The Type of All Sets


 

 

10.2.4 The Type of All Sets

set is a type operator that applies to any list without duplicated elements. This code again uses some advanced material from TBoS.

(datatype set

   _____________
   [] : (set A);   
   
   X : A; Y : ((set A) & (without X));
   ===================================
   [X | Y] : (mode (set A) -);
   
   X : (set A);
   _____________
   X : (list A);
   
   X : P; X : Q;
   ====================
   X : (mode (P & Q) -);   

   ________________
   [] : (without X);

   if (not (= X Z))
   Y : (without Z);
   ______________________
   [X | Y] : (without Z);  
   
   ______________________________________________
   (not (element? X Y)) : verified >> Y : (without X);) 

We can show this works and define set union.

[1 2 3] : (set number)
[1 2 3] : (set number)

[1 2 3 1] : (set number)
type error
   
(define un
  {(set A) --> (set A) --> (set A)}
  [] S -> S
  [X | Y] S -> [X | (un Y S)]     where (not (element? X (un Y S)))
  [_ | Y] S -> (un Y S))
un : ((set A) --> ((set A) --> (set A)))
1. Introduction

2. License

3. History

4. The Core Language

4.1 Base Types
4.1.1 Symbols
4.1.2 Strings
4.1.3 Numbers
4.2 Function Applications
4.3 The Top Level
4.4 Arithmetic
4.5 Comments

4.6 Sequences

4.6.1 Lists
4.6.2 Tuples
4.6.3 Vectors

4.7 lambda and let
4.8 Global Assignments
4.9 Higher Order Functions
4.10 Lazy Evaluation
4.11 I/O
4.12 Loading Files
4.13 Streams
4.14 Exceptions
4.15 Hashing
4.16 Property Lists
4.17 Eval

5 Defining Functions

5.1 Partial Functions
5.2 List Handling Functions
5.3 String Handling Functions
5.4 Tuple Handling Functions
5.5 Vector Handling Functions
5.6 Guards
5.7 Backtracking
5.8 Writing in Kλ
5.9 Macros

6. Packages

7. Shen-YACC

7.1 Recognisor Generator
7.2 Semantic Actions

8. Shen Prolog

8.1 Sample Programs

9. Types

9.1 Types and Constructors
9.2 Functions and Types
9.3 Synonyms

10 Sequent Calculus

10.1 Recursive Types

10.2 Exotic Types

10.2.1 Dependent Types
10.2.2 Negative Types
10.2.3 Subtypes
10.2.4 The Type of All Sets

11 Glossary of Functions

12 The Syntax of Shen

Built by Shen Technology (c) Mark Tarver, September 2021