Re: The Klotski Problem from last year's MOOC.
"Douglas Lewit [email protected] [ocaml_beginners]" <[email protected]> Thu, 22 Jun 2017 00:57:21 -0500
| Newsgroups | gmane.comp.lang.ocaml.beginners |
|---|---|
| Message-ID | <CAM0XMJTjXFKLyNkbf2oXDJ2yMwiCr+udHFCosNYRuYi3nbkG7A@mail.gmail.com> |
--f4030435c2047fe2d80552862865 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable My code copied okay--I think--but it didn't paste that well! :-( Oh well. If anyone can provide some feedback, that's great. IF not.... no worries. I'll figure it out. Thanks, Doug. On Wed, Jun 21, 2017 at 9:31 PM, Douglas Lewit [email protected] [ocaml_beginners] <[email protected]> wrote: > > > Hi everyone, > > Was anyone in the group a member of that OCaml MOOC last year that was > hosted by OCaml Pro and Universite Paris Diderot? Anyhow, do you recall > that Klotski Problem that was one of two final projects at the end of the > course? Can someone take a look at my first six functions for that > project. (If you were in the course and if you did the Klotski Project. > Otherwise don't worry about it!) Am I on the right track with this? I'm > not 100% sure that my functions are correct. They DO compile.... but the= n > again just because a program compiles is no guarantee of correctness!!!! = I > appreciate the feedback, and thanks for your time. > > Best, > > Douglas Lewit > > > *(* OCaml functions written by Douglas Lewit of Oakton Community College > and Northeastern Illinois University. *)* > > *exception NotFound ;;* > > *type 'e rel =3D 'e -> 'e list ;;* > > *let rec loop (p : ('a -> bool)) (f : ('a -> 'a)) (x : 'a) : 'a =3D match= p > x with * > * |true -> x * > * |false -> loop = p > f (f x) ;;* > > > *let rec exists (p : ('a -> bool)) (ls : 'a list) : bool =3D match ls wit= h * > * |[] -> false* > * |head :: tail -> if p head= * > * then true= * > * else > exists p tail ;;* > > *let rec find (p : ('a -> bool)) (ls : 'a list) : 'a =3D match ls with * > * |[] -> raise > NotFound * > * |head :: tail -> if p head = * > * then head = * > * else find = p > tail ;;* > > *(* near returns the function, near'. *)* > *let near : int rel =3D (* The default step size here is just 1. *)* > * let rec sequence ?step:(s=3D1) lower upper =3D * > * if lower > upper * > * then [] * > * else lower :: sequence ~step:s (lower + s= ) > upper * > * in * > * let near' (n : int) : int list =3D * > * let lower_bound =3D n - 2 in * > * let upper_bound =3D n + 2 in * > * sequence lower_bound upper_bound * > * in near' (* Here we have an example of "partial > function application". *) ;;* > > *(* flat_map returns the function, flat_map'. *)* > *let flat_map (rel_function : 'e rel) : 'e list -> 'e list =3D * > * let rec flatten (lst : 'e list list) : 'e list =3D * > * match lst with * > * |[] -> [] * > * |head :: tail -> match head with * > * |[] -> flatten tail * > * |hd :: tl -> hd :: flatten (tl :: > tail) * > * in * > * let rec flat_map' (e_list : 'e list) : 'e list list =3D * > * match e_list with * > * |[] -> [] * > * |head :: tail -> (rel_function head) :: flat_map' > tail * > * in* > * (* The variable "x" in my composition function is really a > dummy variable. *)* > * let composition (f : 'a -> 'b) (g : 'c -> 'a) (x : 'c) : '= b > =3D f (g x) * > * in composition flatten flat_map' (* This is an example of > "partial function application" * > * and also function > composition. *) ;;* > > *(* iter_rel should be used like this: (iter_rel near 5) 2 --> the > interpretation is that (iter_rel near 5) 2 should **yield the same > result as flat_map near (flat_map near( flat_map near( flat_map near (nea= r > 2)))). * > * Try it for yourself! By the way, this function was NOT easy to > write!!! *)* > *let iter_rel (rel_func : 'e rel) (i : int) : 'e rel =3D* > * let rec f x n =3D if n =3D 1* > * then x* > * else f (x |> flat_map rel_func) (n - 1)* > * in let g k =3D f (rel_func k) i* > * in g ;;* > > >=20 > --f4030435c2047fe2d80552862865 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/htm= l4/strict.dtd"> <html> <head> </head> =20 <body style=3D"background-color: #fff;"> <span style=3D"display:none"> </span> <!--~-|**|PrettyHtmlStartT|**|-~--> <div id=3D"ygrp-mlmsg" style=3D"position:relative;"> <div id=3D"ygrp-msg" style=3D"z-index: 1;"> <!--~-|**|PrettyHtmlEndT|**|-~--> <div id=3D"ygrp-text" > =20=20=20=20=20=20 =20=20=20=20=20=20 <p><div dir=3D"ltr"><font size=3D"4">My code copied okay--I think--bu= t it didn't paste that well! =C2=A0 =C2=A0:-( =C2=A0 =C2=A0 Oh well.=C2= =A0 If anyone can provide some feedback, that's great.=C2=A0 IF not....= no worries.=C2=A0 I'll figure it out.</font><div><font size=3D"4"><br>= </font></div><div><font size=3D"4">Thanks,</font></div><div><font size=3D"4= "><br></font></div><div><font size=3D"4">Doug.</font></div><div><font size= =3D"4"><br></font></div></div><div class=3D"gmail_extra"><br><div class=3D"= gmail_quote">On Wed, Jun 21, 2017 at 9:31 PM, Douglas Lewit <a href=3D"mail= to:[email protected]">[email protected]</a> [ocaml_beginners] <span dir=3D"= ltr"><<a href=3D"mailto:[email protected]" target=3D"_blan= k">[email protected]</a>></span> wrote:<br><blockquote cla= ss=3D"gmail_quote" style=3D"border-left:1px #ccc solid;"> <u></u> =20 <div style=3D"background-color:#fff;"> <span>=C2=A0</span> <div id=3D"m_5302536371079538634ygrp-mlmsg"> <div id=3D"m_5302536371079538634ygrp-msg"> <div id=3D"m_5302536371079538634ygrp-text"> =20=20=20=20=20=20 =20=20=20=20=20=20 <p></p><div dir=3D"ltr"><font size=3D"4" face=3D"monospace, monospace= ">Hi everyone,</font><div><font size=3D"4" face=3D"monospace, monospace"><b= r></font></div><div><font size=3D"4" face=3D"monospace, monospace">Was anyo= ne in the group a member of that OCaml MOOC last year that was hosted by OC= aml Pro and Universite Paris Diderot?=C2=A0 Anyhow, do you recall that Klot= ski Problem that was one of two final projects at the end of the course?=C2= =A0 Can someone take a look at my first six functions for that project. =C2= =A0(If you were in the course and if you did the Klotski Project.=C2=A0 Oth= erwise don't worry about it!) =C2=A0Am I on the right track with this?= =C2=A0 I'm not 100% sure that my functions are correct.=C2=A0 They DO c= ompile.... but then again just because a program compiles is no guarantee o= f correctness!!!!=C2=A0 I appreciate the feedback, and thanks for your time= .</font></div><div><font size=3D"4" face=3D"monospace, monospace"><br></fon= t></div><div><font size=3D"4" face=3D"monospace, monospace">Best,</font></d= iv><div><font size=3D"4" face=3D"monospace, monospace"><br></font></div><di= v><font size=3D"4" face=3D"monospace, monospace">Douglas Lewit</font></div>= <div><font size=3D"4" face=3D"monospace, monospace"><br></font></div><div><= font size=3D"4" face=3D"monospace, monospace"><br></font></div><div><font s= ize=3D"4"><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>(* = OCaml functions written by Douglas Lewit of Oakton Community College and No= rtheastern Illinois University. *)</b></font></div><div><font face=3D"monos= pace, monospace" color=3D"#0b5394"><b><br></b></font></div><div><font face= =3D"monospace, monospace" color=3D"#0b5394"><b>exception NotFound ;;</b></f= ont></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b><br= ></b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394= "><b>type 'e rel =3D 'e -> 'e list ;;</b></font></div><div><= font face=3D"monospace, monospace" color=3D"#0b5394"><b><br></b></font></di= v><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>let rec loo= p (p : ('a -> bool)) (f : ('a -> 'a)) (x : 'a) : '= ;a =3D match p x with=C2=A0</b></font></div><div><font face=3D"monospace, m= onospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 |= true =C2=A0 =C2=A0 -> x=C2=A0</b></font></div><div><font face=3D"monospa= ce, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 |false =C2=A0 =C2=A0-> loop p f (f x) ;;</b></font></div><div><font = face=3D"monospace, monospace" color=3D"#0b5394"><b><br></b></font></div><di= v><font face=3D"monospace, monospace" color=3D"#0b5394"><b><br></b></font><= /div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>let rec = exists (p : ('a -> bool)) (ls : 'a list) : bool =3D match ls wit= h=C2=A0</b></font></div><div><font face=3D"monospace, monospace" color=3D"#= 0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 |[] =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 -> false= </b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"= ><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 |head :: tail -> if p head=C2=A0</b></font></div><div>= <font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0then true=C2=A0<= /b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394">= <b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0else exists p tail ;;</b></font></div><div><font face=3D"monospace, m= onospace" color=3D"#0b5394"><b><br></b></font></div><div><font face=3D"mono= space, monospace" color=3D"#0b5394"><b>let rec find (p : ('a -> bool= )) (ls : 'a list) : 'a =3D match ls with=C2=A0</b></font></div><div= ><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 |[] =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0-> raise NotFound=C2=A0</b></font></di= v><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 |head= :: tail =C2=A0-> if p head=C2=A0</b></font></div><div><font face=3D"mon= ospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 then head=C2=A0</b></font></div><div><fo= nt face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 else find p tail ;;</b></f= ont></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b><br= ></b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394= "><b>(* near returns the function, near'. *)</b></font></div><div><font= face=3D"monospace, monospace" color=3D"#0b5394"><b>let near : int rel =3D = =C2=A0 (* The default step size here is just 1. *)</b></font></div><div><fo= nt face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0let rec seque= nce ?step:(s=3D1) lower upper =3D=C2=A0</b></font></div><div><font face=3D"= monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0if lower > upper=C2=A0</b></font></div><div><font face=3D"monospac= e, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0th= en []=C2=A0</b></font></div><div><font face=3D"monospace, monospace" color= =3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0else lower :: sequence = ~step:s (lower + s) upper=C2=A0</b></font></div><div><font face=3D"monospac= e, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0in=C2=A0</b></font></div><div>= <font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0let near&= #39; (n : int) : int list =3D=C2=A0</b></font></div><div><font face=3D"mono= space, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0let lower_bound =3D n - 2 in=C2=A0</b></font></div><div><font face=3D"mo= nospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0let upper_bound =3D n + 2 in=C2=A0</b></font></div><div><font face=3D"mo= nospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0sequence lower_bound upper_bound=C2=A0</b></font></div><div><font face= =3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0in near' (* Here= we have an example of "partial function application". *) ;;</b><= /font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b><= br></b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b53= 94"><b>(* flat_map returns the function, flat_map'. *)</b></font></div>= <div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>let flat_map = (rel_function : 'e rel) : 'e list -> 'e list =3D=C2=A0</b></= font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 let rec flatten (lst : = 9;e list list) : 'e list =3D=C2=A0</b></font></div><div><font face=3D"m= onospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 match lst with=C2=A0</b></fon= t></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0|[= ] =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 -> []=C2=A0</b></font></div><div><f= ont face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0|head :: tail ->= match head with=C2=A0</b></font></div><div><font face=3D"monospace, monosp= ace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0|[] =C2=A0 =C2=A0 =C2=A0 =C2=A0 -> flatten tail=C2=A0</b></fon= t></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0|hd :: tl =C2=A0 -&g= t; hd :: flatten (tl :: tail)=C2=A0</b></font></div><div><font face=3D"mono= space, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 in=C2=A0</b></font></div><div><font face=3D"monospace, monosp= ace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = let rec flat_map' (e_list : 'e list) : 'e list list =3D=C2=A0</= b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><= b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 match e_list with=C2=A0</b></font></div><div><font face=3D"monospace, m= onospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0|[] =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 -&= gt; []=C2=A0</b></font></div><div><font face=3D"monospace, monospace" color= =3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0|head :: tail -> (rel_function head) :: flat_map' t= ail=C2=A0</b></font></div><div><font face=3D"monospace, monospace" color=3D= "#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 in</b></font>= </div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 (* The variable "x" in = my composition function is really a dummy variable. *)</b></font></div><div= ><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 let composition (f : 'a -> 'b) (= g : 'c -> 'a) (x : 'c) : 'b =3D f (g x) =C2=A0</b></font= ></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 in composition flatten flat_map&= #39; (* This is an example of "partial function application"=C2= =A0</b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b53= 94"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 and also function composition. *) ;;</b>= </font></div><div><font face=3D"monospace, monospace" color=3D"#0b5394"><b>= <br></b></font></div><div><font face=3D"monospace, monospace" color=3D"#0b5= 394"><b>(* iter_rel should be used like this: (iter_rel near 5) 2 --> th= e interpretation is that (iter_rel near 5) 2 =C2=A0 =C2=A0 =C2=A0should=C2= =A0</b></font><b style=3D"color:rgb(11,83,148);font-family:monospace,monosp= ace;">yield the same result as flat_map near (flat_map near( flat_map near(= flat_map near (near 2)))). =C2=A0</b></div><div><font face=3D"monospace, m= onospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0Try it for yourself!=C2=A0 By t= he way, this function was NOT easy to write!!! *)</b></font></div><div><fon= t face=3D"monospace, monospace" color=3D"#0b5394"><b>let iter_rel (rel_func= : 'e rel) (i : int) : 'e rel =3D</b></font></div><div><font face= =3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 let rec f x n = =3D if n =3D 1</b></font></div><div><font face=3D"monospace, monospace" col= or=3D"#0b5394"><b><span style=3D"white-space:pre-wrap;"> </span> =C2=A0 = =C2=A0then x</b></font></div><div><font face=3D"monospace, monospace" color= =3D"#0b5394"><b><span style=3D"white-space:pre-wrap;"> </span> =C2=A0 =C2= =A0else f (x |> flat_map rel_func) (n - 1)</b></font></div><div><font fa= ce=3D"monospace, monospace" color=3D"#0b5394"><b>=C2=A0 =C2=A0 in let g k = =3D f (rel_func k) i</b></font></div><div><font face=3D"monospace, monospac= e" color=3D"#0b5394"><b>=C2=A0 =C2=A0 =C2=A0 =C2=A0in g ;;</b></font></div>= <div><br></div></font></div><div><font size=3D"4"><br></font></div></div> <p></p> </div> =20=20=20=20=20 =20=20=20=20 <div style=3D"color:#fff;height:0;"></div> </div> =20=20 </div></div></blockquote></div><br></div> </p> </div> =20=20=20=20=20 <!--~-|**|PrettyHtmlStart|**|-~--> <div style=3D"color: #fff; height: 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