Exercise 4 - Closed intervals whose bounds are integers

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Exercise 4 - Closed intervals whose bounds are integers

The non-built-in type  closed_interval  will be used. It is provided by the  Closed_interval  module.

Specification of the type  closed_interval

This type gives each closed interval whose bounds are integers a unique representation. It is equipped with the five following functions: The constructor  cons_closed  constructs the representation of the interval [a,b] from a pair (a,b) of integers where a is less than or equal to b. The accessors  lower_bound  et  upper_bound  access the bounds of an interval. The conversion function  string_of_closed_interval  gives a string representation of an interval. The printing function  print_closed_interval  prints such a string representation. The interface definition of the module (i.e. the content of the  Closed_interval.mli  file) shows the types of these five functions: type closed_interval val cons_closed : int * int -> closed_interval val lower_bound : closed_interval -> int val upper_bound : closed_interval -> int val string_of_closed_interval : closed_interval -> string val print_closed_interval : closed_interval -> unit It also indicates that the type  closed_interval  is abstract. The type definition phrase  type closed_interval  only contains the keyword  type  followed by the name  closed_interval , indeed. After this there is nothing. Therefore the type representation is hidden.

1

Name:  width . Type:  closed_interval -> int . Computes the width of a closed interval.

2

Name:  is_member . Type:  closed_interval -> bool . Checks whether z belongs to i given a pair (z,i) where z is an integer and i is an interval.

3

Name:  cons_closed_1 . Type:  int * int -> closed_interval . Computes the interval whose lower bound is z and width is n given a pair (z,n) where z is an integer and n is a non-negative integer.

4

Name:  are_disjoint . Type:  closed_interval * closed_interval -> bool . Checks whether two intervals are disjoint.

5

Name:  intersection . Type:  closed_interval * closed_interval -> closed_interval . Computes the intersection of two non-disjoint intervals.

6

Name:  is_subset . Type:  closed_interval * closed_interval -> bool . Checks whether i is a proper subset of or is equal to j given a pair (i,j) of intervals (is a subset means is a proper subset of or is equal to).

7

Name:  envelope . Type:  closed_interval * closed_interval -> closed_interval . Computes the envelope of two intervals, that is, the smallest interval that is a superset of both.

8

First create a directory. Create in this directory a file containing all these definitions and whose name has extension  ml . For instance  Use_closed_interval.ml . Write also in this directory the content of the archive  Closed_interval , that is, the files  Closed_interval.cmi  and  Closed_interval.cmo  (in other words: the module  Closed_interval ). Enter this directory. Build a Caml toplevel that contains the compiled implementation of the module  Closed_interval  preloaded at start-up. To do this, execute the Unix command ocamlmktop -o mytoplevel Closed_interval.cmo To run this new toplevel execute the Unix command mytoplevel In this toplevel first evaluate the Caml phrase open Closed_interval This opens the module  Closed_interval , which allows to use short names instead of long ones in dotted notation (that is, for instance  lower_bound  instead of  Closed_interval.lower_bound ). Then evaluate the Caml phrase #use "Use_closed_interval.ml" This reads, compiles and executes all the source phrases from the given file. Now test each of the definitions, that is, apply each of the function values several times.

Exercise 4 - Closed intervals whose bounds are integers

Documentation and user's manual

Table of contents

OCaml programs