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https://github.com/nmvdw/HITs-Examples
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60 lines
1.8 KiB
Coq
60 lines
1.8 KiB
Coq
(** Classes for set operations, so they can be overloaded. *)
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Require Import HoTT.
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(** The operations on sets for which we add names. *)
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Section structure.
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Variable (T A : Type).
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Class hasMembership : Type :=
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member : A -> T -> hProp.
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Class hasMembership_decidable : Type :=
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member_dec : A -> T -> Bool.
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Class hasSubset : Type :=
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subset : T -> T -> hProp.
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Class hasSubset_decidable : Type :=
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subset_dec : T -> T -> Bool.
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Class hasEmpty : Type :=
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empty : T.
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Class hasSingleton : Type :=
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singleton : A -> T.
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Class hasUnion : Type :=
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union : T -> T -> T.
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Class hasIntersection : Type :=
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intersection : T -> T -> T.
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Class hasComprehension : Type :=
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filter : (A -> Bool) -> T -> T.
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End structure.
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Arguments member {_} {_} {_} _ _.
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Arguments subset {_} {_} _ _.
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Arguments member_dec {_} {_} {_} _ _.
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Arguments subset_dec {_} {_} _ _.
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Arguments empty {_} {_}.
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Arguments singleton {_} {_} {_} _.
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Arguments union {_} {_} _ _.
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Arguments intersection {_} {_} _ _.
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Arguments filter {_} {_} {_} _ _.
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Notation "∅" := empty.
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Notation "{| x |}" := (singleton x).
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Infix "∪" := union (at level 8, right associativity).
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Notation "(∪)" := union (only parsing).
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Notation "( X ∪)" := (union X) (only parsing).
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Notation "( ∪ Y )" := (fun X => X ∪ Y) (only parsing).
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Infix "∩" := intersection (at level 8, right associativity).
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Notation "( ∩ )" := intersection (only parsing).
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Notation "( X ∩ )" := (intersection X) (only parsing).
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Notation "( ∩ Y )" := (fun X => X ∩ Y) (only parsing).
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Notation "{| X & ϕ |}" := (filter ϕ X).
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Infix "∈" := member (at level 9, right associativity).
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Infix "⊆" := subset (at level 10, right associativity).
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Infix "∈_d" := member_dec (at level 9, right associativity).
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Infix "⊆_d" := subset_dec (at level 10, right associativity). |