56 lines
1.5 KiB
Text
56 lines
1.5 KiB
Text
/-
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Copyright (c) 2015 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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Author: Jeremy Avigad
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Notation for intervals and some properties.
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The mnemonic: o = open, c = closed, u = unbounded. For example, Iou a b is '(a, ∞).
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-/
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import .order data.set
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open set
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namespace intervals
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variables {A : Type} [order_pair A]
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definition Ioo (a b : A) : set A := {x | a < x ∧ x < b}
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definition Ioc (a b : A) : set A := {x | a < x ∧ x ≤ b}
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definition Ico (a b : A) : set A := {x | a ≤ x ∧ x < b}
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definition Icc (a b : A) : set A := {x | a ≤ x ∧ x ≤ b}
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definition Iou (a : A) : set A := {x | a < x}
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definition Icu (a : A) : set A := {x | a ≤ x}
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definition Iuo (b : A) : set A := {x | x < b}
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definition Iuc (b : A) : set A := {x | x ≤ b}
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notation `'` `(` a `, ` b `)` := Ioo a b
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notation `'` `(` a `, ` b `]` := Ioc a b
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notation `'[` a `, ` b `)` := Ico a b
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notation `'[` a `, ` b `]` := Icc a b
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notation `'` `(` a `, ` `∞` `)` := Iou a
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notation `'[` a `, ` `∞` `)` := Icu a
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notation `'` `(` `-∞` `, ` b `)` := Iuo b
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notation `'` `(` `-∞` `, ` b `]` := Iuc b
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variables a b c d e f : A
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/- some examples:
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check '(a, b)
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check '(a, b]
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check '[a, b)
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check '[a, b]
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check '(a, ∞)
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check '[a, ∞)
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check '(-∞, b)
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check '(-∞, b]
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check '{a, b, c}
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check '(a, b] ∩ '(c, ∞)
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check '(-∞, b) \ ('(c, d) ∪ '[e, ∞))
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-/
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proposition Iou_inter_Iuo : '(a, ∞) ∩ '(-∞, b) = '(a, b) := rfl
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end intervals
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