97 lines
3.1 KiB
Text
97 lines
3.1 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 data.bool
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open eq.ops bool
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namespace set
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definition set (T : Type) :=
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T → bool
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definition mem {T : Type} (x : T) (s : set T) :=
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(s x) = tt
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infix `∈` := mem
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definition eqv {T : Type} (A B : set T) : Prop :=
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∀x, x ∈ A ↔ x ∈ B
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infixl `∼`:50 := eqv
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theorem eqv_refl {T : Type} (A : set T) : A ∼ A :=
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take x, iff.rfl
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theorem eqv_symm {T : Type} {A B : set T} (H : A ∼ B) : B ∼ A :=
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take x, iff.symm (H x)
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theorem eqv_trans {T : Type} {A B C : set T} (H1 : A ∼ B) (H2 : B ∼ C) : A ∼ C :=
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take x, iff.trans (H1 x) (H2 x)
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definition empty {T : Type} : set T :=
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λx, ff
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notation `∅` := empty
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theorem mem_empty {T : Type} (x : T) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, absurd H ff_ne_tt
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definition univ {T : Type} : set T :=
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λx, tt
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theorem mem_univ {T : Type} (x : T) : x ∈ univ :=
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rfl
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definition inter {T : Type} (A B : set T) : set T :=
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λx, A x && B x
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infixl `∩` := inter
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theorem mem_inter {T : Type} (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_id {T : Type} (A : set T) : A ∩ A ∼ A :=
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take x, band.id (A x) ▸ iff.rfl
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theorem inter_empty_right {T : Type} (A : set T) : A ∩ ∅ ∼ ∅ :=
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take x, band.ff_right (A x) ▸ iff.rfl
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theorem inter_empty_left {T : Type} (A : set T) : ∅ ∩ A ∼ ∅ :=
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take x, band.ff_left (A x) ▸ iff.rfl
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theorem inter_comm {T : Type} (A B : set T) : A ∩ B ∼ B ∩ A :=
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take x, band.comm (A x) (B x) ▸ iff.rfl
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theorem inter_assoc {T : Type} (A B C : set T) : (A ∩ B) ∩ C ∼ A ∩ (B ∩ C) :=
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take x, band.assoc (A x) (B x) (C x) ▸ iff.rfl
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definition union {T : Type} (A B : set T) : set T :=
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λx, A x || B x
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infixl `∪` := union
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theorem mem_union {T : Type} (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_id {T : Type} (A : set T) : A ∪ A ∼ A :=
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take x, bor.id (A x) ▸ iff.rfl
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theorem union_empty_right {T : Type} (A : set T) : A ∪ ∅ ∼ A :=
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take x, bor.ff_right (A x) ▸ iff.rfl
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theorem union_empty_left {T : Type} (A : set T) : ∅ ∪ A ∼ A :=
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take x, bor.ff_left (A x) ▸ iff.rfl
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theorem union_comm {T : Type} (A B : set T) : A ∪ B ∼ B ∪ A :=
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take x, bor.comm (A x) (B x) ▸ iff.rfl
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theorem union_assoc {T : Type} (A B C : set T) : (A ∪ B) ∪ C ∼ A ∪ (B ∪ C) :=
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take x, bor.assoc (A x) (B x) (C x) ▸ iff.rfl
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end set
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