65 lines
2.1 KiB
Text
65 lines
2.1 KiB
Text
----------------------------------------------------------------------------------------------------
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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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--- Author: Jeremy Avigad, Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic.axioms.funext data.bool
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using eq_ops bool
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namespace set
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definition set (T : Type) := T → bool
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definition mem {T : Type} (x : T) (s : set T) := (s x) = tt
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infix `∈` := mem
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section
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parameter {T : Type}
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definition empty : set T := λx, ff
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notation `∅` := empty
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theorem mem_empty (x : T) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, absurd H ff_ne_tt
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definition univ : set T := λx, tt
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theorem mem_univ (x : T) : x ∈ univ := refl _
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definition inter (A B : set T) : set T := λx, A x && B x
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infixl `∩` := inter
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theorem mem_inter (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B) :=
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iff_intro
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(assume H, and_intro (band_eq_tt_elim_left H) (band_eq_tt_elim_right H))
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(assume H,
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have e1 : A x = tt, from and_elim_left H,
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have e2 : B x = tt, from and_elim_right H,
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show A x && B x = tt, from e1⁻¹ ▸ e2⁻¹ ▸ band_tt_left tt)
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theorem inter_comm (A B : set T) : A ∩ B = B ∩ A :=
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funext (λx, band_comm (A x) (B x))
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theorem inter_assoc (A B C : set T) : (A ∩ B) ∩ C = A ∩ (B ∩ C) :=
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funext (λx, band_assoc (A x) (B x) (C x))
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definition union (A B : set T) : set T := λx, A x || B x
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infixl `∪` := union
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theorem mem_union (x : T) (A B : set T) : x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B) :=
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iff_intro
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(assume H, bor_to_or H)
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(assume H, or_elim H
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(assume Ha : A x = tt,
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show A x || B x = tt, from Ha⁻¹ ▸ bor_tt_left (B x))
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(assume Hb : B x = tt,
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show A x || B x = tt, from Hb⁻¹ ▸ bor_tt_right (A x)))
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theorem union_comm (A B : set T) : A ∪ B = B ∪ A :=
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funext (λx, bor_comm (A x) (B x))
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theorem union_assoc (A B C : set T) : (A ∪ B) ∪ C = A ∪ (B ∪ C) :=
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funext (λx, bor_assoc (A x) (B x) (C x))
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end
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end set
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