48 lines
984 B
Plaintext
48 lines
984 B
Plaintext
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import Common.Set.OrderedPair
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import Mathlib.Data.Set.Basic
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import Mathlib.Data.Set.Prod
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/-! # Relations
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A representation of a relation, i.e. a set of ordered pairs. Like `Set`, it is
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assumed a relation is homogeneous.
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-/
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namespace Set
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/--
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A relation type as defined by Enderton.
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We choose to use tuples to represent our ordered pairs, as opposed to
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Kuratowski's definition of a set.
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Not to be confused with the Lean-provided `Rel`.
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-/
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abbrev Relation (α : Type _) := Set (α × α)
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namespace Relation
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/--
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The domain of a `Relation`.
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-/
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def dom (R : Relation α) : Set α := Prod.fst '' R
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/--
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The range of a `Relation`.
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-/
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def ran (R : Relation α) : Set α := Prod.snd '' R
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/--
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The field of a `Relation`.
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-/
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def fld (R : Relation α) : Set α := dom R ∪ ran R
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/--
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Convert a `Relation` into an equivalent representation using `OrderedPair`s.
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-/
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def toOrderedPairs (R : Relation α) : Set (Set (Set α)) :=
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R.image (fun (x, y) => OrderedPair x y)
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end Relation
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end Set
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