chore(library/data/real): replace theorems with more general versions from algebra
This commit is contained in:
Rob Lewis
Leonardo de Moura
parent
b23e23061f
commit
b94d0a948d
4 changed files with 54 additions and 634 deletions
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/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad
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Partially ordered additive groups, modeled on Isabelle's library. We could refine the structures,
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but we would have to declare more inheritance paths.
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-/
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import logic.eq data.unit data.sigma data.prod
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import algebra.function algebra.binary
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import algebra.group algebra.order
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open eq eq.ops -- note: ⁻¹ will be overloaded
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namespace algebra
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variable {A : Type}
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/- partially ordered monoids, such as the natural numbers -/
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structure ordered_cancel_comm_monoid [class] (A : Type) extends add_comm_monoid A,
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add_left_cancel_semigroup A, add_right_cancel_semigroup A, order_pair A :=
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(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
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(le_of_add_le_add_left : ∀a b c, le (add a b) (add a c) → le b c)
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section
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variables [s : ordered_cancel_comm_monoid A]
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variables {a b c d e : A}
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include s
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theorem add_le_add_left (H : a ≤ b) (c : A) : c + a ≤ c + b :=
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!ordered_cancel_comm_monoid.add_le_add_left H c
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theorem add_le_add_right (H : a ≤ b) (c : A) : a + c ≤ b + c :=
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(add.comm c a) ▸ (add.comm c b) ▸ (add_le_add_left H c)
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theorem add_le_add (Hab : a ≤ b) (Hcd : c ≤ d) : a + c ≤ b + d :=
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le.trans (add_le_add_right Hab c) (add_le_add_left Hcd b)
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theorem add_lt_add_left (H : a < b) (c : A) : c + a < c + b :=
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have H1 : c + a ≤ c + b, from add_le_add_left (le_of_lt H) c,
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have H2 : c + a ≠ c + b, from
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take H3 : c + a = c + b,
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have H4 : a = b, from add.left_cancel H3,
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ne_of_lt H H4,
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sorry--lt_of_le_of_ne H1 H2
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theorem add_lt_add_right (H : a < b) (c : A) : a + c < b + c :=
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begin
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rewrite [add.comm, {b + _}add.comm],
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exact (add_lt_add_left H c)
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end
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theorem le_add_of_nonneg_right (H : b ≥ 0) : a ≤ a + b :=
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begin
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have H1 : a + b ≥ a + 0, from add_le_add_left H a,
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rewrite add_zero at H1,
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exact H1
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end
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theorem le_add_of_nonneg_left (H : b ≥ 0) : a ≤ b + a :=
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begin
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have H1 : 0 + a ≤ b + a, from add_le_add_right H a,
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rewrite zero_add at H1,
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exact H1
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end
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theorem add_lt_add (Hab : a < b) (Hcd : c < d) : a + c < b + d :=
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lt.trans (add_lt_add_right Hab c) (add_lt_add_left Hcd b)
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theorem add_lt_add_of_le_of_lt (Hab : a ≤ b) (Hcd : c < d) : a + c < b + d :=
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lt_of_le_of_lt (add_le_add_right Hab c) (add_lt_add_left Hcd b)
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theorem add_lt_add_of_lt_of_le (Hab : a < b) (Hcd : c ≤ d) : a + c < b + d :=
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lt_of_lt_of_le (add_lt_add_right Hab c) (add_le_add_left Hcd b)
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theorem lt_add_of_pos_right (H : b > 0) : a < a + b := !add_zero ▸ add_lt_add_left H a
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theorem lt_add_of_pos_left (H : b > 0) : a < b + a := !zero_add ▸ add_lt_add_right H a
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-- here we start using le_of_add_le_add_left.
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theorem le_of_add_le_add_left (H : a + b ≤ a + c) : b ≤ c :=
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!ordered_cancel_comm_monoid.le_of_add_le_add_left H
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theorem le_of_add_le_add_right (H : a + b ≤ c + b) : a ≤ c :=
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le_of_add_le_add_left (show b + a ≤ b + c, begin rewrite [add.comm, {b + _}add.comm], exact H end)
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theorem lt_of_add_lt_add_left (H : a + b < a + c) : b < c :=
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have H1 : b ≤ c, from le_of_add_le_add_left (le_of_lt H),
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have H2 : b ≠ c, from
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assume H3 : b = c, lt.irrefl _ (H3 ▸ H),
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sorry --lt_of_le_of_ne H1 H2
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theorem lt_of_add_lt_add_right (H : a + b < c + b) : a < c :=
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lt_of_add_lt_add_left ((add.comm a b) ▸ (add.comm c b) ▸ H)
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theorem add_le_add_left_iff (a b c : A) : a + b ≤ a + c ↔ b ≤ c :=
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iff.intro le_of_add_le_add_left (assume H, add_le_add_left H _)
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theorem add_le_add_right_iff (a b c : A) : a + b ≤ c + b ↔ a ≤ c :=
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iff.intro le_of_add_le_add_right (assume H, add_le_add_right H _)
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theorem add_lt_add_left_iff (a b c : A) : a + b < a + c ↔ b < c :=
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iff.intro lt_of_add_lt_add_left (assume H, add_lt_add_left H _)
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theorem add_lt_add_right_iff (a b c : A) : a + b < c + b ↔ a < c :=
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iff.intro lt_of_add_lt_add_right (assume H, add_lt_add_right H _)
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-- here we start using properties of zero.
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theorem add_nonneg (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a + b :=
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!zero_add ▸ (add_le_add Ha Hb)
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theorem add_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a + b :=
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!zero_add ▸ (add_lt_add Ha Hb)
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theorem add_pos_of_pos_of_nonneg (Ha : 0 < a) (Hb : 0 ≤ b) : 0 < a + b :=
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!zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
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theorem add_pos_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 < a + b :=
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!zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
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theorem add_nonpos (Ha : a ≤ 0) (Hb : b ≤ 0) : a + b ≤ 0 :=
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!zero_add ▸ (add_le_add Ha Hb)
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theorem add_neg (Ha : a < 0) (Hb : b < 0) : a + b < 0 :=
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!zero_add ▸ (add_lt_add Ha Hb)
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theorem add_neg_of_neg_of_nonpos (Ha : a < 0) (Hb : b ≤ 0) : a + b < 0 :=
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!zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
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theorem add_neg_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : a + b < 0 :=
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!zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
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-- TODO: add nonpos version (will be easier with simplifier)
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theorem add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg
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(Ha : 0 ≤ a) (Hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
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iff.intro
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(assume Hab : a + b = 0,
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have Ha' : a ≤ 0, from
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calc
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a = a + 0 : by rewrite add_zero
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... ≤ a + b : add_le_add_left Hb
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... = 0 : Hab,
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have Haz : a = 0, from le.antisymm Ha' Ha,
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have Hb' : b ≤ 0, from
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calc
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b = 0 + b : by rewrite zero_add
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... ≤ a + b : add_le_add_right Ha
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... = 0 : Hab,
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have Hbz : b = 0, from le.antisymm Hb' Hb,
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and.intro Haz Hbz)
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(assume Hab : a = 0 ∧ b = 0,
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obtain Ha' Hb', from Hab,
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by rewrite [Ha', Hb', add_zero])
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theorem le_add_of_nonneg_of_le (Ha : 0 ≤ a) (Hbc : b ≤ c) : b ≤ a + c :=
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!zero_add ▸ add_le_add Ha Hbc
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theorem le_add_of_le_of_nonneg (Hbc : b ≤ c) (Ha : 0 ≤ a) : b ≤ c + a :=
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!add_zero ▸ add_le_add Hbc Ha
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theorem lt_add_of_pos_of_le (Ha : 0 < a) (Hbc : b ≤ c) : b < a + c :=
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!zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
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theorem lt_add_of_le_of_pos (Hbc : b ≤ c) (Ha : 0 < a) : b < c + a :=
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!add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
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theorem add_le_of_nonpos_of_le (Ha : a ≤ 0) (Hbc : b ≤ c) : a + b ≤ c :=
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!zero_add ▸ add_le_add Ha Hbc
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theorem add_le_of_le_of_nonpos (Hbc : b ≤ c) (Ha : a ≤ 0) : b + a ≤ c :=
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!add_zero ▸ add_le_add Hbc Ha
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theorem add_lt_of_neg_of_le (Ha : a < 0) (Hbc : b ≤ c) : a + b < c :=
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!zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
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theorem add_lt_of_le_of_neg (Hbc : b ≤ c) (Ha : a < 0) : b + a < c :=
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!add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
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theorem lt_add_of_nonneg_of_lt (Ha : 0 ≤ a) (Hbc : b < c) : b < a + c :=
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!zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
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theorem lt_add_of_lt_of_nonneg (Hbc : b < c) (Ha : 0 ≤ a) : b < c + a :=
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!add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
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theorem lt_add_of_pos_of_lt (Ha : 0 < a) (Hbc : b < c) : b < a + c :=
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!zero_add ▸ add_lt_add Ha Hbc
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theorem lt_add_of_lt_of_pos (Hbc : b < c) (Ha : 0 < a) : b < c + a :=
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!add_zero ▸ add_lt_add Hbc Ha
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theorem add_lt_of_nonpos_of_lt (Ha : a ≤ 0) (Hbc : b < c) : a + b < c :=
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!zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
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theorem add_lt_of_lt_of_nonpos (Hbc : b < c) (Ha : a ≤ 0) : b + a < c :=
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!add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
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theorem add_lt_of_neg_of_lt (Ha : a < 0) (Hbc : b < c) : a + b < c :=
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!zero_add ▸ add_lt_add Ha Hbc
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theorem add_lt_of_lt_of_neg (Hbc : b < c) (Ha : a < 0) : b + a < c :=
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!add_zero ▸ add_lt_add Hbc Ha
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end
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-- TODO: add properties of max and min
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/- partially ordered groups -/
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structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_pair A :=
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(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
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theorem ordered_comm_group.le_of_add_le_add_left [s : ordered_comm_group A] {a b c : A} (H : a + b ≤ a + c) : b ≤ c :=
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assert H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _,
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by rewrite *neg_add_cancel_left at H'; exact H'
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definition ordered_comm_group.to_ordered_cancel_comm_monoid [instance] [coercion] [reducible]
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[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
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⦃ ordered_cancel_comm_monoid, s,
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add_left_cancel := @add.left_cancel A s,
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add_right_cancel := @add.right_cancel A s,
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le_of_add_le_add_left := @ordered_comm_group.le_of_add_le_add_left A s ⦄
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section
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variables [s : ordered_comm_group A] (a b c d e : A)
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include s
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theorem neg_le_neg {a b : A} (H : a ≤ b) : -b ≤ -a :=
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have H1 : 0 ≤ -a + b, from !add.left_inv ▸ !(add_le_add_left H),
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!add_neg_cancel_right ▸ !zero_add ▸ add_le_add_right H1 (-b)
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theorem le_of_neg_le_neg {a b : A} (H : -b ≤ -a) : a ≤ b :=
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neg_neg a ▸ neg_neg b ▸ neg_le_neg H
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theorem neg_le_neg_iff_le : -a ≤ -b ↔ b ≤ a :=
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iff.intro le_of_neg_le_neg neg_le_neg
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theorem nonneg_of_neg_nonpos {a : A} (H : -a ≤ 0) : 0 ≤ a :=
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le_of_neg_le_neg (neg_zero⁻¹ ▸ H)
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theorem neg_nonpos_of_nonneg {a : A} (H : 0 ≤ a) : -a ≤ 0 :=
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neg_zero ▸ neg_le_neg H
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theorem neg_nonpos_iff_nonneg : -a ≤ 0 ↔ 0 ≤ a :=
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iff.intro nonneg_of_neg_nonpos neg_nonpos_of_nonneg
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theorem nonpos_of_neg_nonneg {a : A} (H : 0 ≤ -a) : a ≤ 0 :=
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le_of_neg_le_neg (neg_zero⁻¹ ▸ H)
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theorem neg_nonneg_of_nonpos {a : A} (H : a ≤ 0) : 0 ≤ -a :=
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neg_zero ▸ neg_le_neg H
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theorem neg_nonneg_iff_nonpos : 0 ≤ -a ↔ a ≤ 0 :=
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iff.intro nonpos_of_neg_nonneg neg_nonneg_of_nonpos
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theorem neg_lt_neg {a b : A} (H : a < b) : -b < -a :=
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have H1 : 0 < -a + b, from !add.left_inv ▸ !(add_lt_add_left H),
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!add_neg_cancel_right ▸ !zero_add ▸ add_lt_add_right H1 (-b)
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theorem lt_of_neg_lt_neg {a b : A} (H : -b < -a) : a < b :=
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neg_neg a ▸ neg_neg b ▸ neg_lt_neg H
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theorem neg_lt_neg_iff_lt : -a < -b ↔ b < a :=
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iff.intro lt_of_neg_lt_neg neg_lt_neg
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theorem pos_of_neg_neg {a : A} (H : -a < 0) : 0 < a :=
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lt_of_neg_lt_neg (neg_zero⁻¹ ▸ H)
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theorem neg_neg_of_pos {a : A} (H : 0 < a) : -a < 0 :=
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neg_zero ▸ neg_lt_neg H
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theorem neg_neg_iff_pos : -a < 0 ↔ 0 < a :=
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iff.intro pos_of_neg_neg neg_neg_of_pos
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theorem neg_of_neg_pos {a : A} (H : 0 < -a) : a < 0 :=
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lt_of_neg_lt_neg (neg_zero⁻¹ ▸ H)
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theorem neg_pos_of_neg {a : A} (H : a < 0) : 0 < -a :=
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neg_zero ▸ neg_lt_neg H
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theorem neg_pos_iff_neg : 0 < -a ↔ a < 0 :=
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iff.intro neg_of_neg_pos neg_pos_of_neg
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theorem le_neg_iff_le_neg : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff_le
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theorem neg_le_iff_neg_le : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff_le
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theorem lt_neg_iff_lt_neg : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff_lt
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theorem neg_lt_iff_neg_lt : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff_lt
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theorem sub_nonneg_iff_le : 0 ≤ a - b ↔ b ≤ a := !sub_self ▸ !add_le_add_right_iff
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theorem sub_nonpos_iff_le : a - b ≤ 0 ↔ a ≤ b := !sub_self ▸ !add_le_add_right_iff
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theorem sub_pos_iff_lt : 0 < a - b ↔ b < a := !sub_self ▸ !add_lt_add_right_iff
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theorem sub_neg_iff_lt : a - b < 0 ↔ a < b := !sub_self ▸ !add_lt_add_right_iff
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theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c :=
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have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_add_left_iff),
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!neg_add_cancel_left ▸ H
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theorem add_le_iff_le_sub_left : a + b ≤ c ↔ b ≤ c - a :=
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by rewrite [sub_eq_add_neg, {c+_}add.comm]; apply add_le_iff_le_neg_add
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theorem add_le_iff_le_sub_right : a + b ≤ c ↔ a ≤ c - b :=
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have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_add_right_iff),
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!add_neg_cancel_right ▸ H
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theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c :=
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assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
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by rewrite neg_add_cancel_left at H; exact H
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theorem le_add_iff_sub_left_le : a ≤ b + c ↔ a - b ≤ c :=
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by rewrite [sub_eq_add_neg, {a+_}add.comm]; apply le_add_iff_neg_add_le
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theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b :=
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assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff),
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by rewrite add_neg_cancel_right at H; exact H
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theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c :=
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assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff),
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begin rewrite neg_add_cancel_left at H, exact H end
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theorem add_lt_iff_lt_neg_add_right : a + b < c ↔ a < -b + c :=
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by rewrite add.comm; apply add_lt_iff_lt_neg_add_left
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theorem add_lt_iff_lt_sub_left : a + b < c ↔ b < c - a :=
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begin
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rewrite [sub_eq_add_neg, {c+_}add.comm],
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apply add_lt_iff_lt_neg_add_left
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end
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theorem add_lt_add_iff_lt_sub_right : a + b < c ↔ a < c - b :=
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assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff),
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by rewrite add_neg_cancel_right at H; exact H
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|
||||
theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c :=
|
||||
assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff),
|
||||
by rewrite neg_add_cancel_left at H; exact H
|
||||
|
||||
theorem lt_add_iff_neg_add_lt_right : a < b + c ↔ -c + a < b :=
|
||||
by rewrite add.comm; apply lt_add_iff_neg_add_lt_left
|
||||
|
||||
theorem lt_add_iff_sub_lt_left : a < b + c ↔ a - b < c :=
|
||||
by rewrite [sub_eq_add_neg, {a + _}add.comm]; apply lt_add_iff_neg_add_lt_left
|
||||
|
||||
theorem lt_add_iff_sub_lt_right : a < b + c ↔ a - c < b :=
|
||||
by rewrite add.comm; apply lt_add_iff_sub_lt_left
|
||||
|
||||
-- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
|
||||
theorem le_iff_le_of_sub_eq_sub {a b c d : A} (H : a - b = c - d) : a ≤ b ↔ c ≤ d :=
|
||||
calc
|
||||
a ≤ b ↔ a - b ≤ 0 : iff.symm (sub_nonpos_iff_le a b)
|
||||
... = (c - d ≤ 0) : by rewrite H
|
||||
... ↔ c ≤ d : sub_nonpos_iff_le c d
|
||||
|
||||
theorem lt_iff_lt_of_sub_eq_sub {a b c d : A} (H : a - b = c - d) : a < b ↔ c < d :=
|
||||
calc
|
||||
a < b ↔ a - b < 0 : iff.symm (sub_neg_iff_lt a b)
|
||||
... = (c - d < 0) : by rewrite H
|
||||
... ↔ c < d : sub_neg_iff_lt c d
|
||||
|
||||
theorem sub_le_sub_left {a b : A} (H : a ≤ b) (c : A) : c - b ≤ c - a :=
|
||||
add_le_add_left (neg_le_neg H) c
|
||||
|
||||
theorem sub_le_sub_right {a b : A} (H : a ≤ b) (c : A) : a - c ≤ b - c := add_le_add_right H (-c)
|
||||
|
||||
theorem sub_le_sub {a b c d : A} (Hab : a ≤ b) (Hcd : c ≤ d) : a - d ≤ b - c :=
|
||||
add_le_add Hab (neg_le_neg Hcd)
|
||||
|
||||
theorem sub_lt_sub_left {a b : A} (H : a < b) (c : A) : c - b < c - a :=
|
||||
add_lt_add_left (neg_lt_neg H) c
|
||||
|
||||
theorem sub_lt_sub_right {a b : A} (H : a < b) (c : A) : a - c < b - c := add_lt_add_right H (-c)
|
||||
|
||||
theorem sub_lt_sub {a b c d : A} (Hab : a < b) (Hcd : c < d) : a - d < b - c :=
|
||||
add_lt_add Hab (neg_lt_neg Hcd)
|
||||
|
||||
theorem sub_lt_sub_of_le_of_lt {a b c d : A} (Hab : a ≤ b) (Hcd : c < d) : a - d < b - c :=
|
||||
add_lt_add_of_le_of_lt Hab (neg_lt_neg Hcd)
|
||||
|
||||
theorem sub_lt_sub_of_lt_of_le {a b c d : A} (Hab : a < b) (Hcd : c ≤ d) : a - d < b - c :=
|
||||
add_lt_add_of_lt_of_le Hab (neg_le_neg Hcd)
|
||||
|
||||
theorem sub_le_self (a : A) {b : A} (H : b ≥ 0) : a - b ≤ a :=
|
||||
calc
|
||||
a - b = a + -b : rfl
|
||||
... ≤ a + 0 : add_le_add_left (neg_nonpos_of_nonneg H)
|
||||
... = a : by rewrite add_zero
|
||||
|
||||
theorem sub_lt_self (a : A) {b : A} (H : b > 0) : a - b < a :=
|
||||
calc
|
||||
a - b = a + -b : rfl
|
||||
... < a + 0 : add_lt_add_left (neg_neg_of_pos H)
|
||||
... = a : by rewrite add_zero
|
||||
end
|
||||
|
||||
structure decidable_linear_ordered_comm_group [class] (A : Type)
|
||||
extends add_comm_group A, decidable_linear_order A :=
|
||||
(add_le_add_left : ∀ a b, le a b → ∀ c, le (add c a) (add c b))
|
||||
|
||||
private theorem add_le_add_left' (A : Type) (s : decidable_linear_ordered_comm_group A) (a b : A) :
|
||||
a ≤ b → (∀ c : A, c + a ≤ c + b) :=
|
||||
decidable_linear_ordered_comm_group.add_le_add_left a b
|
||||
|
||||
definition decidable_linear_ordered_comm_group.to_ordered_comm_group [instance] [reducible] [coercion]
|
||||
(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
|
||||
⦃ordered_comm_group, s,
|
||||
le_of_lt := @le_of_lt A s,
|
||||
add_le_add_left := add_le_add_left' A s,
|
||||
lt_of_le_of_lt := @lt_of_le_of_lt A s,
|
||||
lt_of_lt_of_le := @lt_of_lt_of_le A s⦄
|
||||
|
||||
section
|
||||
variables [s : decidable_linear_ordered_comm_group A]
|
||||
variables {a b c d e : A}
|
||||
include s
|
||||
|
||||
theorem eq_zero_of_neg_eq (H : -a = a) : a = 0 :=
|
||||
lt.by_cases
|
||||
(assume H1 : a < 0,
|
||||
have H2: a > 0, from H ▸ neg_pos_of_neg H1,
|
||||
absurd H1 (lt.asymm H2))
|
||||
(assume H1 : a = 0, H1)
|
||||
(assume H1 : a > 0,
|
||||
have H2: a < 0, from H ▸ neg_neg_of_pos H1,
|
||||
absurd H1 (lt.asymm H2))
|
||||
|
||||
definition abs (a : A) : A := if 0 ≤ a then a else -a
|
||||
|
||||
theorem abs_of_nonneg (H : a ≥ 0) : abs a = a := if_pos H
|
||||
|
||||
theorem abs_of_pos (H : a > 0) : abs a = a := if_pos (le_of_lt H)
|
||||
|
||||
theorem abs_of_neg (H : a < 0) : abs a = -a := if_neg (not_le_of_gt H)
|
||||
|
||||
theorem abs_zero : abs 0 = (0:A) := abs_of_nonneg (le.refl _)
|
||||
|
||||
theorem abs_of_nonpos (H : a ≤ 0) : abs a = -a :=
|
||||
decidable.by_cases
|
||||
(assume H1 : a = 0, by rewrite [H1, abs_zero, neg_zero])
|
||||
(assume H1 : a ≠ 0,
|
||||
have H2 : a < 0, from lt_of_le_of_ne H H1,
|
||||
abs_of_neg H2)
|
||||
|
||||
theorem abs_neg (a : A) : abs (-a) = abs a :=
|
||||
or.elim (le.total 0 a)
|
||||
(assume H1 : 0 ≤ a, by rewrite [abs_of_nonpos (neg_nonpos_of_nonneg H1), neg_neg, abs_of_nonneg H1])
|
||||
(assume H1 : a ≤ 0, by rewrite [abs_of_nonneg (neg_nonneg_of_nonpos H1), abs_of_nonpos H1])
|
||||
|
||||
theorem abs_nonneg (a : A) : abs a ≥ 0 :=
|
||||
or.elim (le.total 0 a)
|
||||
(assume H : 0 ≤ a, by rewrite (abs_of_nonneg H); exact H)
|
||||
(assume H : a ≤ 0,
|
||||
calc
|
||||
0 ≤ -a : neg_nonneg_of_nonpos H
|
||||
... = abs a : abs_of_nonpos H)
|
||||
|
||||
theorem abs_abs (a : A) : abs (abs a) = abs a := abs_of_nonneg !abs_nonneg
|
||||
|
||||
theorem le_abs_self (a : A) : a ≤ abs a :=
|
||||
or.elim (le.total 0 a)
|
||||
(assume H : 0 ≤ a, abs_of_nonneg H ▸ !le.refl)
|
||||
(assume H : a ≤ 0, le.trans H !abs_nonneg)
|
||||
|
||||
theorem neg_le_abs_self (a : A) : -a ≤ abs a :=
|
||||
!abs_neg ▸ !le_abs_self
|
||||
|
||||
theorem eq_zero_of_abs_eq_zero (H : abs a = 0) : a = 0 :=
|
||||
have H1 : a ≤ 0, from H ▸ le_abs_self a,
|
||||
have H2 : -a ≤ 0, from H ▸ abs_neg a ▸ le_abs_self (-a),
|
||||
le.antisymm H1 (nonneg_of_neg_nonpos H2)
|
||||
|
||||
theorem abs_eq_zero_iff_eq_zero (a : A) : abs a = 0 ↔ a = 0 :=
|
||||
iff.intro eq_zero_of_abs_eq_zero (assume H, congr_arg abs H ⬝ !abs_zero)
|
||||
|
||||
theorem abs_pos_of_pos (H : a > 0) : abs a > 0 :=
|
||||
by rewrite (abs_of_pos H); exact H
|
||||
|
||||
theorem abs_pos_of_neg (H : a < 0) : abs a > 0 :=
|
||||
!abs_neg ▸ abs_pos_of_pos (neg_pos_of_neg H)
|
||||
|
||||
theorem abs_pos_of_ne_zero (H : a ≠ 0) : abs a > 0 :=
|
||||
or.elim (lt_or_gt_of_ne H) abs_pos_of_neg abs_pos_of_pos
|
||||
|
||||
theorem abs_sub (a b : A) : abs (a - b) = abs (b - a) :=
|
||||
by rewrite [-neg_sub, abs_neg]
|
||||
|
||||
theorem abs.by_cases {P : A → Prop} {a : A} (H1 : P a) (H2 : P (-a)) : P (abs a) :=
|
||||
or.elim (le.total 0 a)
|
||||
(assume H : 0 ≤ a, (abs_of_nonneg H)⁻¹ ▸ H1)
|
||||
(assume H : a ≤ 0, (abs_of_nonpos H)⁻¹ ▸ H2)
|
||||
|
||||
theorem abs_le_of_le_of_neg_le (H1 : a ≤ b) (H2 : -a ≤ b) : abs a ≤ b :=
|
||||
abs.by_cases H1 H2
|
||||
|
||||
theorem abs_lt_of_lt_of_neg_lt (H1 : a < b) (H2 : -a < b) : abs a < b :=
|
||||
abs.by_cases H1 H2
|
||||
|
||||
-- the triangle inequality
|
||||
section
|
||||
private lemma aux1 {a b : A} (H1 : a + b ≥ 0) (H2 : a ≥ 0) : abs (a + b) ≤ abs a + abs b :=
|
||||
decidable.by_cases
|
||||
(assume H3 : b ≥ 0,
|
||||
calc
|
||||
abs (a + b) ≤ abs (a + b) : le.refl
|
||||
... = a + b : by rewrite (abs_of_nonneg H1)
|
||||
... = abs a + b : by rewrite (abs_of_nonneg H2)
|
||||
... = abs a + abs b : by rewrite (abs_of_nonneg H3))
|
||||
(assume H3 : ¬ b ≥ 0,
|
||||
assert H4 : b ≤ 0, from le_of_lt (lt_of_not_ge H3),
|
||||
calc
|
||||
abs (a + b) = a + b : by rewrite (abs_of_nonneg H1)
|
||||
... = abs a + b : by rewrite (abs_of_nonneg H2)
|
||||
... ≤ abs a + 0 : add_le_add_left H4
|
||||
... ≤ abs a + -b : add_le_add_left (neg_nonneg_of_nonpos H4)
|
||||
... = abs a + abs b : by rewrite (abs_of_nonpos H4))
|
||||
|
||||
private lemma aux2 {a b : A} (H1 : a + b ≥ 0) : abs (a + b) ≤ abs a + abs b :=
|
||||
or.elim (le.total b 0)
|
||||
(assume H2 : b ≤ 0,
|
||||
have H3 : ¬ a < 0, from
|
||||
assume H4 : a < 0,
|
||||
have H5 : a + b < 0, from !add_zero ▸ add_lt_add_of_lt_of_le H4 H2,
|
||||
not_lt_of_ge H1 H5,
|
||||
aux1 H1 (le_of_not_gt H3))
|
||||
(assume H2 : 0 ≤ b,
|
||||
begin
|
||||
have H3 : abs (b + a) ≤ abs b + abs a,
|
||||
begin
|
||||
rewrite add.comm at H1,
|
||||
exact aux1 H1 H2
|
||||
end,
|
||||
rewrite [add.comm, {abs a + _}add.comm],
|
||||
exact H3
|
||||
end)
|
||||
|
||||
theorem abs_add_le_abs_add_abs (a b : A) : abs (a + b) ≤ abs a + abs b :=
|
||||
or.elim (le.total 0 (a + b))
|
||||
(assume H2 : 0 ≤ a + b, aux2 H2)
|
||||
(assume H2 : a + b ≤ 0,
|
||||
assert H3 : -a + -b = -(a + b), by rewrite neg_add,
|
||||
assert H4 : -(a + b) ≥ 0, from iff.mp' (neg_nonneg_iff_nonpos (a+b)) H2,
|
||||
have H5 : -a + -b ≥ 0, begin rewrite -H3 at H4, exact H4 end,
|
||||
calc
|
||||
abs (a + b) = abs (-a + -b) : by rewrite [-abs_neg, neg_add]
|
||||
... ≤ abs (-a) + abs (-b) : aux2 H5
|
||||
... = abs a + abs b : by rewrite *abs_neg)
|
||||
end
|
||||
|
||||
theorem abs_sub_abs_le_abs_sub (a b : A) : abs a - abs b ≤ abs (a - b) :=
|
||||
have H1 : abs a - abs b + abs b ≤ abs (a - b) + abs b, from
|
||||
calc
|
||||
abs a - abs b + abs b = abs a : by rewrite sub_add_cancel
|
||||
... = abs (a - b + b) : by rewrite sub_add_cancel
|
||||
... ≤ abs (a - b) + abs b : abs_add_le_abs_add_abs,
|
||||
algebra.le_of_add_le_add_right H1
|
||||
end
|
||||
|
||||
end algebra
|
||||
|
|
@ -300,5 +300,25 @@ theorem pnat_cancel' (n m : ℕ+) : (n * n * m)⁻¹ * (rat_of_pnat n * rat_of_p
|
|||
rewrite [rat.mul.comm, *inv_mul_eq_mul_inv, simp, *inv_cancel_left, *rat.one_mul]
|
||||
end
|
||||
|
||||
definition pceil (a : ℚ) : ℕ+ := pnat.pos (ubound a) !ubound_pos
|
||||
|
||||
theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) (Ha : a > 0) : n⁻¹ ≤ 1 / a :=
|
||||
begin
|
||||
apply rat.le.trans,
|
||||
apply inv_ge_of_le H,
|
||||
apply div_le_div_of_le,
|
||||
apply Ha,
|
||||
apply ubound_ge
|
||||
end
|
||||
|
||||
theorem inv_pceil_div (a b : ℚ) (Ha : a > 0) (Hb : b > 0) : (pceil (a / b))⁻¹ ≤ b / a :=
|
||||
begin
|
||||
rewrite -(@div_div' (b / a)),
|
||||
apply div_le_div_of_le,
|
||||
apply div_pos_of_pos,
|
||||
apply pos_div_of_pos_of_pos Hb Ha,
|
||||
rewrite [(div_div_eq_mul_div (ne_of_gt Hb) (ne_of_gt Ha)), rat.one_mul],
|
||||
apply ubound_ge
|
||||
end
|
||||
|
||||
end pnat
|
||||
|
|
|
|||
|
|
@ -21,10 +21,6 @@ local notation 1 := rat.of_num 1
|
|||
-------------------------------------
|
||||
-- theorems to add to (ordered) field and/or rat
|
||||
|
||||
theorem div_two (a : ℚ) : (a + a) / (1 + 1) = a := sorry
|
||||
|
||||
theorem two_pos : (1 : ℚ) + 1 > 0 := rat.add_pos rat.zero_lt_one rat.zero_lt_one
|
||||
|
||||
theorem find_midpoint {a b : ℚ} (H : a > b) : ∃ c : ℚ, a > b + c :=
|
||||
exists.intro ((a - b) / (1 + 1))
|
||||
(have H2 [visible] : a + a > (b + b) + (a - b), from calc
|
||||
|
|
@ -32,32 +28,19 @@ theorem find_midpoint {a b : ℚ} (H : a > b) : ∃ c : ℚ, a > b + c :=
|
|||
... = b + a + b - b : rat.add_sub_cancel
|
||||
... = (b + b) + (a - b) : sorry, -- simp
|
||||
have H3 [visible] : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
|
||||
from div_lt_div_of_lt_of_pos H2 two_pos,
|
||||
from div_lt_div_of_lt_of_pos H2 dec_trivial,
|
||||
by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
|
||||
|
||||
theorem add_sub_comm (a b c d : ℚ) : a + b - (c + d) = (a - c) + (b - d) := sorry
|
||||
|
||||
theorem div_helper (a b : ℚ) : (1 / (a * b)) * a = 1 / b := sorry
|
||||
|
||||
theorem distrib_three_right (a b c d : ℚ) : (a + b + c) * d = a * d + b * d + c * d := sorry
|
||||
|
||||
theorem mul_le_mul_of_mul_div_le (a b c d : ℚ) : a * (b / c) ≤ d → b * a ≤ d * c := sorry
|
||||
|
||||
definition pceil (a : ℚ) : ℕ+ := pnat.pos (ubound a) !ubound_pos
|
||||
|
||||
theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) : n⁻¹ ≤ 1 / a := sorry
|
||||
|
||||
theorem inv_pceil_div (a b : ℚ) (Ha : a > 0) (Hb : b > 0) : (pceil (a / b))⁻¹ ≤ b / a := sorry
|
||||
|
||||
|
||||
theorem s_mul_assoc_lemma_4 {n : ℕ+} {ε q : ℚ} (Hε : ε > 0) (Hq : q > 0) (H : n ≥ pceil (q / ε)) :
|
||||
q * n⁻¹ ≤ ε :=
|
||||
begin
|
||||
let H2 := pceil_helper H,
|
||||
let H2 := pceil_helper H (pos_div_of_pos_of_pos Hq Hε),
|
||||
let H3 := mul_le_of_le_div (pos_div_of_pos_of_pos Hq Hε) H2,
|
||||
rewrite -(one_mul ε),
|
||||
apply mul_le_mul_of_mul_div_le,
|
||||
assumption
|
||||
repeat assumption
|
||||
end
|
||||
|
||||
-------------------------------------
|
||||
|
|
@ -80,10 +63,8 @@ theorem squeeze {a b : ℚ} (H : ∀ j : ℕ+, a ≤ b + j⁻¹ + j⁻¹ + j⁻
|
|||
exact absurd !H (not_le_of_gt Ha)
|
||||
end
|
||||
|
||||
|
||||
theorem squeeze_2 {a b : ℚ} (H : ∀ ε : ℚ, ε > 0 → a ≥ b - ε) : a ≥ b := sorry
|
||||
|
||||
|
||||
theorem rewrite_helper (a b c d : ℚ) : a * b - c * d = a * (b - d) + (a - c) * d :=
|
||||
sorry
|
||||
|
||||
|
|
@ -106,11 +87,9 @@ theorem factor_lemma (a b c d e : ℚ) : abs (a + b + c - (d + (b + e))) = abs (
|
|||
|
||||
theorem factor_lemma_2 (a b c d : ℚ) : (a + b) + (c + d) = (a + c) + (d + b) := sorry
|
||||
|
||||
|
||||
-------------------------------------
|
||||
-- The only sorry's after this point are for the simplifier.
|
||||
|
||||
--------------------------------------
|
||||
--------------------------------------
|
||||
-- define cauchy sequences and equivalence. show equivalence actually is one
|
||||
|
||||
|
|
@ -206,7 +185,8 @@ theorem pnat_bound {ε : ℚ} (Hε : ε > 0) : ∃ p : ℕ+, p⁻¹ ≤ ε :=
|
|||
existsi (pceil (1 / ε)),
|
||||
rewrite -(rat.div_div (rat.ne_of_gt Hε)) at {2},
|
||||
apply pceil_helper,
|
||||
apply le.refl
|
||||
apply le.refl,
|
||||
apply div_pos_of_pos Hε
|
||||
end
|
||||
|
||||
theorem bdd_of_eq_var {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
|
||||
|
|
@ -500,7 +480,7 @@ theorem s_mul_assoc {s t u : seq} (Hs : regular s) (Ht : regular t) (Hu : regula
|
|||
apply Hs,
|
||||
apply Ht,
|
||||
apply Hu,
|
||||
rewrite [*s_mul_assoc_lemma_3, -distrib_three_right],
|
||||
rewrite [*s_mul_assoc_lemma_3, -rat.distrib_three_right],
|
||||
apply s_mul_assoc_lemma_4,
|
||||
apply a,
|
||||
repeat apply rat.add_pos,
|
||||
|
|
@ -608,6 +588,7 @@ theorem mul_bound_helper {s t : seq} (Hs : regular s) (Ht : regular t) (a b c :
|
|||
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),
|
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apply rat.le_of_lt,
|
||||
apply rat.add_pos,
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||||
apply rat.mul_pos,
|
||||
|
|
@ -618,7 +599,11 @@ theorem mul_bound_helper {s t : seq} (Hs : regular s) (Ht : regular t) (a b c :
|
|||
apply rat.add_pos,
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repeat apply inv_pos,
|
||||
apply rat_of_pnat_is_pos,
|
||||
rewrite div_helper,
|
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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,
|
||||
|
|
@ -786,55 +771,30 @@ theorem diff_equiv_zero_of_equiv {s t : seq} (Hs : regular s) (Ht : regular t) (
|
|||
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
|
||||
let Hst := reg_mul_reg Hs Ht,
|
||||
let Hsu := reg_mul_reg Hs Hu,
|
||||
let Hnu := reg_neg_reg Hu,
|
||||
let Hstu := reg_add_reg Hst Hsu,
|
||||
let Hsnu := reg_mul_reg Hs Hnu,
|
||||
let Htnu := reg_add_reg Ht Hnu,
|
||||
-- let Hstsu := reg_add_reg Hst Hsnu,
|
||||
apply equiv_of_diff_equiv_zero,
|
||||
apply Hst,
|
||||
apply Hsu,
|
||||
apply equiv.trans,
|
||||
apply reg_add_reg,
|
||||
apply Hst,
|
||||
apply reg_neg_reg Hsu,
|
||||
rotate 1,
|
||||
apply zero_is_reg,
|
||||
apply equiv.symm,
|
||||
apply add_well_defined,
|
||||
rotate 2,
|
||||
apply reg_mul_reg Hs Ht,
|
||||
apply reg_neg_reg Hsu,
|
||||
apply equiv.refl,
|
||||
apply mul_neg_equiv_neg_mul,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply equiv.symm,
|
||||
apply s_distrib,
|
||||
repeat assumption,
|
||||
rotate 1,
|
||||
apply reg_add_reg Hst Hsnu,
|
||||
apply Hst,
|
||||
apply Hsnu,
|
||||
apply reg_add_reg Hst Hsnu,
|
||||
apply reg_mul_reg Hs,
|
||||
apply reg_add_reg Ht Hnu,
|
||||
apply zero_is_reg,
|
||||
apply mul_zero_equiv_zero,
|
||||
rotate 2,
|
||||
apply diff_equiv_zero_of_equiv,
|
||||
repeat assumption
|
||||
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
|
||||
let Hsu := reg_mul_reg Hs Hu,
|
||||
let Hus := reg_mul_reg Hu Hs,
|
||||
let Htu := reg_mul_reg Ht Hu,
|
||||
let Hut := reg_mul_reg Hu Ht,
|
||||
apply equiv.trans,
|
||||
rotate 3,
|
||||
apply s_mul_comm,
|
||||
|
|
@ -845,7 +805,7 @@ theorem mul_well_defined_half2 {s t u : seq} (Hs : regular s) (Ht : regular t) (
|
|||
apply Ht,
|
||||
rotate 1,
|
||||
apply s_mul_comm,
|
||||
repeat assumption
|
||||
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)
|
||||
|
|
|
|||
|
|
@ -11,7 +11,8 @@ At this point, we no longer proceed constructively: this file makes heavy use of
|
|||
Here, we show that ℝ is complete.
|
||||
-/
|
||||
|
||||
import data.real.basic data.real.order data.real.division data.rat data.nat data.pnat logic.axioms.classical
|
||||
import data.real.basic data.real.order data.real.division data.rat data.nat data.pnat
|
||||
import logic.axioms.classical
|
||||
open -[coercions] rat
|
||||
local notation 0 := rat.of_num 0
|
||||
local notation 1 := rat.of_num 1
|
||||
|
|
|
|||
Loading…
Reference in a new issue