feat(library/algebra): update algebraic hierarchy to be more constructive
This commit is contained in:
Rob Lewis
parent
7342f342a9
commit
4b67cd1f97
3 changed files with 164 additions and 65 deletions
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@ -116,63 +116,31 @@ wf.rec_on x H
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/- structures with a weak and a strict order -/
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structure order_pair [class] (A : Type) extends weak_order A, has_lt A :=
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(lt_iff_le_and_ne : ∀a b, lt a b ↔ (le a b ∧ a ≠ b))
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(le_of_lt : ∀ a b, lt a b → le a b)
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(lt_of_lt_of_le : ∀ a b c, lt a b → le b c → lt a c)
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(lt_of_le_of_lt : ∀ a b c, le a b → lt b c → lt a c)
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(lt_irrefl : ∀ a, ¬ lt a a)
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--lt_iff_le_and_ne : a < b ↔ (a ≤ b ∧ a ≠ b)
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section
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variable [s : order_pair A]
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variables {a b c : A}
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include s
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theorem lt_iff_le_and_ne : a < b ↔ (a ≤ b ∧ a ≠ b) :=
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!order_pair.lt_iff_le_and_ne
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theorem le_of_lt : a < b → a ≤ b := !order_pair.le_of_lt
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theorem le_of_lt (H : a < b) : a ≤ b :=
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and.elim_left (iff.mp lt_iff_le_and_ne H)
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theorem lt_of_lt_of_le [trans] : a < b → b ≤ c → a < c := !order_pair.lt_of_lt_of_le
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theorem lt_of_le_of_ne (H1 : a ≤ b) (H2 : a ≠ b) : a < b :=
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iff.mp (iff.symm lt_iff_le_and_ne) (and.intro H1 H2)
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theorem lt_of_le_of_lt [trans] : a ≤ b → b < c → a < c := !order_pair.lt_of_le_of_lt
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private theorem lt_irrefl (s' : order_pair A) (a : A) : ¬ a < a :=
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assume H : a < a,
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have H1 : a ≠ a, from and.elim_right (iff.mp !lt_iff_le_and_ne H),
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H1 rfl
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private theorem lt_irrefl (s' : order_pair A) (a : A) : ¬ a < a := !order_pair.lt_irrefl
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private theorem lt_trans (s' : order_pair A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
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have le_ab : a ≤ b, from le_of_lt lt_ab,
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have le_bc : b ≤ c, from le_of_lt lt_bc,
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have le_ac : a ≤ c, from le.trans le_ab le_bc,
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
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have eq_ab : a = b, from le.antisymm le_ab le_ba,
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have ne_ab : a ≠ b, from and.elim_right (iff.mp lt_iff_le_and_ne lt_ab),
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ne_ab eq_ab,
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show a < c, from lt_of_le_of_ne le_ac ne_ac
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lt_of_lt_of_le lt_ab (le_of_lt lt_bc)
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definition order_pair.to_strict_order [instance] [coercion] [reducible] : strict_order A :=
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⦃ strict_order, s, lt_irrefl := lt_irrefl s, lt_trans := lt_trans s ⦄
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theorem lt_of_lt_of_le [trans] : a < b → b ≤ c → a < c :=
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assume lt_ab : a < b,
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assume le_bc : b ≤ c,
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have le_ac : a ≤ c, from le.trans (le_of_lt lt_ab) le_bc,
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
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have eq_ab : a = b, from le.antisymm (le_of_lt lt_ab) le_ba,
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show false, from ne_of_lt lt_ab eq_ab,
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show a < c, from lt_of_le_of_ne le_ac ne_ac
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theorem lt_of_le_of_lt [trans] : a ≤ b → b < c → a < c :=
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assume le_ab : a ≤ b,
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assume lt_bc : b < c,
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have le_ac : a ≤ c, from le.trans le_ab (le_of_lt lt_bc),
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_cb : c ≤ b, from eq_ac ▸ le_ab,
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have eq_bc : b = c, from le.antisymm (le_of_lt lt_bc) le_cb,
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show false, from ne_of_lt lt_bc eq_bc,
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show a < c, from lt_of_le_of_ne le_ac ne_ac
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theorem gt_of_gt_of_ge [trans] (H1 : a > b) (H2 : b ≥ c) : a > c := lt_of_le_of_lt H2 H1
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@ -187,8 +155,9 @@ section
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lt.irrefl _ (lt_of_le_of_lt H H1)
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end
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structure strong_order_pair [class] (A : Type) extends order_pair A :=
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structure strong_order_pair [class] (A : Type) extends weak_order A, has_lt A := --order_pair A :=
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(le_iff_lt_or_eq : ∀a b, le a b ↔ lt a b ∨ a = b)
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(lt_irrefl : ∀ a, ¬ lt a a)
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theorem le_iff_lt_or_eq [s : strong_order_pair A] {a b : A} : a ≤ b ↔ a < b ∨ a = b :=
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!strong_order_pair.le_iff_lt_or_eq
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@ -196,6 +165,60 @@ theorem le_iff_lt_or_eq [s : strong_order_pair A] {a b : A} : a ≤ b ↔ a < b
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theorem lt_or_eq_of_le [s : strong_order_pair A] {a b : A} (le_ab : a ≤ b) : a < b ∨ a = b :=
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iff.mp le_iff_lt_or_eq le_ab
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theorem le_of_lt_or_eq [s : strong_order_pair A] {a b : A} (lt_or_eq : a < b ∨ a = b) : a ≤ b :=
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iff.mp' le_iff_lt_or_eq lt_or_eq
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private theorem lt_irrefl' [s : strong_order_pair A] (a : A) : ¬ a < a :=
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!strong_order_pair.lt_irrefl
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private theorem le_of_lt' [s : strong_order_pair A] (a b : A) : a < b → a ≤ b :=
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take Hlt, le_of_lt_or_eq (or.inl Hlt)
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private theorem lt_iff_le_and_ne [s : strong_order_pair A] {a b : A} : a < b ↔ (a ≤ b ∧ a ≠ b) :=
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iff.intro
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(take Hlt, and.intro (le_of_lt_or_eq (or.inl Hlt)) (take Hab, absurd (Hab ▸ Hlt) !lt_irrefl'))
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(take Hand,
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have Hor : a < b ∨ a = b, from lt_or_eq_of_le (and.left Hand),
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or_resolve_left Hor (and.right Hand))
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theorem lt_of_le_of_ne [s : strong_order_pair A] {a b : A} : a ≤ b → a ≠ b → a < b :=
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take H1 H2, iff.mp' lt_iff_le_and_ne (and.intro H1 H2)
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private theorem ne_of_lt' [s : strong_order_pair A] {a b : A} (H : a < b) : a ≠ b :=
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and.right ((iff.mp lt_iff_le_and_ne) H)
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private theorem lt_of_lt_of_le' [s : strong_order_pair A] (a b c : A) : a < b → b ≤ c → a < c :=
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assume lt_ab : a < b,
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assume le_bc : b ≤ c,
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have le_ac : a ≤ c, from le.trans (le_of_lt' _ _ lt_ab) le_bc,
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
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have eq_ab : a = b, from le.antisymm (le_of_lt' _ _ lt_ab) le_ba,
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show false, from ne_of_lt' lt_ab eq_ab,
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show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
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theorem lt_of_le_of_lt' [s : strong_order_pair A] (a b c : A) : a ≤ b → b < c → a < c :=
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assume le_ab : a ≤ b,
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assume lt_bc : b < c,
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have le_ac : a ≤ c, from le.trans le_ab (le_of_lt' _ _ lt_bc),
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_cb : c ≤ b, from eq_ac ▸ le_ab,
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have eq_bc : b = c, from le.antisymm (le_of_lt' _ _ lt_bc) le_cb,
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show false, from ne_of_lt' lt_bc eq_bc,
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show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
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definition strong_order_pair.to_order_pair [instance] [coercion] [reducible] [s : strong_order_pair A]
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: order_pair A :=
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⦃ order_pair, s,
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lt_irrefl := lt_irrefl',
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le_of_lt := le_of_lt',
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lt_of_le_of_lt := lt_of_le_of_lt',
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lt_of_lt_of_le := lt_of_lt_of_le'
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⦄
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-- We can also construct a strong order pair by defining a strict order, and then defining
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-- x ≤ y ↔ x < y ∨ x = y
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@ -235,13 +258,55 @@ iff.intro
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iff.mp !strict_order_with_le.le_iff_lt_or_eq (and.elim_left H),
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show a < b, from or_resolve_left H1 (and.elim_right H))
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private theorem le_of_lt' (s : strict_order_with_le A) (a b : A) : a < b → a ≤ b :=
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take Hlt, and.left (iff.mp (lt_iff_le_ne s _ _) Hlt)
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private theorem lt_trans (s : strict_order_with_le A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
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have le_ab : a ≤ b, from le_of_lt' s _ _ lt_ab,
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have le_bc : b ≤ c, from le_of_lt' s _ _ lt_bc,
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have le_ac : a ≤ c, from le_trans s _ _ _ le_ab le_bc,
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
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have eq_ab : a = b, from le_antisymm s a b le_ab le_ba,
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have ne_ab : a ≠ b, from and.elim_right ((iff.mp (lt_iff_le_ne s a b)) lt_ab),
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ne_ab eq_ab,
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show a < c, from (iff.mp' !lt_iff_le_ne) (and.intro le_ac ne_ac)
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theorem lt_of_lt_of_le' (s : strict_order_with_le A) (a b c : A) : a < b → b ≤ c → a < c :=
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assume lt_ab : a < b,
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assume le_bc : b ≤ c,
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have le_ac : a ≤ c, from le_trans s _ _ _ (le_of_lt' s _ _ lt_ab) le_bc,
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
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have eq_ab : a = b, from le_antisymm s _ _ (le_of_lt' s _ _ lt_ab) le_ba,
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show false, from ne_of_lt lt_ab eq_ab,
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show a < c, from iff.mp' (lt_iff_le_ne s _ _) (and.intro le_ac ne_ac)
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theorem lt_of_le_of_lt'' (s : strict_order_with_le A) (a b c : A) : a ≤ b → b < c → a < c :=
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assume le_ab : a ≤ b,
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assume lt_bc : b < c,
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have le_ac : a ≤ c, from le_trans s _ _ _ le_ab (le_of_lt' s _ _ lt_bc),
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have ne_ac : a ≠ c, from
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assume eq_ac : a = c,
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have le_cb : c ≤ b, from eq_ac ▸ le_ab,
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have eq_bc : b = c, from le_antisymm s _ _ (le_of_lt' s _ _ lt_bc) le_cb,
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show false, from ne_of_lt lt_bc eq_bc,
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show a < c, from iff.mp' (lt_iff_le_ne s _ _) (and.intro le_ac ne_ac)
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definition strict_order_with_le.to_order_pair [instance] [coercion] [reducible] [s : strict_order_with_le A] :
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strong_order_pair A :=
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⦃ strong_order_pair, s,
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le_refl := le_refl s,
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le_trans := le_trans s,
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le_antisymm := le_antisymm s,
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lt_iff_le_and_ne := lt_iff_le_ne s ⦄
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le_antisymm := le_antisymm s ⦄
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--le_of_lt := le_of_lt' s,
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--lt_of_le_of_lt := lt_of_le_of_lt' s,
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--lt_of_lt_of_le := lt_of_lt_of_le' s ⦄
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--lt_iff_le_and_ne := lt_iff_le_ne s ⦄
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/- linear orders -/
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@ -250,6 +315,10 @@ structure linear_order_pair [class] (A : Type) extends order_pair A, linear_weak
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structure linear_strong_order_pair [class] (A : Type) extends strong_order_pair A,
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linear_weak_order A
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definition linear_strong_order_pair.to_linear_order_pair [instance] [coercion] [reducible]
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[s : linear_strong_order_pair A] : linear_order_pair A :=
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⦃ linear_order_pair, s, strong_order_pair.to_order_pair⦄
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section
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variable [s : linear_strong_order_pair A]
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variables (a b c : A)
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@ -270,10 +339,6 @@ section
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(assume H, H1 H)
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(assume H, or.elim H (assume H', H2 H') (assume H', H3 H'))
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definition linear_strong_order_pair.to_linear_order_pair [instance] [coercion] [reducible]
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: linear_order_pair A :=
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⦃ linear_order_pair, s ⦄
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theorem le_of_not_gt {a b : A} (H : ¬ a > b) : a ≤ b :=
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lt.by_cases (assume H', absurd H' H) (assume H', H' ▸ !le.refl) (assume H', le_of_lt H')
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@ -22,12 +22,23 @@ structure ordered_cancel_comm_monoid [class] (A : Type) extends add_comm_monoid
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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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(add_lt_add_left : ∀a b, lt a b → ∀c, lt (add c a) (add c b))
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(lt_of_add_lt_add_left : ∀a b c, lt (add a b) (add a c) → lt 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_lt_add_left (H : a < b) (c : A) : c + a < c + b :=
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!ordered_cancel_comm_monoid.add_lt_add_left H c
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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 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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@ -37,19 +48,15 @@ section
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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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/- 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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lt_of_le_of_ne H1 H2
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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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@ -86,10 +93,11 @@ section
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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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!ordered_cancel_comm_monoid.lt_of_add_lt_add_left H
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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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lt_of_le_of_ne H1 H2
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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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@ -208,17 +216,26 @@ end
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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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(add_lt_add_left : ∀a b, lt a b → ∀ c, lt (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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--(lt_of_add_lt_add_left : ∀a b c, lt (add a b) (add a c) → lt b c)
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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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theorem ordered_comm_group.lt_of_add_lt_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_lt_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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le_of_add_le_add_left := @ordered_comm_group.le_of_add_le_add_left A s,
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lt_of_add_lt_add_left := @ordered_comm_group.lt_of_add_lt_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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@ -397,7 +414,21 @@ section
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end
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structure decidable_linear_ordered_comm_group [class] (A : Type)
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extends ordered_comm_group A, decidable_linear_order A
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extends add_comm_group A, decidable_linear_order 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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(add_lt_add_left : ∀ a b, lt a b → ∀ c, lt (add c a) (add c b))
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private theorem add_le_add_left' (A : Type) (s : decidable_linear_ordered_comm_group A) (a b : A) :
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a ≤ b → (∀ c : A, c + a ≤ c + b) :=
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decidable_linear_ordered_comm_group.add_le_add_left a b
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definition decidable_linear_ordered_comm_group.to_ordered_comm_group [instance] [reducible] [coercion]
|
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(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
|
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⦃ordered_comm_group, s,
|
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le_of_lt := @le_of_lt A s,
|
||||
add_le_add_left := add_le_add_left' A s,
|
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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
|
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variables [s : decidable_linear_ordered_comm_group A]
|
||||
|
|
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|
|
@ -197,7 +197,8 @@ definition ordered_ring.to_ordered_semiring [instance] [coercion] [reducible] [s
|
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mul_le_mul_of_nonneg_left := @ordered_ring.mul_le_mul_of_nonneg_left A s,
|
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mul_le_mul_of_nonneg_right := @ordered_ring.mul_le_mul_of_nonneg_right A s,
|
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mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left A s,
|
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mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right A s ⦄
|
||||
mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right A s,
|
||||
lt_of_add_lt_add_left := @lt_of_add_lt_add_left A s⦄
|
||||
|
||||
section
|
||||
variable [s : ordered_ring A]
|
||||
|
|
@ -262,7 +263,8 @@ end
|
|||
|
||||
-- TODO: we can eliminate mul_pos_of_pos, but now it is not worth the effort to redeclare the
|
||||
-- class instance
|
||||
structure linear_ordered_ring [class] (A : Type) extends ordered_ring A, linear_strong_order_pair A
|
||||
structure linear_ordered_ring [class] (A : Type) extends ordered_ring A, linear_strong_order_pair A :=
|
||||
(zero_lt_one : lt zero one)
|
||||
|
||||
-- print fields linear_ordered_semiring
|
||||
|
||||
|
|
@ -279,7 +281,8 @@ definition linear_ordered_ring.to_linear_ordered_semiring [instance] [coercion]
|
|||
mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right A s,
|
||||
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left A s,
|
||||
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right A s,
|
||||
le_total := linear_ordered_ring.le_total ⦄
|
||||
le_total := linear_ordered_ring.le_total,
|
||||
lt_of_add_lt_add_left := @lt_of_add_lt_add_left A s ⦄
|
||||
|
||||
structure linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_ring A, comm_monoid A
|
||||
|
||||
|
|
@ -336,7 +339,7 @@ section
|
|||
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
|
||||
|
||||
theorem zero_le_one : 0 ≤ (1:A) := one_mul 1 ▸ mul_self_nonneg 1
|
||||
theorem zero_lt_one : 0 < (1:A) := lt_of_le_of_ne zero_le_one zero_ne_one
|
||||
theorem zero_lt_one : 0 < (1:A) := linear_ordered_ring.zero_lt_one A
|
||||
|
||||
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
|
||||
(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=
|
||||
|
|
|
|||
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Reference in a new issue