) +
|
||
n⁻¹ * ((a⁻¹ + c⁻¹) * (rat_of_pnat (K t))) ≤ j⁻¹, begin
|
||
rewrite -rat.left_distrib,
|
||
apply rat.le.trans,
|
||
apply rat.mul_le_mul_of_nonneg_right,
|
||
apply pceil_helper Hn,
|
||
repeat (apply rat.mul_pos | apply rat.add_pos | apply inv_pos | apply rat_of_pnat_is_pos),
|
||
apply rat.le_of_lt,
|
||
apply rat.add_pos,
|
||
apply rat.mul_pos,
|
||
apply rat_of_pnat_is_pos,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
apply rat.mul_pos,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
apply rat_of_pnat_is_pos,
|
||
have H : (rat_of_pnat (K s) * (b⁻¹ + c⁻¹) + (a⁻¹ + c⁻¹) * rat_of_pnat (K t)) ≠ 0, begin
|
||
apply rat.ne_of_gt,
|
||
repeat (apply rat.mul_pos | apply rat.add_pos | apply inv_pos | apply rat_of_pnat_is_pos)
|
||
end,
|
||
rewrite (rat.div_helper H),
|
||
apply rat.le.refl
|
||
end,
|
||
apply rat.add_le_add,
|
||
rewrite [-rat.mul_sub_left_distrib, abs_mul],
|
||
apply rat.le.trans,
|
||
apply rat.mul_le_mul,
|
||
apply canon_bound,
|
||
apply Hs,
|
||
apply Ht,
|
||
apply abs_nonneg,
|
||
apply rat.le_of_lt,
|
||
apply rat_of_pnat_is_pos,
|
||
rewrite [*inv_mul_eq_mul_inv, -rat.right_distrib, -rat.mul.assoc, rat.mul.comm],
|
||
apply rat.mul_le_mul_of_nonneg_left,
|
||
apply rat.le.refl,
|
||
apply rat.le_of_lt,
|
||
apply inv_pos,
|
||
rewrite [-rat.mul_sub_right_distrib, abs_mul],
|
||
apply rat.le.trans,
|
||
apply rat.mul_le_mul,
|
||
apply Hs,
|
||
apply canon_bound,
|
||
apply Ht,
|
||
apply abs_nonneg,
|
||
apply add_invs_nonneg,
|
||
rewrite [*inv_mul_eq_mul_inv, -rat.right_distrib, mul.comm _ n⁻¹, rat.mul.assoc],
|
||
apply rat.mul_le_mul,
|
||
apply rat.le.refl,
|
||
apply rat.le.refl,
|
||
apply rat.le_of_lt,
|
||
apply rat.mul_pos,
|
||
apply rat.add_pos,
|
||
repeat apply inv_pos,
|
||
apply rat_of_pnat_is_pos,
|
||
apply rat.le_of_lt,
|
||
apply inv_pos
|
||
end
|
||
|
||
theorem s_distrib {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u) :
|
||
smul s (sadd t u) ≡ sadd (smul s t) (smul s u) :=
|
||
begin
|
||
apply eq_of_bdd,
|
||
repeat (assumption | apply reg_add_reg | apply reg_mul_reg),
|
||
intros,
|
||
let exh1 := λ a b c, mul_bound_helper Hs Ht a b c (2 * j),
|
||
apply exists.elim,
|
||
apply exh1,
|
||
rotate 3,
|
||
intros N1 HN1,
|
||
let exh2 := λ d e f, mul_bound_helper Hs Hu d e f (2 * j),
|
||
apply exists.elim,
|
||
apply exh2,
|
||
rotate 3,
|
||
intros N2 HN2,
|
||
existsi max N1 N2,
|
||
intros n Hn,
|
||
rewrite [↑sadd at *, ↑smul, rewrite_helper3, -add_halves j, -*pnat.mul.assoc at *],
|
||
apply rat.le.trans,
|
||
apply abs_add_le_abs_add_abs,
|
||
apply rat.add_le_add,
|
||
apply HN1,
|
||
apply le.trans,
|
||
apply max_left N1 N2,
|
||
apply Hn,
|
||
apply HN2,
|
||
apply le.trans,
|
||
apply max_right N1 N2,
|
||
apply Hn
|
||
end
|
||
|
||
theorem mul_zero_equiv_zero {s t : seq} (Hs : regular s) (Ht : regular t) (Htz : t ≡ zero) :
|
||
smul s t ≡ zero :=
|
||
begin
|
||
apply eq_of_bdd_var,
|
||
apply reg_mul_reg Hs Ht,
|
||
apply zero_is_reg,
|
||
intro ε Hε,
|
||
let Bd := bdd_of_eq_var Ht zero_is_reg Htz (ε / (Kq s))
|
||
(pos_div_of_pos_of_pos Hε (Kq_bound_pos Hs)),
|
||
apply exists.elim Bd,
|
||
intro N HN,
|
||
existsi N,
|
||
intro n Hn,
|
||
rewrite [↑equiv at Htz, ↑zero at *, rat.sub_zero, ↑smul, abs_mul],
|
||
apply rat.le.trans,
|
||
apply rat.mul_le_mul,
|
||
apply Kq_bound Hs,
|
||
let HN' := λ n, (!rat.sub_zero ▸ HN n),
|
||
apply HN',
|
||
apply le.trans Hn,
|
||
apply pnat.mul_le_mul_left,
|
||
apply abs_nonneg,
|
||
apply rat.le_of_lt (Kq_bound_pos Hs),
|
||
rewrite (rat.mul_div_cancel' (ne.symm (rat.ne_of_lt (Kq_bound_pos Hs)))),
|
||
apply rat.le.refl
|
||
end
|
||
|
||
theorem neg_bound_eq_bound (s : seq) : K (sneg s) = K s :=
|
||
by rewrite [↑K, ↑sneg, abs_neg]
|
||
|
||
theorem neg_bound2_eq_bound2 (s t : seq) : K₂ s (sneg t) = K₂ s t :=
|
||
by rewrite [↑K₂, neg_bound_eq_bound]
|
||
|
||
theorem sneg_def (s : seq) : (λ (n : ℕ+), -(s n)) = sneg s := rfl
|
||
|
||
theorem mul_neg_equiv_neg_mul {s t : seq} : smul s (sneg t) ≡ sneg (smul s t) :=
|
||
begin
|
||
rewrite [↑equiv, ↑smul],
|
||
intros,
|
||
rewrite [↑sneg, *sub_neg_eq_add, -neg_mul_eq_mul_neg, rat.add.comm, *sneg_def,
|
||
*neg_bound2_eq_bound2, rat.sub_self, abs_zero],
|
||
apply add_invs_nonneg
|
||
end
|
||
|
||
theorem equiv_of_diff_equiv_zero {s t : seq} (Hs : regular s) (Ht : regular t)
|
||
(H : sadd s (sneg t) ≡ zero) : s ≡ t :=
|
||
begin
|
||
have simp : ∀ a b c d e : ℚ, a + b + c + (d + e) = (b + d) + a + e + c, from sorry,
|
||
apply eq_of_bdd Hs Ht,
|
||
intros,
|
||
let He := bdd_of_eq H,
|
||
existsi 2 * (2 * (2 * j)),
|
||
intros n Hn,
|
||
rewrite (rewrite_helper5 _ _ (s (2 * n)) (t (2 * n))),
|
||
apply rat.le.trans,
|
||
apply abs_add_three,
|
||
apply rat.le.trans,
|
||
apply add_le_add_three,
|
||
apply Hs,
|
||
rewrite [↑sadd at He, ↑sneg at He, ↑zero at He],
|
||
let He' := λ a b c, !rat.sub_zero ▸ (He a b c),
|
||
apply (He' _ _ Hn),
|
||
apply Ht,
|
||
rewrite [simp, add_halves, -(add_halves j), -(add_halves (2 * j)), -*rat.add.assoc],
|
||
apply rat.add_le_add_right,
|
||
apply add_le_add_three,
|
||
repeat (apply rat.le.trans; apply inv_ge_of_le Hn; apply inv_two_mul_le_inv)
|
||
end
|
||
|
||
theorem s_sub_cancel (s : seq) : sadd s (sneg s) ≡ zero :=
|
||
begin
|
||
rewrite [↑equiv, ↑sadd, ↑sneg, ↑zero],
|
||
intros,
|
||
rewrite [rat.sub_zero, rat.sub_self, abs_zero],
|
||
apply add_invs_nonneg
|
||
end
|
||
|
||
theorem diff_equiv_zero_of_equiv {s t : seq} (Hs : regular s) (Ht : regular t) (H : s ≡ t) :
|
||
sadd s (sneg t) ≡ zero :=
|
||
begin
|
||
let Hnt := reg_neg_reg Ht,
|
||
let Hsnt := reg_add_reg Hs Hnt,
|
||
let Htnt := reg_add_reg Ht Hnt,
|
||
apply equiv.trans,
|
||
rotate 4,
|
||
apply s_sub_cancel t,
|
||
rotate 2,
|
||
apply zero_is_reg,
|
||
apply add_well_defined,
|
||
repeat assumption,
|
||
apply equiv.refl,
|
||
repeat assumption
|
||
end
|
||
|
||
theorem mul_well_defined_half1 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Etu : t ≡ u) : smul s t ≡ smul s u :=
|
||
begin
|
||
apply equiv_of_diff_equiv_zero,
|
||
rotate 2,
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply equiv.symm,
|
||
apply add_well_defined,
|
||
rotate 4,
|
||
apply equiv.refl,
|
||
apply mul_neg_equiv_neg_mul,
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply equiv.symm,
|
||
apply s_distrib,
|
||
rotate 3,
|
||
apply mul_zero_equiv_zero,
|
||
rotate 2,
|
||
apply diff_equiv_zero_of_equiv,
|
||
repeat (assumption | apply reg_mul_reg | apply reg_neg_reg | apply reg_add_reg |
|
||
apply zero_is_reg)
|
||
end
|
||
|
||
theorem mul_well_defined_half2 {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Est : s ≡ t) : smul s u ≡ smul t u :=
|
||
begin
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply s_mul_comm,
|
||
apply equiv.trans,
|
||
rotate 3,
|
||
apply mul_well_defined_half1,
|
||
rotate 2,
|
||
apply Ht,
|
||
rotate 1,
|
||
apply s_mul_comm,
|
||
repeat (assumption | apply reg_mul_reg)
|
||
end
|
||
|
||
theorem mul_well_defined {s t u v : seq} (Hs : regular s) (Ht : regular t) (Hu : regular u)
|
||
(Hv : regular v) (Esu : s ≡ u) (Etv : t ≡ v) : smul s t ≡ smul u v :=
|
||
begin
|
||
apply equiv.trans,
|
||
exact reg_mul_reg Hs Ht,
|
||
exact reg_mul_reg Hs Hv,
|
||
exact reg_mul_reg Hu Hv,
|
||
apply mul_well_defined_half1,
|
||
repeat assumption,
|
||
apply mul_well_defined_half2,
|
||
repeat assumption
|
||
end
|
||
|
||
theorem neg_well_defined {s t : seq} (Est : s ≡ t) : sneg s ≡ sneg t :=
|
||
begin
|
||
rewrite [↑sneg, ↑equiv at *],
|
||
intros,
|
||
rewrite [-abs_neg, neg_sub, sub_neg_eq_add, rat.add.comm],
|
||
apply Est
|
||
end
|
||
|
||
theorem one_is_reg : regular one :=
|
||
begin
|
||
rewrite [↑regular, ↑one],
|
||
intros,
|
||
rewrite [rat.sub_self, abs_zero],
|
||
apply add_invs_nonneg
|
||
end
|
||
|
||
theorem s_one_mul {s : seq} (H : regular s) : smul one s ≡ s :=
|
||
begin
|
||
rewrite ↑equiv,
|
||
intros,
|
||
rewrite [↑smul, ↑one, rat.one_mul],
|
||
apply rat.le.trans,
|
||
apply H,
|
||
apply rat.add_le_add_right,
|
||
apply inv_mul_le_inv
|
||
end
|
||
|
||
theorem s_mul_one {s : seq} (H : regular s) : smul s one ≡ s :=
|
||
begin
|
||
apply equiv.trans,
|
||
apply reg_mul_reg H one_is_reg,
|
||
rotate 2,
|
||
apply s_mul_comm,
|
||
apply s_one_mul H,
|
||
apply reg_mul_reg one_is_reg H,
|
||
apply H
|
||
end
|
||
|
||
theorem zero_nequiv_one : ¬ zero ≡ one :=
|
||
begin
|
||
intro Hz,
|
||
rewrite [↑equiv at Hz, ↑zero at Hz, ↑one at Hz],
|
||
let H := Hz (2 * 2),
|
||
rewrite [rat.zero_sub at H, abs_neg at H, add_halves at H],
|
||
have H' : pone⁻¹ ≤ 2⁻¹, from calc
|
||
pone⁻¹ = 1 : by rewrite -pone_inv
|
||
... = abs 1 : abs_of_pos zero_lt_one
|
||
... ≤ 2⁻¹ : H,
|
||
let H'' := ge_of_inv_le H',
|
||
apply absurd (one_lt_two) (not_lt_of_ge H'')
|
||
end
|
||
|
||
---------------------------------------------
|
||
-- create the type of regular sequences and lift theorems
|
||
|
||
record reg_seq : Type :=
|
||
(sq : seq) (is_reg : regular sq)
|
||
|
||
definition requiv (s t : reg_seq) := (reg_seq.sq s) ≡ (reg_seq.sq t)
|
||
definition requiv.refl (s : reg_seq) : requiv s s := equiv.refl (reg_seq.sq s)
|
||
definition requiv.symm (s t : reg_seq) (H : requiv s t) : requiv t s :=
|
||
equiv.symm (reg_seq.sq s) (reg_seq.sq t) H
|
||
definition requiv.trans (s t u : reg_seq) (H : requiv s t) (H2 : requiv t u) : requiv s u :=
|
||
equiv.trans _ _ _ (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) H H2
|
||
|
||
definition radd (s t : reg_seq) : reg_seq :=
|
||
reg_seq.mk (sadd (reg_seq.sq s) (reg_seq.sq t))
|
||
(reg_add_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
|
||
infix `+` := radd
|
||
|
||
definition rmul (s t : reg_seq) : reg_seq :=
|
||
reg_seq.mk (smul (reg_seq.sq s) (reg_seq.sq t))
|
||
(reg_mul_reg (reg_seq.is_reg s) (reg_seq.is_reg t))
|
||
infix `*` := rmul
|
||
|
||
definition rneg (s : reg_seq) : reg_seq :=
|
||
reg_seq.mk (sneg (reg_seq.sq s)) (reg_neg_reg (reg_seq.is_reg s))
|
||
prefix `-` := rneg
|
||
|
||
definition radd_well_defined {s t u v : reg_seq} (H : requiv s u) (H2 : requiv t v) :
|
||
requiv (s + t) (u + v) :=
|
||
add_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) (reg_seq.is_reg v) H H2
|
||
|
||
definition rmul_well_defined {s t u v : reg_seq} (H : requiv s u) (H2 : requiv t v) :
|
||
requiv (s * t) (u * v) :=
|
||
mul_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u) (reg_seq.is_reg v) H H2
|
||
|
||
definition rneg_well_defined {s t : reg_seq} (H : requiv s t) : requiv (-s) (-t) :=
|
||
neg_well_defined H
|
||
|
||
theorem requiv_is_equiv : equivalence requiv :=
|
||
mk_equivalence requiv requiv.refl requiv.symm requiv.trans
|
||
|
||
definition reg_seq.to_setoid [instance] : setoid reg_seq :=
|
||
⦃setoid, r := requiv, iseqv := requiv_is_equiv⦄
|
||
|
||
definition r_zero : reg_seq :=
|
||
reg_seq.mk (zero) (zero_is_reg)
|
||
|
||
definition r_one : reg_seq :=
|
||
reg_seq.mk (one) (one_is_reg)
|
||
|
||
theorem r_add_comm (s t : reg_seq) : requiv (s + t) (t + s) :=
|
||
s_add_comm (reg_seq.sq s) (reg_seq.sq t)
|
||
|
||
theorem r_add_assoc (s t u : reg_seq) : requiv (s + t + u) (s + (t + u)) :=
|
||
s_add_assoc (reg_seq.sq s) (reg_seq.sq t) (reg_seq.sq u) (reg_seq.is_reg s) (reg_seq.is_reg u)
|
||
|
||
theorem r_zero_add (s : reg_seq) : requiv (r_zero + s) s :=
|
||
s_zero_add (reg_seq.sq s) (reg_seq.is_reg s)
|
||
|
||
theorem r_add_zero (s : reg_seq) : requiv (s + r_zero) s :=
|
||
s_add_zero (reg_seq.sq s) (reg_seq.is_reg s)
|
||
|
||
theorem r_neg_cancel (s : reg_seq) : requiv (-s + s) r_zero :=
|
||
s_neg_cancel (reg_seq.sq s) (reg_seq.is_reg s)
|
||
|
||
theorem r_mul_comm (s t : reg_seq) : requiv (s * t) (t * s) :=
|
||
s_mul_comm (reg_seq.sq s) (reg_seq.sq t)
|
||
|
||
theorem r_mul_assoc (s t u : reg_seq) : requiv (s * t * u) (s * (t * u)) :=
|
||
s_mul_assoc (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u)
|
||
|
||
theorem r_mul_one (s : reg_seq) : requiv (s * r_one) s :=
|
||
s_mul_one (reg_seq.is_reg s)
|
||
|
||
theorem r_one_mul (s : reg_seq) : requiv (r_one * s) s :=
|
||
s_one_mul (reg_seq.is_reg s)
|
||
|
||
theorem r_distrib (s t u : reg_seq) : requiv (s * (t + u)) (s * t + s * u) :=
|
||
s_distrib (reg_seq.is_reg s) (reg_seq.is_reg t) (reg_seq.is_reg u)
|
||
|
||
theorem r_zero_nequiv_one : ¬ requiv r_zero r_one :=
|
||
zero_nequiv_one
|
||
|
||
----------------------------------------------
|
||
-- take quotients to get ℝ and show it's a comm ring
|
||
|
||
namespace real
|
||
definition real := quot reg_seq.to_setoid
|
||
notation `ℝ` := real
|
||
|
||
definition add (x y : ℝ) : ℝ :=
|
||
(quot.lift_on₂ x y (λ a b, quot.mk (a + b))
|
||
(take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
|
||
quot.sound (radd_well_defined Hab Hcd)))
|
||
infix `+` := add
|
||
|
||
definition mul (x y : ℝ) : ℝ :=
|
||
(quot.lift_on₂ x y (λ a b, quot.mk (a * b))
|
||
(take a b c d : reg_seq, take Hab : requiv a c, take Hcd : requiv b d,
|
||
quot.sound (rmul_well_defined Hab Hcd)))
|
||
infix `*` := mul
|
||
|
||
definition neg (x : ℝ) : ℝ :=
|
||
(quot.lift_on x (λ a, quot.mk (-a)) (take a b : reg_seq, take Hab : requiv a b,
|
||
quot.sound (rneg_well_defined Hab)))
|
||
prefix `-` := neg
|
||
|
||
definition zero : ℝ := quot.mk r_zero
|
||
--notation 0 := zero
|
||
|
||
definition one : ℝ := quot.mk r_one
|
||
|
||
theorem add_comm (x y : ℝ) : x + y = y + x :=
|
||
quot.induction_on₂ x y (λ s t, quot.sound (r_add_comm s t))
|
||
|
||
theorem add_assoc (x y z : ℝ) : x + y + z = x + (y + z) :=
|
||
quot.induction_on₃ x y z (λ s t u, quot.sound (r_add_assoc s t u))
|
||
|
||
theorem zero_add (x : ℝ) : zero + x = x :=
|
||
quot.induction_on x (λ s, quot.sound (r_zero_add s))
|
||
|
||
theorem add_zero (x : ℝ) : x + zero = x :=
|
||
quot.induction_on x (λ s, quot.sound (r_add_zero s))
|
||
|
||
theorem neg_cancel (x : ℝ) : -x + x = zero :=
|
||
quot.induction_on x (λ s, quot.sound (r_neg_cancel s))
|
||
|
||
theorem mul_assoc (x y z : ℝ) : x * y * z = x * (y * z) :=
|
||
quot.induction_on₃ x y z (λ s t u, quot.sound (r_mul_assoc s t u))
|
||
|
||
theorem mul_comm (x y : ℝ) : x * y = y * x :=
|
||
quot.induction_on₂ x y (λ s t, quot.sound (r_mul_comm s t))
|
||
|
||
theorem one_mul (x : ℝ) : one * x = x :=
|
||
quot.induction_on x (λ s, quot.sound (r_one_mul s))
|
||
|
||
theorem mul_one (x : ℝ) : x * one = x :=
|
||
quot.induction_on x (λ s, quot.sound (r_mul_one s))
|
||
|
||
theorem distrib (x y z : ℝ) : x * (y + z) = x * y + x * z :=
|
||
quot.induction_on₃ x y z (λ s t u, quot.sound (r_distrib s t u))
|
||
|
||
theorem distrib_l (x y z : ℝ) : (x + y) * z = x * z + y * z :=
|
||
by rewrite [mul_comm, distrib, {x * _}mul_comm, {y * _}mul_comm] -- this shouldn't be necessary
|
||
|
||
theorem zero_ne_one : ¬ zero = one :=
|
||
take H : zero = one,
|
||
absurd (quot.exact H) (r_zero_nequiv_one)
|
||
|
||
definition comm_ring [reducible] : algebra.comm_ring ℝ :=
|
||
begin
|
||
fapply algebra.comm_ring.mk,
|
||
exact add,
|
||
exact add_assoc,
|
||
exact zero,
|
||
exact zero_add,
|
||
exact add_zero,
|
||
exact neg,
|
||
exact neg_cancel,
|
||
exact add_comm,
|
||
exact mul,
|
||
exact mul_assoc,
|
||
apply one,
|
||
apply one_mul,
|
||
apply mul_one,
|
||
apply distrib,
|
||
apply distrib_l,
|
||
apply mul_comm
|
||
end
|
||
|
||
end real
|