├── .gitattributes ├── .github └── FUNDING.yml ├── .gitignore ├── .ocamlformat ├── LICENSE ├── README.org ├── chapter1 ├── chapter1.md ├── dune └── dune.inc ├── chapter10 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter11 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter12 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter13 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter14 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter15 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter16 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter17 └── README.md ├── chapter18 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter19 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter2 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter20 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter21 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter22 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter23 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter24 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter25 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter26 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter27 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter3 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter30 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter4 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter5 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter6 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter7 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter8 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── chapter9 ├── README.md ├── dune ├── dune.inc └── prelude.ml ├── dune ├── dune-project ├── ocaml-ctfp.install ├── ocaml-ctfp.opam └── playground ├── .merlin ├── dune ├── dune-project ├── playground.md └── playground.md.out.expected /.gitattributes: -------------------------------------------------------------------------------- 1 | *.md linguist-language=OCaml 2 | -------------------------------------------------------------------------------- /.github/FUNDING.yml: -------------------------------------------------------------------------------- 1 | # These are supported funding model platforms 2 | 3 | github: [ArulselvanMadhavan] 4 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | _opam/** 2 | **/_build/** 3 | **/.merlin -------------------------------------------------------------------------------- /.ocamlformat: -------------------------------------------------------------------------------- 1 | profile=janestreet 2 | margin=70 -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.org: -------------------------------------------------------------------------------- 1 | * OCaml version of Category Theory For Programmers 2 | 3 | ** How do I get the book? 4 | - Currently, the book can be download by cloning this fork - 5 | https://github.com/ArulselvanMadhavan/milewski-ctfp-pdf and 6 | running 7 | #+BEGIN_SRC bash 8 | cd milewski-ctfp-pdf 9 | git checkout ocaml 10 | cd src 11 | make ocaml 12 | #+END_SRC 13 | 14 | ** Acknowledgements 15 | - Thanks to Marcello Seri @mseri for doing the code review. If @mseri 16 | hadn't offered to do the code review, this project would not have started 17 | - Thanks to @XVilka for promoting this effort in reddit, ocaml 18 | forums and offering helpful links 19 | - All the kind people in OCaml discord who hang out in the 20 | #beginners channel answering questions 21 | 22 | ** Helpful Resources 23 | - Resources I found useful as I started translating the book in OCaml: 24 | 1) https://github.com/freebroccolo/ocaml-cats 25 | 2) https://github.com/ocamllabs/higher 26 | 27 | -------------------------------------------------------------------------------- /chapter1/chapter1.md: -------------------------------------------------------------------------------- 1 | # Chapter 1 - The essence of composition 2 | ### Utilities (Feel free to skip. Used to compile code below) 3 | ```ocaml 4 | # let (>>) : 'a 'b 'c. ('b -> 'c) -> ('a -> 'b) -> 'a -> 'c = 5 | fun g f x -> g (f x) 6 | val ( >> ) : ('b -> 'c) -> ('a -> 'b) -> 'a -> 'c = 7 | # let f : string -> int = String.length 8 | val f : string -> int = 9 | ``` 10 | ### Code snippets 11 | * A generic function from type a to type b. 12 | ```ocaml 13 | module type Polymorphic_Function_F = sig 14 | type a 15 | type b 16 | val f : a -> b 17 | end 18 | ``` 19 | * Another polymorphic function from type b to type c. 20 | ```ocaml 21 | module type Polymorphic_Function_G = sig 22 | type b 23 | type c 24 | val g : b -> c 25 | end 26 | ``` 27 | * Compose Function definition is not part of the standard library. We can define a custom /compose/ function. 28 | ```ocaml 29 | module Compose_Example = functor(F: Polymorphic_Function_F)(G: Polymorphic_Function_G with type b = F.b) -> struct 30 | (** OCaml doesn't have a compose operator. So, creating one. **) 31 | let (>>) g f x = g (f x) 32 | let compose : 'a -> 'c = G.g >> F.f 33 | end 34 | ``` 35 | * Compose Three functions 36 | ```OCaml 37 | module Compose_Three_GF = functor(F:Polymorphic_Function_F)(G:Polymorphic_Function_G with type b = F.b)(H:Polymorphic_Function_H with type c = G.c) -> struct 38 | let compose : 'a -> 'd = H.h >> (G.g >> F.f) 39 | end 40 | 41 | module Compose_Three_HG = functor(F:Polymorphic_Function_F)(G:Polymorphic_Function_G with type b = F.b)(H:Polymorphic_Function_H with type c = G.c) -> struct 42 | let compose : 'a -> 'd = (H.h >> G.g) >> F.f 43 | end 44 | 45 | Compose_Three_GF.compose = Compose_Three_HG.compose 46 | ``` 47 | * Identity Function 48 | ```ocaml 49 | # let id x = x 50 | val id : 'a -> 'a = 51 | ``` 52 | * Compose and Identity 53 | ```OCaml 54 | f >> id 55 | id >> f 56 | ``` 57 | -------------------------------------------------------------------------------- /chapter1/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml chapter1.md))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps chapter1.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter1/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x chapter1.md)) 4 | (action (progn 5 | (run ocaml-mdx test --direction=to-ml %{x}) 6 | (diff? %{x} %{x}.corrected) 7 | ))) 8 | -------------------------------------------------------------------------------- /chapter10/README.md: -------------------------------------------------------------------------------- 1 | # Natural Transformations 2 | ### Utilities used by the code below 3 | ```ocaml 4 | type ('a, 'b) const = Const of 'a 5 | module type Functor = sig 6 | type 'a t 7 | val fmap : ('a -> 'b) -> 'a t -> 'b t 8 | end 9 | let compose f g x = f (g x) 10 | module type Contravariant = sig 11 | type 'a t 12 | val contramap : ('b -> 'a) -> 'a t -> 'b t 13 | end 14 | ``` 15 | - Functor is a mapping between categories that preserve their structure. 16 | - Embeds source category in target category by using the source as the blueprint. 17 | - Functor 18 | - may collapse the whole source category into one object in the target category. 19 | - map every object to a different object 20 | - map every morphism to different morphism 21 | - Natural transformation help us compare these realizations. 22 | - Mappings of functors. 23 | - Natural isomorphism is defined as a natural transformation whose components are all isomorphisms. 24 | ### Polymorphic Functions 25 | - Endofunctors - Type constructors that map types to types. Source and target category are the same. 26 | ```OCaml 27 | val alpha : 'a . 'a f -> 'a g 28 | ``` 29 | - Without universal quantification. 30 | ```OCaml 31 | val alpha : 'a f -> 'a g 32 | ``` 33 | - Polymorphic functions like this automatically satisfy the naturality condition. 34 | ```OCaml 35 | val alpha : 'a f -> 'a g 36 | ``` 37 | - Free theorems are the naturality conditions, in the case of natural transformations expressed using parametric polymorphism. 38 | ```ocaml 39 | # let safe_head = function 40 | | [] -> None 41 | | x :: xs -> Some x 42 | val safe_head : 'a list -> 'a option = 43 | ``` 44 | - Verifying the Naturality condition(Pseudo OCaml, since function equality cannot be expressed) 45 | ```OCaml 46 | compose (fmap f) safe_head = compose safe_head (fmap f) 47 | ``` 48 | - Cases to handle 49 | ```OCaml 50 | (* Starting with empty list *) 51 | let fmap f (safe_head []) = fmap f None = None 52 | ``` 53 | ```OCaml 54 | let safe_head (fmap f []) = safe_head [] = None 55 | ``` 56 | - Non empty list 57 | ```OCaml 58 | let fmap f (safe_head (x :: xs)) = fmap f (Some x)= Some (f x) 59 | ``` 60 | ```OCaml 61 | let safe_head (fmap f (x :: xs)) = safe_head (f x :: f xs) = Some (f x) 62 | ``` 63 | - Implemenation of fmap for lists 64 | ```ocaml 65 | # let rec fmap f = function 66 | | [] -> [] 67 | | (x :: xs) -> f x :: fmap f xs 68 | val fmap : ('a -> 'b) -> 'a list -> 'b list = 69 | ``` 70 | - Implementation of fmap for option type 71 | ```ocaml 72 | # let rec fmap f = function 73 | | None -> None 74 | | Some x -> Some (f x) 75 | val fmap : ('a -> 'b) -> 'a option -> 'b option = 76 | ``` 77 | - Natural transformation to or from the *Const* functor looks just like a function that's polymorphic in either the return type or argument type. 78 | ```ocaml 79 | (** OCaml requires mutually recursive functions to be defined together *) 80 | let rec length : 'a list -> (int, 'a) const = function 81 | | [] -> Const 0 82 | | (_ :: xs) -> Const (1 + un_const (length xs)) 83 | and un_const : 'c 'a. ('c, 'a) const -> 'c = function | Const c -> c 84 | ``` 85 | - unconst defined separately 86 | ```ocaml 87 | let un_const : 'c 'a. ('c, 'a) const -> 'c = function | Const c -> c 88 | ``` 89 | - In practice, length is defined as 90 | ```OCaml 91 | val length : 'a list -> int 92 | ``` 93 | - Parametrically polymorphic function from `Const` functor 94 | ```ocaml 95 | # let scam : 'a. ('int, 'a) const -> 'a option = function 96 | | Const a -> None 97 | val scam : ('int, 'a) const -> 'a option = 98 | ``` 99 | - Reader type 100 | ```ocaml 101 | type ('e, 'a) reader = Reader of ('e -> 'a) 102 | ``` 103 | - Functor instance for Reader type 104 | ```ocaml 105 | module Reader_Functor(T : sig type e end) : Functor = struct 106 | type 'a t = (T.e, 'a) reader 107 | let fmap : 'a 'b. ('a -> 'b) -> 'a t -> 'b t = fun f -> function 108 | | Reader r -> Reader (compose f r) 109 | end 110 | ``` 111 | - Natural Transformation from Reader () a -> Maybe a 112 | ```OCaml 113 | val alpha : (unit, 'a) reader -> 'a option 114 | ``` 115 | - dumb version 116 | ```ocaml 117 | # let dumb : 'a. (unit, 'a) reader -> 'a option = function 118 | | Reader _ -> None 119 | val dumb : (unit, 'a) reader -> 'a option = 120 | ``` 121 | - obvious implementation 122 | ```ocaml 123 | # let obvious : 'a. (unit, 'a) reader -> 'a option = function 124 | | Reader f -> Some (f ()) 125 | val obvious : (unit, 'a) reader -> 'a option = 126 | ``` 127 | ### Beyond Naturality 128 | - Parametrically polymorphic function between two functors is always a natural transformation. 129 | - Function types are covariant in their return type. 130 | - Function types are contravariant in their argument type. 131 | - Contravariant example 132 | ```ocaml 133 | type ('r, 'a) op = Op of ('a -> 'r) 134 | ``` 135 | - Contravariant instance 136 | ```ocaml 137 | module Op_Contravariant(T : sig type r end): Contravariant = struct 138 | type 'a t = (T.r, 'a) op 139 | let contramap : ('b -> 'a) -> 'a t -> 'b t = fun f -> function 140 | | Op g -> Op (compose g f) 141 | end 142 | ``` 143 | - predToStr 144 | ```ocaml 145 | # let pred_to_str = function 146 | | Op f -> Op (fun x -> if f x then "T" else "F") 147 | val pred_to_str : (bool, 'a) op -> (string, 'a) op = 148 | ``` 149 | - Contravariant functors satisfy the opposite naturality condition. 150 | ```OCaml 151 | compose (contramap f) pred_to_str = compose pred_to_str (contramap f) 152 | ``` 153 | - Op Bool contramap signature 154 | ```ocaml 155 | module Op_Bool = Op_Contravariant(struct type r = bool end) 156 | let op_bool_contramap : ('b -> 'a) -> 'a Op_Bool.t -> 'b Op_Bool.t = Op_Bool.contramap 157 | ``` 158 | - Type constructors that are neither covariant or contravariant. 159 | ```OCaml 160 | 'a -> 'a 161 | ``` 162 | ```OCaml 163 | ('a -> 'a) -> 'a f 164 | ``` 165 | - Dinatural transformation is a generalization of the natural transformations. 166 | ### Functor Category 167 | - There is one category of functors for each pair of categories, C and D. 168 | - Objects in this category are functors from C to D. 169 | - Morphisms = Natural transformations between C and D. 170 | - Composition of natural transformations is associative. 171 | - Functor category between C and D is Fun(C, D) or |C, D| or D^C 172 | ### Revisiting abstractions 173 | - Category 174 | - objects and morphisms 175 | - Cat - Higher order category that contains other categories as objects. 176 | - Morphisms in Cat are functors. 177 | - Hom-Set in Cat is a set of functors. Cat(C,D) - Set of functors between C and D categories. 178 | - Functor Category |C,D| is *also* a set of functors between C and D. 179 | - Functor Category is also a category. So it must be part of Cat. 180 | - Cat is a full blown Cartesian Closed Category in which there is an exponential object D^C for any pair of categories. This exponential object is a category - Functor Category Fun(C, D) 181 | ### 2-Categories 182 | - Cat - Higher order category with categories as objects and functors as morphisms. 183 | - Hom-Set in Cat is a set of functors. 184 | - Set of functors form a category. 185 | - Functors are morphisms in Cat. Natural transformations are morphisms between morphisms. 186 | - 2-category contains 187 | - objects - Categories 188 | - 1 morphisms - Functors between categories 189 | - 2 morphisms between morphisms - Natural transformation between functors. 190 | -------------------------------------------------------------------------------- /chapter10/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter10/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter10/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter11/README.md: -------------------------------------------------------------------------------- 1 | # Declarative Programming 2 | ### Utilities (Needed for compiling the code. Skip this section) 3 | ```ocaml 4 | let compose g f x = g (f x) 5 | ``` 6 | - Snippets marked as Pseudo OCaml are not compiled by mdx 7 | ### Declarative Programming in compose 8 | - Compose (declarative) 9 | ```OCaml 10 | (* Assume g and f are already defined *) 11 | let h = compose g f 12 | ``` 13 | - Compose (imperative) 14 | ```OCaml 15 | let h = fun x -> 16 | let y = f x in 17 | g y 18 | ``` 19 | -------------------------------------------------------------------------------- /chapter11/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter11/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter11/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter12/README.md: -------------------------------------------------------------------------------- 1 | # Limits and Colimits 2 | ### Utilities used by the code below 3 | ```ocaml 4 | let compose f g x = f (g x) 5 | ``` 6 | - In CT, everything is related to everything and can be viewed from many angles. 7 | ### Generalizing products 8 | - 2 category 9 | - contains two objects. 10 | - No morphisms between them. 11 | - Only identity morphsisms. 12 | - Selecting two objects in C is the same as defining a functor(D) from 2 to C. 13 | - Selecting a candidate object c 14 | - A constant functor from 2 to C 15 | - Selection of c in C is done using the constant functor. 16 | - Natural transformation between constant functor (to c) and D 17 | - NT component - 'c' to 'a' is a morphism - p 18 | - NT component - 'c' to 'b' is a morphism - q 19 | ```OCaml 20 | let p1 = compose p m 21 | let q1 = compose q m 22 | ``` 23 | - Contramap definition 24 | ```ocaml 25 | # let contramap : ('c_prime -> 'c) -> ('c -> 'limD) -> ('c_prime -> 'limD) = fun f u -> compose u f 26 | val contramap : ('c_prime -> 'c) -> ('c -> 'limD) -> 'c_prime -> 'limD = 27 | 28 | ``` 29 | - Presheaves: Contravariant functor from any category C to Set 30 | - Natural Isomorphism: Natural Transformation whose every component is an isomorphism. 31 | ### Examples of limits 32 | - Example of a limit - Terminal object 33 | - Terminal object is a limit generated by an empty category. 34 | - Functor from an empty category selects no object, so a cone shrinks to just the apex. 35 | - Equalizer - A limit generated by a two-element category with two parallel morphisms going between them 36 | - 37 | ```OCaml 38 | val f : 'a -> 'b 39 | val g : 'a -> 'b 40 | ``` 41 | ```OCaml 42 | val p : 'c -> 'a 43 | val q : 'c -> 'b 44 | ``` 45 | ```OCaml 46 | q = compose f p 47 | q = compose g p 48 | ``` 49 | ```OCaml 50 | (** Pseudo OCaml expressing function equality **) 51 | compose f p = compose g p 52 | ``` 53 | - Example: a - two dimensional plane parameterized by coordinates x and y 54 | - b - real line 55 | ```ocaml 56 | let f (x, y) = 2 * y + x 57 | let g (x, y) = y - x 58 | ``` 59 | - 60 | ```ocaml 61 | let p t = (t, (-2) * t) 62 | ``` 63 | ```OCaml 64 | (** Pseudo OCaml expressing function equality **) 65 | compose f p' = compose g p' 66 | ``` 67 | ```ocaml 68 | let p' () = (0, 0) 69 | ``` 70 | ```OCaml 71 | let p' = compose p h 72 | ``` 73 | ```ocaml 74 | let h () = 0 75 | ``` 76 | - Pullback - Has two morphisms that we want to equate but their domains are different. 77 | ```OCaml 78 | val f : 'a -> 'b 79 | val g : 'c -> 'b 80 | ``` 81 | - cospan 82 | ```OCaml 83 | val p : 'd -> 'a 84 | val q : 'd -> 'c 85 | val r : 'd -> 'b 86 | ``` 87 | - Commutativity conditions 88 | ```OCaml 89 | compose g q = compose f p 90 | ``` 91 | ```ocaml 92 | let f x = 1.23 93 | ``` 94 | ### Colimits 95 | - Dual of limits 96 | - Co-cone - Invert the direction of all arrows in a cone 97 | - Universal co-cone - Co-limit 98 | - Example of a co-limit - co-product 99 | - Terminal object - limit 100 | - Initial object - co-limit 101 | - Dual of the pullback - pushout ex: 1 <- 2 -> 3 102 | ### Continuity 103 | - Continuous Functor F from C to C' preserves limits 104 | - Functors preserve almost everything. 105 | - Uniqueness condition may be the only affected property 106 | - Hom-functor C^op x C -> Set 107 | ```ocaml 108 | module type Contravariant = sig 109 | type 'a t 110 | val contramap : ('b -> 'a) -> 'a t -> 'b t 111 | end 112 | type 'a tostring = ToString of ('a -> string) 113 | 114 | module ToStringInstance : Contravariant = struct 115 | type 'a t = 'a tostring 116 | let contramap f (ToString g) = ToString (compose g f) 117 | end 118 | ``` 119 | ```OCaml 120 | ('b 'c either) tostring ~ ('b -> string, 'c -> string) 121 | ``` 122 | ```OCaml 123 | 'r -> ('a, 'b) ~ ('r -> 'a, 'r -> 'b) 124 | ``` 125 | 126 | -------------------------------------------------------------------------------- /chapter12/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter12/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter12/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter13/README.md: -------------------------------------------------------------------------------- 1 | # Free Monoids 2 | ## Utilities used by the code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | let compose f g x = f (g x) 9 | ``` 10 | - Categories - strongly typed languages 11 | - Monoids - untyped languages 12 | - Monoid - Category with a single object, where all logic is encoded in rules of morphism composition. 13 | - Categorical model of monoid is fully equivalent to set theoretical definition of monoid. 14 | - Generators of the free monoid - basic set of elements needed to generate free monoid 15 | - Free construction - keep generating all possible combination of elements, and perform the minimum number of identifications - just enough to uphold the laws 16 | ### Free Monoid in OCaml 17 | ```ocaml 18 | module type Monoid = sig 19 | type m 20 | val mempty : m 21 | val mappend : m -> m -> m 22 | end 23 | ``` 24 | - Monoid instance for list 25 | ```ocaml 26 | module ListMonoid(T1 : sig type a end) : (Monoid with type m = T1.a list) = struct 27 | type m = T1.a list 28 | let mempty = [] 29 | let mappend xs ys = List.append xs ys 30 | end;; 31 | ``` 32 | ```OCaml 33 | 2 * 3 = 6 34 | List.append [2] [3] = [2; 3] 35 | ``` 36 | ### Free monoid universal construction 37 | ```OCaml 38 | let h (a * b) = h a * h b 39 | ``` 40 | - Homomorphism from lists of integers to integers 41 | ```OCaml 42 | List.append [2] [3] = [2; 3] 43 | ``` 44 | - becomes multiplication 45 | ```ocaml 46 | 2 * 3 = 6 47 | ``` 48 | - Let p be the function that identifies the set of generators inside the X-ray image of monoid m. 49 | ```ocaml 50 | module type FreeMonoidRep = functor (F : Functor) -> sig 51 | type x 52 | type m 53 | val p : x -> m F.t 54 | end 55 | ``` 56 | - Similar function on a different monoid n can be 57 | ```ocaml 58 | module type FreeMonoidRep = functor (F : Functor) -> sig 59 | type x 60 | type n 61 | val q : x -> n F.t 62 | end 63 | ``` 64 | - Homomorphism between monoids 65 | ```OCaml 66 | val h : m -> n 67 | ``` 68 | - Building q through p 69 | ```OCaml 70 | val q = compose uh p 71 | ``` 72 | -------------------------------------------------------------------------------- /chapter13/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter13/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter13/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter14/README.md: -------------------------------------------------------------------------------- 1 | # Representable Functors 2 | ## Utilities used by the code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | 9 | module type Contravariant = sig 10 | type 'a t 11 | val contramap : ('b -> 'a) -> 'a t -> 'b t 12 | end 13 | 14 | module type Profunctor = sig 15 | type ('a,'b) p 16 | val dimap : ('a -> 'b) -> ('c -> 'd) -> ('b, 'c) p -> ('a, 'd) p 17 | val lmap : ('a -> 'b) -> ('b, 'c) p -> ('a, 'c) p 18 | val rmap : ('b -> 'c) -> ('a, 'b) p -> ('a, 'c) p 19 | end 20 | 21 | let flip f b a = f a b 22 | let compose f g x = f (g x) 23 | ``` 24 | - Set theory - assembly language of mathematics 25 | - *Set* category of all sets 26 | - morphisms between any two objects in a category form a set (hom-set) 27 | - All the information about sets can be encoded in *isormorphic* functions between them. 28 | - Interaction between programming and math goes both ways. 29 | ## Hom Functor 30 | - There's family of mappings from a category to Set 31 | - These mappings are functors. 32 | - hom-functor is reader functor 33 | ```ocaml 34 | type ('a, 'x) reader = 'a -> 'x 35 | ``` 36 | - C(a,-) - CoVariant Functor instance 37 | ```ocaml 38 | module ReaderFunctor(T : sig type r end) : Functor = struct 39 | type 'a t = (T.r, 'a) reader 40 | let fmap f h = fun a -> f (h a) 41 | end 42 | ``` 43 | - C(-,a) - Contravariant Functor 44 | ```ocaml 45 | type ('a, 'x) op = 'x -> 'a 46 | ``` 47 | - Contravariant instance 48 | ```ocaml 49 | module OpContravariant(T : sig type r end) : Contravariant = struct 50 | type 'a t = (T.r, 'a) op 51 | let contramap f h = fun b -> h (f b) 52 | end 53 | ``` 54 | - C(-, -) - If we allows both arguments to change 55 | ```ocaml 56 | module ProfunctorArrow : Profunctor = struct 57 | type ('a, 'b) p = 'a -> 'b 58 | let dimap f g p = compose g (compose p f) 59 | let lmap f p = (flip compose) f p 60 | let rmap = compose 61 | end 62 | ``` 63 | - The mapping of objects from any category to hom-sets is functorial. 64 | - C^op x C -> Set 65 | ## Representable Functors 66 | - Structure preserving mapping to Set is called a *representation*. 67 | - Any functor F that is naturally isomorphic to hom-functor for some choice of a is called representable functor. 68 | - For F to be representable, 69 | - an object a in C 70 | - Nat Transformation(between functors) - alpha - C(a, -) to F 71 | - Nat Transformation - beta - F to C(a, -) 72 | - Composition of these two Nat Trans is Identity Nat Trans 73 | - alphaAtx :: C(a, x) -> Fx 74 | ```ocaml 75 | module type NT_AX_FX = sig 76 | type a 77 | type 'x t 78 | val alpha : (a -> 'x) -> 'x t 79 | end 80 | ``` 81 | - Pseudo OCaml expressing function equality 82 | ```OCaml 83 | compose (F.fmap f) NT.alpha = compose NT.alpha (F.fmap f) 84 | ``` 85 | - Replacing fmap with function composition (since fmap on f is a reader functor on the right hand side) 86 | ```OCaml 87 | F.fmap f (N.alpha h) = N.alpha (compose f h) 88 | ``` 89 | - Other Nat Transformation that goes opposite direction 90 | ```ocaml 91 | module type NT_FX_AX = sig 92 | type a 93 | type 'x t 94 | val beta : 'x t -> (a -> 'x) 95 | end 96 | ``` 97 | - alpha . beta = id = beta . alpha 98 | - Yoneda Lemma: Nat Trans from C(a, -) to any Set-valued functor always exists but it's not always invertible. 99 | - Alpha example 100 | ```ocaml 101 | module NT_Impl(F: Functor with type 'a t = 'a list) : NT_AX_FX with type a = int and type 'x t = 'x list = struct 102 | type a = int 103 | type 'x t = 'x list 104 | let alpha: 'x. (int -> 'x) -> 'x list = fun h -> F.fmap h [12] 105 | end 106 | ``` 107 | - Naturality condition is equivalent to the composiability of map. 108 | ```OCaml 109 | F.fmap f (F.fmap h [12]) = F.fmap (compose f h) [12] 110 | ``` 111 | - Beta with example on List and Int 112 | ```ocaml 113 | module type NT_ListX_IntX = sig 114 | type a = int 115 | type 'x t = 'x list 116 | val beta : 'x t -> (a -> 'x) 117 | end 118 | ``` 119 | - Beta can't be implemented for List and Int combination. (When List is empty we can't return a value of type x) 120 | - So, List functor is not Representable. 121 | - Rep. functors are containers for memoized results of function calls 122 | - Representing type 'a' of C(a, -) is the key type to access values of a function. 123 | - *alpha* is called tabulate and *beta* is called index. 124 | - Representable functor 125 | ```ocaml 126 | module type Representable = sig 127 | type 'x t 128 | type rep (* Representing type 'a' *) 129 | val tabulate : (rep -> 'x) -> 'x t 130 | val index : 'x t -> (rep -> 'x) 131 | end 132 | ``` 133 | - Stream type 134 | ```ocaml 135 | type 'a stream = | Cons of 'a * 'a stream Lazy.t 136 | ``` 137 | - Representable functor Instance 138 | ```ocaml 139 | module StreamRepresentable : Representable = struct 140 | type rep = int 141 | type 'x t = 'x stream 142 | let rec tabulate f = Cons ((f 0), lazy (tabulate (compose f succ))) 143 | let rec index (Cons (b, bs)) n = if n = 0 then b else index (Lazy.force bs) (n - 1) 144 | end 145 | ``` 146 | - Single memoization scheme to cover a whole family of functions with arbitrary return types. 147 | - Representability of contravariant functors 148 | - Functor that are based on product types can be represented with sum types. 149 | - Sum-type functors in general are not representable. 150 | - Representable functor gives us two different implementations of the same thing - one a function, one a data structure. 151 | - Two representations are implemented differently and have different perf characteristics. 152 | - Being able to provide different representation of the same underlying computations is very valuable, in practice. 153 | -------------------------------------------------------------------------------- /chapter14/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter14/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter14/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter15/README.md: -------------------------------------------------------------------------------- 1 | # The Yoneda Lemma 2 | ## Utilities used by code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | ``` 9 | ## Yoneda Lemma 10 | - Yoneda Lemma is a statement about categories that doesn't have a parallel in other branches of mathematics 11 | - Arbitrary category C, Functor F from C to Set 12 | - Some Set-valued functors are "representable" 13 | - YL: All set-valued functors can be obtained from hom-functors through natural transformations and it explicitly enumerates all such natural transformations. 14 | - Naturality condition can be quite restrictive 15 | - YL: NT between a hom-functor and any other functor F is completely determined by specifying the value of its single component at just one point. 16 | - Hom-functor: C(a,x) , C(a, f) 17 | - Set-valued functor F 18 | - alpha - NT between these two functors 19 | - Nat Trans |C(a,-), F| = Fa 20 | - Reader = Hom-Functor 21 | ```ocaml 22 | type ('a, 'x) reader = 'a -> 'x 23 | ``` 24 | - Reader Functor Instance (Maps morphisms by precomposition 25 | ```ocaml 26 | module ReaderFunctor(T : sig type a end):Functor = struct 27 | type 'x t = (T.a, 'x) reader 28 | let fmap : ('x -> 'y) -> 'x t -> 'y t = fun f r -> fun a -> f (r a) 29 | end 30 | ``` 31 | - YL: Reader functor can be naturally mapped to any other functor 32 | - A Polymorphic function `alpha` 33 | ```ocaml 34 | module type NT_AX_FX = sig 35 | type a 36 | type 'x t 37 | val alpha : (a -> 'x) -> 'x t 38 | end 39 | ``` 40 | - Polymorphic function 41 | ```OCaml 42 | val alpha : (a -> 'x) -> 'x t 43 | ``` 44 | - YL can produce a container of F a when given a polymorphic function `alpha`. (Pseudo OCaml) 45 | ```OCaml 46 | alpha id : 'a f 47 | ``` 48 | - Converse is also true 49 | ```OCaml 50 | val fa : 'a f 51 | ``` 52 | - Converse implementation (Define a polymorphic function from F a . (Pseudo OCaml) 53 | ```OCaml 54 | alpha h = F.fmap h fa 55 | ``` 56 | - Advantage of having a multiple representations is that one might be easier to compose than the other or one might be efficient than the other. 57 | - CPS = YL applied to Identity Functor 58 | ### Co-Yoneda 59 | - YL on C^op 60 | - Nat|C(-, a), F) = F a 61 | ``` 62 | forall: 'x . ('x -> 'a) -> 'x t = 'a f 63 | ``` 64 | -------------------------------------------------------------------------------- /chapter15/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter15/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter15/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter16/README.md: -------------------------------------------------------------------------------- 1 | # Yoneda Embedding 2 | ## Utilities used by code below 3 | ```ocaml 4 | let id a = a 5 | ``` 6 | - C(a, -) - covariant functor from C to Set 7 | - For every a, when we try to get a mapping from C(a, -) to Set, they end up being objects in the functor category. 8 | - [C, Set] Functor Category from C to Set 9 | - A morphism in C is C(a, b) 10 | - A morphism in [C, Set] is a natural transformation. 11 | - Mapping of morphism to Natural Transformations is interesting 12 | - a is mapped to C(a, -) 13 | - b is mapped to C(b, -) 14 | - YL: [C, Set](C(a,-), F) = Fa 15 | - [C, Set](C(a, -), C(b,-)) = C(b, a) 16 | - NT between two hom-functors we got using YL gave us a contravariant functor 17 | - Mapping from C to [C, Set] is a contravariant, fully faithful functor 18 | - Faithful functor - Injective on hom-sets - No coalescing of morphisms 19 | - Full functor - surjective on hom-sets - maps one hom-set onto the other hom-set, fully covering the latter. 20 | - Fully faithful functor is a bijection on hom-sets. 21 | ### Embedding 22 | - The contravariant functor that maps objects in C to [C, Set] defines the Yoneda Embedding 23 | - It embeds C inside [C, Set] 24 | - Co-Yoneda Embedding 25 | - Embedding of category C in the category of presheaves 26 | - [C, Set](C(-, a), C(-, b)) = C(a, b) 27 | ### Application to OCaml 28 | - Yoneda Embedding - isomorphism between natural transformations amongst reader functors and functions 29 | ```ocaml 30 | module type BtoA = sig 31 | type a 32 | type b 33 | val btoa : b -> a 34 | end 35 | (* Define the Yoneda embedding *) 36 | module Yoneda_Embedding(E: BtoA) = struct 37 | let fromY : (E.a -> 'x) -> E.b -> 'x = fun f b -> f (E.btoa b) 38 | end 39 | ``` 40 | - To recover converter 41 | ```OCaml 42 | module YE = Yoneda_Embedding(BtoAImpl) 43 | YE.fromY id (* output type : BtoA.b -> BtoA.a *) 44 | ``` 45 | ### Preorder example 46 | - YE to preorder category 47 | - A set with Preorder relation gives rise to a category. 48 | - hom-set in this category is either an empty set or a one-element set. 49 | ## Naturality 50 | - YL establishes the isomorphism between the set of NT and an object in Set. 51 | - Set of NT between any functors is a hom-set in the category [C, Set] 52 | - [C, Set](C(a,-), F) = Fa 53 | - Natural isomorphism is an invertible NT between two functors. 54 | -------------------------------------------------------------------------------- /chapter16/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter16/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter16/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter17/README.md: -------------------------------------------------------------------------------- 1 | # It's all about Morphisms 2 | - Review of building blocks of hom-sets 3 | - Adjunctions are defined in terms of isomorphisms of hom-sets 4 | ## Functors 5 | - mappings of morphisms 6 | - objects tell us which pairs of morphisms are composable. 7 | ## Commuting diagrams 8 | - Used for expressing properties of morphisms 9 | - Forms the basis of all universal constructions 10 | ## Natural Transformations 11 | - NT - mapping from morphisms to commuting squares 12 | - Naturality square - Two opposing sides are the mappings of a morphism f under F and G 13 | - The other sides are the components of the NT 14 | - Functor modifies the content of a container without changing its shape 15 | - NT repackages the content in a different container without touching the contents. 16 | - A limit is defined in terms of cones. 17 | - Universal cone is the NT between the contravariant hom-functor 18 | ## Natural Isomorphisms 19 | - National Isomorphims - A NT whose every component is reversible 20 | ## Hom-Sets 21 | - Fix the source and target objects, the morphisms between the two form a boring set 22 | - Difference between members of a hom-set lies in the way they compose with other morphisms 23 | - Ff =/ Fg 24 | ## Hom-set Isomorphism 25 | - Iso between hom-sets 26 | - Plain isomorphism between them is not interesting. 27 | - Meaningful iso should take into account the composition 28 | - composition involves more than one hom-set 29 | - Iso that span whole collection of hom-sets, and we need to impose some compatibility conditions that interoperate with composition 30 | -------------------------------------------------------------------------------- /chapter18/README.md: -------------------------------------------------------------------------------- 1 | # Adjunctions 2 | ## Utilities used by code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | module type Representable = sig 9 | type 'x t 10 | type rep (* Representing type 'a' *) 11 | val tabulate : (rep -> 'x) -> 'x t 12 | val index : 'x t -> (rep -> 'x) 13 | val fmap : ('x -> 'y) -> 'x t -> 'y t 14 | end 15 | let idty : 'a -> 'a = fun a -> a 16 | ``` 17 | ## Equality vs Iso 18 | - Equality is too strong 19 | - Isomorphism - two things can be same without actually being equal as long as there is an invertible morphism 20 | ```ocaml 21 | let swap (a, b) = (b, a) 22 | ``` 23 | ## Adjunction and Unit/Counit Pair 24 | - Categories being isomorphic is expressed in terms of functors between them. 25 | - Cat C and D are isomorphic, 26 | - if there exists a functor R from C to D 27 | - if there exists a functor L from D to C 28 | - Composition of two functors R compose L and L compose R. 29 | - R compose L = I_D 30 | - L compose R = I_C 31 | - Equality of functors can be expressed via natural isomorphisms. 32 | - unit :: I_D -> R compose L 33 | - counit :: L compose R -> I_C 34 | - L - Left adjoint 35 | - R - Right adjoint 36 | - Compact Notation for Adjunction: L -| R 37 | - unit lets us introduce composition R compose L in place of an Identity functor on D 38 | - counit lets us eliminate composition L compose R and replace it with an Identity functor on C. 39 | - unit in OCaml 40 | ```ocaml 41 | module type Unit_Example = sig 42 | type 'a m 43 | val return : 'a -> 'A m 44 | end 45 | ``` 46 | - extract in OCaml 47 | ```ocaml 48 | module type Counit_Example = sig 49 | type 'c w 50 | val extract : 'c w -> 'c 51 | end 52 | ``` 53 | - 'm' is the endofunctor corresponding to R compose L and w is the endofunctor corresponding to L compose R. 54 | - unit is a polymorphic function that creates a default box around a value of arbitrary type. 55 | - counit does the opposite. 56 | - Pair of adjoint functors define a monad and comonad 57 | - Adjunctions of Endofunctors. 58 | ```ocaml 59 | (* L is Functor F and R is Representable Functor U *) 60 | module type Adjunction = functor (F : Functor) (U : Representable) -> sig 61 | val unit : 'a -> ('a F.t) U.t 62 | val counit : ('a U.t) F.t -> 'a 63 | end 64 | ``` 65 | ## Adjunctions and Hom-Sets 66 | - Adjunctions in terms of natural isomorphisms of hom-sets. 67 | - Unique morphism - mapping some set to hom-set 68 | - Factorization can be described in terms of NT 69 | - Factorization involves commuting diagrams since it defines a morphism being equal to composition of two morphisms. 70 | - Universal construction - involves factorization - morphism to commuting diagram and then to a unique morphism. 71 | - If this morphism is invertible and it can be extended to all hom-sets then we have an adjunction. 72 | - Alternative defintion: 73 | - L :: D -> C and R :: C -> D 74 | - d - source object in D; c - target object in C 75 | - Ld -> map d to C 76 | - hom-set between Ld and c = C(Ld, c) 77 | - Rc -> map c to R 78 | - hom-set between d and Rc = D(d, Rc) 79 | - L is left-adjoint to R iff there is an isomorphism of hom-sets: 80 | - C(Ld, c) is isomorphic to D(d, Rc) 81 | - Naturality means d can be varied across D and c can be varied across C. 82 | ```ocaml 83 | module type Adjunction_HomSet = functor (F : Functor)(U : Representable) -> sig 84 | val left_adjunct : ('a F.t -> 'b) -> ('a -> 'b U.t) 85 | val right_adjunct : ('a -> 'b U.t) -> ('a F.t -> 'b) 86 | end 87 | ``` 88 | - Equivalence between unit/counit and left_adjunct/right_adjunct 89 | - Complete Adjunction Definition 90 | ```ocaml 91 | (* Putting it all together to show the equivalence between unit/counit and left_adjunct/right_adjunct *) 92 | module type Adjunction = functor (F : Functor)(U : Representable) -> sig 93 | val unit : 'a -> ('a F.t) U.t 94 | val counit : ('a U.t) F.t -> 'a 95 | val left_adjunct : ('a F.t -> 'b) -> ('a -> 'b U.t) 96 | val right_adjunct : ('a -> 'b U.t) -> ('a F.t -> 'b) 97 | end 98 | 99 | (* Adjunction via unit/counit *) 100 | module type Adjunction_Unit_Counit = functor(F: Functor)(U: Representable) -> sig 101 | val unit : 'a -> ('a F.t) U.t 102 | val counit : ('a U.t) F.t -> 'a 103 | end 104 | (* Adjunction via left and right adjoints *) 105 | module type Adjunction_Hom_Set = functor (F:Functor)(U:Representable) -> sig 106 | val left_adjunct : ('a F.t -> 'b) -> 'a -> 'b U.t 107 | val right_adjunct : ('a -> 'b U.t) -> 'a F.t -> 'b 108 | end 109 | 110 | (* Implementing unit/counit from left and right adjoint definitions *) 111 | module Adjunction_From_Hom_Set(A : Adjunction_Hom_Set) : Adjunction = functor(F : Functor)(U : Representable) -> struct 112 | type 't f = 't F.t 113 | type 't u = 't U.t 114 | module M = A(F)(U) 115 | include M 116 | let unit : 'a -> 'a f u = fun a -> M.left_adjunct idty a 117 | let counit : ('a u) f -> 'a = fun fua -> M.right_adjunct idty fua 118 | end 119 | 120 | (* Implementing left and right adjunct from unit/counit Definitions *) 121 | module Adjunction_From_Unit_Counit(A:Adjunction_Unit_Counit):Adjunction = functor(F:Functor)(U:Representable) -> struct 122 | type 't f = 't F.t 123 | type 't u = 't U.t 124 | module M = A(F)(U) 125 | include M 126 | let left_adjunct f = fun a -> (U.fmap f) (M.unit a) 127 | let right_adjunct f = fun fa -> M.counit (F.fmap f fa) 128 | end 129 | ``` 130 | - Why is the Right adjoint a representable functor 131 | - Right category D is Set 132 | - R is representable if we can find an object rep in C such that C(rep, _) is naturally isomorphic to R 133 | - rep = L() 134 | - C(L(), c) is naturally isomorphic to Set((), Rc) 135 | - C(L(), _) is naturally isomorphic to R 136 | - So, R is representable 137 | ### Product from Adjunction 138 | - factorizer 139 | ```ocaml 140 | let factorizer p q = fun x -> (p x, q x) 141 | ``` 142 | - Pseudo OCaml expressing function equality 143 | ```OCaml 144 | compose fst (factorizer p q) = p 145 | compose snd (factorizer p q) = q 146 | ``` 147 | - Product category 148 | - C and D 149 | - C x D - product category - pairs of objects from C and D - pairs of morphisms from C and D 150 | - Left aadjoint functor is the diagonal functor 151 | - Right adjoint functor is the Product bifunctor 152 | ```OCaml 153 | int * bool ~ (int, bool) 154 | ``` 155 | ## Exponential form Adjunction 156 | - Function object a => b 157 | - g :: z * a -> b ; h :: z -> (a => b) 158 | - Mapping between hom-sets 159 | - Two adjoint functors are endofunctors 160 | - eval function is the counit of this adjunction: (a => b) x a -> b 161 | - If a functor has an adjoint, then it is unique up to isomorphism 162 | -------------------------------------------------------------------------------- /chapter18/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter18/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter18/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter19/README.md: -------------------------------------------------------------------------------- 1 | # Free/Forgetful Adjunctions 2 | ## Utilities used by code below 3 | ## Introduction 4 | - A free functor is the left adjoint of a forgetful functor 5 | - Forgetful functor - that forgets some structure 6 | - A functor mapping from C to Set. 7 | - A set corresponding to some object in C is called underlying set. 8 | - Monoid category has objects that have underlying sets 9 | - Forgetful functor U -> maps from Mon to Set 10 | - U maps monoid homomorphisms to functions between sets. 11 | - Mon to Set 12 | - Adjunction: Mon(Fx, m) = Set(x, Um) 13 | - Fx is the maximum monoid that can be built on the basis of x. 14 | ## Monoid with list type 15 | - 'a list is a free monoid where 'a represents the set of generators 16 | - Appending is associative and unital 17 | ```ocaml 18 | (* OCaml's string type is an immutable array of bytes *) 19 | type string' = char list 20 | ``` 21 | - Isomorphism with list of units 22 | ```ocaml 23 | let to_nat : unit list -> int = List.length 24 | let to_lst : int -> unit list = fun n -> List.init n ~f:(fun _ -> ()) 25 | ``` 26 | ## Intuitions 27 | - Functors and functions are lossy in nature. 28 | - Functors may collapse multiple objects and morphisms 29 | - Functions may collapse multiple elements of a set. 30 | - Monomorphisms - Injective(non-collapsing) 31 | - Epimorphisms - Surjective(covering the whole codomain) 32 | - Free -| Forgetful adjunction 33 | - Constrained category C is on the left 34 | - Less constrained category D is on the right 35 | - Morhphisms in C are fewer because they have to preserve additional structure. 36 | - Image of F must consist of structure-free objects, so that there is no structure to preserve by morphisms. 37 | - These objects of F are called free objects. 38 | - C(Fd, c) = D(d, Uc) 39 | - Free monoids instead of performing the operation, they accumulate the arguments that were passed to it. 40 | - Free constructions: Accumulate expression trees before evaluating them. 41 | -------------------------------------------------------------------------------- /chapter19/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter19/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter19/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter2/README.md: -------------------------------------------------------------------------------- 1 | # Types and Functions 2 | * Declare a variable and assign a type 3 | ```ocaml 4 | module type Chapter2_DeclareVariable = sig 5 | val x : int 6 | end 7 | ``` 8 | * Declare a function and assign a type 9 | ```ocaml 10 | module type Chapter2_DeclareFunction = sig 11 | val f : bool -> bool 12 | end 13 | ``` 14 | * OCaml doesn't have null. Throw exceptions to introduce runtime errors 15 | ```ocaml 16 | module Chapter2_Bottom : Chapter2_DeclareFunction = 17 | struct 18 | let f (b:bool):bool = failwith "Not Implemented" 19 | end 20 | ``` 21 | * Bottom is also a member of bool -> bool 22 | ```ocaml 23 | module Chapter2_Bottom : Chapter2_DeclareFunction = 24 | struct 25 | let f : bool -> bool = fun _ -> failwith "Not implemented" 26 | end 27 | ``` 28 | * Functions that return bottom are called /Partial/ 29 | * /Hask/ can be treated as /Set/ 30 | * Factorial Function in OCaml 31 | ```ocaml 32 | # let fact n = 33 | List.fold (List.range 1 n) ~init:1 ~f:( * ) 34 | val fact : int -> int = 35 | ``` 36 | * Absurd in OCaml 37 | ```ocaml 38 | type void 39 | let rec absurd (x:void) = absurd x 40 | ``` 41 | * Function taking unit and returning any type 42 | ```ocaml 43 | # let f44 () : int = 44 44 | val f44 : unit -> int = 45 | ``` 46 | * f\_int 47 | ```ocaml 48 | # let f_int (x:int) = () 49 | val f_int : int -> unit = 50 | ``` 51 | * f\_int Generic param 52 | ```ocaml 53 | # let f_int (_:int) = () 54 | val f_int : int -> unit = 55 | ``` 56 | * A generic/polymorphic unit function 57 | ```ocaml 58 | # let unit _ = () 59 | val unit : 'a -> unit = 60 | ``` 61 | * Bool as ADT 62 | ```ocaml 63 | type bool = false | true 64 | ``` 65 | -------------------------------------------------------------------------------- /chapter2/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter2/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter2/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | open Core 5 | 6 | let () = Printexc.record_backtrace false 7 | -------------------------------------------------------------------------------- /chapter20/README.md: -------------------------------------------------------------------------------- 1 | # Monads 2 | ## Utilities used by code below 3 | ```ocaml 4 | let ( <.> ) f g x = f (g x) 5 | let flip f x y = f y x 6 | type ('w, 'a) writer = Writer of ('a * 'w) 7 | module type Functor = sig 8 | type 'a t 9 | val fmap : ('a -> 'b) -> 'a t -> 'b t 10 | end 11 | module type Monoid = sig 12 | type a 13 | val mempty : a 14 | val mappend : a -> a -> a 15 | end 16 | let up_case = fun s -> Writer(String.uppercase_ascii s, "up_case ") 17 | ``` 18 | ## Introduction 19 | ```ocaml 20 | (* Depends on OCaml library Base - #require "base" *) 21 | module Vlen(F : Functor with type 'a t = 'a list) = struct 22 | 23 | let summable = (module Float:Base.Container_intf.Summable with type t = float) 24 | 25 | let vlen = Float.sqrt <.> (List.sum summable ~f:Fn.id) <.> (F.fmap (flip Float.int_pow 2)) 26 | end 27 | ``` 28 | ## Kleisli Category 29 | - Writer's Functor Instance 30 | ```ocaml 31 | module WriterInstance (W : sig type w end) : Functor with type 'a t = (W.w, 'a) writer = struct 32 | type 'a t = (W.w, 'a) writer 33 | let fmap f (Writer (a, w)) = Writer (f a, w) 34 | end 35 | ``` 36 | - Composing embellished functions 37 | ```OCaml 38 | 'a -> ('w, 'b) writer 39 | ``` 40 | - Kleisli Category K has the same objects as C but its morphisms are different. 41 | - A morphism between a and b in K is implemented as a morphism a -> m b in C 42 | - Functor m that has a composition, which is associative, and has an identity arrow for every object, is called a monad 43 | ```ocaml 44 | module type Monad = 45 | sig 46 | type 'a m 47 | val ( >=> ) : ('a -> 'b m) -> ('b -> 'c m) -> 'a -> 'c m 48 | val return : 'a -> 'a m 49 | end 50 | ``` 51 | - There are other ways of defining a monad 52 | - Monad is a way of composing embellished functions 53 | - Monad Instance for Writer 54 | ```ocaml 55 | module WriterMonad(W : Monoid): Monad with type 'a m = (W.a, 'a) writer = struct 56 | type 'a m = (W.a, 'a) writer 57 | 58 | let (>=>) f g = fun a -> 59 | let Writer (b, w) = f a in 60 | let Writer (c, w') = g b in 61 | Writer (c, W.mappend w w') 62 | 63 | let return a = Writer (a, W.mempty) 64 | end 65 | ``` 66 | - Tell for Writer 67 | ```ocaml 68 | let tell w = Writer ((), w) 69 | ``` 70 | ## Fish Anatomy 71 | - Step 1 72 | ```OCaml 73 | let (>=>) f g = fun a -> ... 74 | ``` 75 | - Step 2 76 | ```OCaml 77 | let (>=>) f g = fun a -> 78 | let mb = f a in 79 | ... 80 | ``` 81 | - Step 3 82 | ```OCaml 83 | val (>>=) : 'a m -> ('a -> 'b m) -> 'b m 84 | ``` 85 | - Monad using bind 86 | ```ocaml 87 | module type Monad_Bind = 88 | sig 89 | type 'a m 90 | val ( >>= ) : 'a m -> ('a -> 'b m) -> 'b m 91 | val return : 'a -> 'a m 92 | end 93 | ``` 94 | - Writer using bind 95 | ```ocaml 96 | module WriterMonadBind(W : Monoid) = struct 97 | let (>>=) (Writer (a, w)) f = 98 | let Writer (b, w') = f a in 99 | Writer (b, W.mappend w w') 100 | end 101 | ``` 102 | - join in ocaml 103 | ```OCaml 104 | val join : ('a m) m -> 'a m 105 | ``` 106 | - Rewrite bind as 107 | ```ocaml 108 | module BindUsingFunctionAndJoin(F : Functor) = struct 109 | type 'a m = 'a F.t 110 | 111 | (** Warning: The use of a compiler directive "%identity" 112 | here is only to make the type signature of 113 | join work without providing an implementation. 114 | This is only for the sake of the example and 115 | should **never** be relied upon. **) 116 | external join : 'a m m -> 'a m = "%identity" 117 | 118 | let (>>=) ma f = join (F.fmap f ma) 119 | end 120 | ``` 121 | - Third option for defining a Monad 122 | ```ocaml 123 | module type Monad_Join = functor (F : Functor) -> sig 124 | type 'a m = 'a F.t 125 | val join : 'a m m -> 'a m 126 | val return : 'a -> 'a m 127 | end 128 | ``` 129 | - fmap in terms of bind and return 130 | ```ocaml 131 | module Fmap_Using_Monad(M : Monad_Bind) = struct 132 | let fmap f ma = M.(>>=) ma (fun a -> M.return (f a)) 133 | end 134 | ``` 135 | - join for the Writer monad 136 | ```ocaml 137 | module Writer_Join(W : Monoid) = struct 138 | let join (Writer (Writer (a, w'), w)) = Writer (a, W.mappend w w') 139 | end 140 | ``` 141 | ## do Notation 142 | - Writer example 143 | ```ocaml 144 | let to_words = fun s -> Writer (String.split_on_char ' ' s, "to_words") 145 | 146 | module Writer_Process(W : Monad with type 'a m = (string, 'a) writer) = 147 | struct 148 | let process = W.(up_case >=> to_words) 149 | end 150 | ``` 151 | - OCaml do Notation 152 | ```ocaml 153 | module Process_Do(W : Monad_Bind with type 'a m = (string, 'a) writer) = 154 | struct 155 | 156 | (* Needs OCaml compiler >= 4.08 *) 157 | let (let*) = W.(>>=) 158 | 159 | let process s = 160 | let* up_str = up_case s in 161 | to_words up_str 162 | end 163 | ``` 164 | - Upcase 165 | ```ocaml 166 | let up_case = fun s -> Writer(String.uppercase_ascii s, "up_case ") 167 | ``` 168 | - Do block is 169 | ```ocaml 170 | module Process_Bind_Without_Do(W : Monad_Bind with type 'a m = (string, 'a) writer) = 171 | struct 172 | let process s = W.(up_case s >>= (fun up_str -> to_words up_str)) 173 | end 174 | ``` 175 | - Up Case 176 | ```OCaml 177 | let* up_str <- up_case s 178 | ``` 179 | - Using teller towords 180 | ```ocaml 181 | module Process_Tell(W : Monad_Bind with type 'a m = (string, 'a) writer) = 182 | struct 183 | (* Needs OCaml compiler >= 4.08 *) 184 | let (let*) = W.(>>=) 185 | 186 | let tell w = Writer ((), w) 187 | 188 | let process s = 189 | let* up_str = up_case s in 190 | let* _ = tell "to_words " in 191 | to_words up_str 192 | end;; 193 | ``` 194 | - With bind 195 | ```ocaml 196 | module Process_Bind_Without_Do(W : Monad_Bind with type 'a m = (string, 'a) writer) = 197 | struct 198 | let tell w = Writer((), w) 199 | let process s = W.(up_case s >>= (fun up_str -> 200 | tell "to_words" >>= (fun _ -> 201 | to_words up_str))) 202 | end;; 203 | ``` 204 | - Special operator (>>) 205 | ```ocaml 206 | module Monad_Ops(M : Monad_Bind) = struct 207 | let (>>) m k = M.(m >>= (fun _ -> k)) 208 | end 209 | ``` 210 | - Process with special ops 211 | ```ocaml 212 | module Process_Bind_Without_Do(W : Monad_Bind with type 'a m = (string, 'a) writer) = 213 | struct 214 | open Monad_Ops(W) 215 | let tell w = Writer((), w) 216 | let process s = W.(up_case s >>= (fun up_str -> 217 | tell "to_words" >> 218 | to_words up_str)) 219 | end 220 | ``` 221 | -------------------------------------------------------------------------------- /chapter20/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter20/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter20/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top";; 2 | #require "str";; 3 | 4 | open Base 5 | 6 | let () = Printexc.record_backtrace false 7 | -------------------------------------------------------------------------------- /chapter21/README.md: -------------------------------------------------------------------------------- 1 | # Monads and Effects 2 | ## Utilities used by code below 3 | ```ocaml 4 | let flip f x y = f y x 5 | let compose f g x = f (g x) 6 | module type Functor = sig 7 | type 'a t 8 | val fmap : ('a -> 'b) -> 'a t -> 'b t 9 | end 10 | module type MonadJoin = sig 11 | type 'a t 12 | include Functor with type 'a t := 'a t 13 | val join : 'a t t -> 'a t 14 | val return : 'a -> 'a t 15 | end 16 | module type MonadBind = 17 | sig 18 | type 'a m 19 | val ( >>= ) : 'a m -> ('a -> 'b m) -> 'b m 20 | val return : 'a -> 'a m 21 | end 22 | module type Monoid = sig 23 | type t 24 | val mempty : t 25 | val mappend : t -> t -> t 26 | end 27 | type 'a io = IO of (unit -> 'a) 28 | let put_str s = IO (fun () -> print_string s) 29 | ``` 30 | ## Introduction 31 | - Partiality - may not terminate 32 | - Nondeterminism - return many results 33 | - Side effects - access/modify state 34 | - Exceptions - Partial functions 35 | - Continuations - ability to save state of the program and then restore it 36 | - Interactive input 37 | - Interactive output 38 | ## Partiality 39 | ```ocaml 40 | module ListMonad = functor (F : Functor with type 'a t = 'a list) -> (struct 41 | include F 42 | type 'a t = 'a F.t 43 | let join = List.concat 44 | let return a = [a] 45 | end : MonadJoin) 46 | ``` 47 | - Bind using join and fmap 48 | ```ocaml 49 | module ListFunctor : Functor with type 'a t = 'a list = struct 50 | type 'a t = 'a list 51 | let fmap = List.map 52 | end 53 | module L = ListMonad(ListFunctor) 54 | let ( >>= ) xs k = L.(compose join (fmap k) xs) 55 | ``` 56 | - Triples in OCaml 57 | ```ocaml 58 | (* This requires the "gen" library, 59 | after having installed them, execute 60 | #require "gen";; *) 61 | 62 | module Pythagorean = struct 63 | 64 | let (let*) = flip Gen.flat_map 65 | 66 | let (let+) x f = Gen.map f x 67 | 68 | let guard b = if b then Gen.return () else Gen.empty 69 | 70 | let triples = 71 | let* z = Gen.init (fun i -> i + 1) in 72 | let* x = Gen.init ~limit:z (fun i -> i + 1) in 73 | let* y = Gen.init ~limit:z (fun i -> i + x) in 74 | let+ _ = guard (x * x + y * y = z * z) in 75 | Gen.return (x, y, z) 76 | end 77 | ``` 78 | - guard for List 79 | ```ocaml 80 | let guard = function 81 | | true -> [()] 82 | | false -> [] 83 | ``` 84 | - Triples alternate 85 | ```ocaml 86 | (* This requires the "gen" library, 87 | after having installed them, execute 88 | #require "gen";; *) 89 | 90 | module Pythagorean = struct 91 | 92 | let (let*) = flip Gen.flat_map 93 | 94 | let (let+) x f = Gen.map f x 95 | 96 | let guard b = if b then Gen.return () else Gen.empty 97 | 98 | let triples = 99 | let* z = Gen.init (fun i -> i + 1) in 100 | let* x = Gen.init ~limit:z (fun i -> i + 1) in 101 | let* y = Gen.init ~limit:z (fun i -> i + x) in 102 | if (x * x + y * y = z * z) then 103 | Gen.return (x, y, z) else Gen.empty 104 | end 105 | ``` 106 | ### Reader 107 | - Reader type 108 | ```ocaml 109 | type ('e, 'a) reader = Reader of ('e -> 'a) 110 | ``` 111 | - run_reader 112 | ```ocaml 113 | let run_reader (Reader f) e = f e 114 | ``` 115 | - Bind implementation for reader 116 | ```OCaml 117 | let (>>=) ra k = Reader (fun e -> ...) 118 | ``` 119 | ```OCaml 120 | let (>>=) ra k = Reader (fun e -> 121 | let a = run_reader ra e in 122 | ...) 123 | ``` 124 | ```OCaml 125 | let (>>=) ra k = Reader (fun e -> 126 | let a = run_reader ra e in 127 | let rb = k a in 128 | ...) 129 | ``` 130 | ```ocaml 131 | let (>>=) ra k = Reader (fun e -> 132 | let a = run_reader ra e in 133 | let rb = k a in 134 | run_reader rb e) 135 | ``` 136 | ```ocaml 137 | module ReaderMonad(T : sig type t end) : MonadBind = struct 138 | type 'a m = (T.t, 'a) reader 139 | let return a = Reader (fun e -> a) 140 | let (>>=) ra k = Reader (fun e -> run_reader (k (run_reader ra e)) e) 141 | end 142 | ``` 143 | ### Write Only State 144 | ```ocaml 145 | type ('w, 'a) writer = Writer of ('a * 'w);; 146 | ``` 147 | ```ocaml 148 | let run_writer (Writer (a, w)) = (a, w) 149 | ``` 150 | - Writer Instance 151 | ```ocaml 152 | module WriterMonad(W : Monoid):MonadBind = struct 153 | type 'a m = (W.t, 'a) writer 154 | 155 | let return a = Writer (a, W.mempty) 156 | 157 | let (>>=) (Writer (a, w)) k = 158 | let (a', w') = run_writer (k a) in 159 | Writer (a', W.mappend w w') 160 | end 161 | ``` 162 | ### State 163 | - Combines Reader and Writer 164 | ```ocaml 165 | type ('s, 'a) state = State of ('s -> ('a * 's)) 166 | ``` 167 | - runstate 168 | ```ocaml 169 | let run_state (State f) s = f s 170 | ``` 171 | - bind 172 | ```ocaml 173 | let (>>=) sa k = State (fun s -> 174 | let (a, s') = run_state sa s in 175 | let sb = k a in 176 | run_state sb s') 177 | ``` 178 | - Monad Instance 179 | ```ocaml 180 | module StateMonad(S : sig type t end) : MonadBind = struct 181 | 182 | type 'a m = (S.t, 'a) state 183 | 184 | let (>>=) sa k = State (fun s -> 185 | let (a, s') = run_state sa s in 186 | let sb = k a in 187 | run_state sb s') 188 | 189 | let return a = State (fun s -> (a, s)) 190 | end 191 | ``` 192 | ```ocaml 193 | let get = State (fun s -> (s, s)) 194 | ``` 195 | ```ocaml 196 | let put s' = State (fun s -> ((), s')) 197 | ``` 198 | ### Exceptions 199 | - Partial functions(throws exceptions) can be converted to total functions 200 | - Ex: Maybe, Either 201 | ```ocaml 202 | module OptionMonad : MonadBind = struct 203 | type 'a m = 'a option 204 | 205 | let (>>=) = function 206 | | Some a -> fun k -> k a 207 | | None -> fun _ -> None 208 | 209 | let return a = Some a 210 | end 211 | ``` 212 | ### Continuations 213 | - Continuation type 214 | ```ocaml 215 | type ('r, 'a) cont = Cont of (('a -> 'r) -> 'r);; 216 | ``` 217 | - runCont 218 | ```ocaml 219 | let run_cont (Cont k) h = k h 220 | ``` 221 | - bind for Cont Monad 222 | ```OCaml 223 | val (>>=) : (('a -> 'r) -> 'r) -> ('a -> (('b -> 'r) -> 'r)) -> (('b -> 'r) -> 'r) 224 | ``` 225 | - Step1 226 | ```OCaml 227 | let (>>=) ka kab = Cont (fun hb -> ...) 228 | ``` 229 | - Step2 230 | ```OCaml 231 | run_cont ka (fun a -> ...) 232 | ``` 233 | - Step3 234 | ```OCaml 235 | run_cont ka (fun a -> 236 | let kb = kab a in 237 | run_cont kb hb) 238 | ``` 239 | - Monad Instance 240 | ```ocaml 241 | module ContMonad(R:sig type t end) : MonadBind = struct 242 | type 'a m = (R.t, 'a) cont 243 | 244 | let return a = Cont (fun ha -> ha a) 245 | 246 | let (>>=) ka kab = Cont (fun hb -> 247 | run_cont ka (fun a -> 248 | run_cont (kab a) hb)) 249 | end 250 | ``` 251 | ### Interactive Input 252 | ```ocaml 253 | module type TerminalIO = sig 254 | (* OCaml doesn't have a built-in IO type*) 255 | type 'a io = IO of (unit -> 'a) 256 | 257 | val get_char : unit -> char io 258 | end 259 | ``` 260 | - main 261 | ```OCaml 262 | val main : unit io 263 | ``` 264 | - As Kleisli arrow 265 | ```OCaml 266 | val main : unit -> unit io 267 | ``` 268 | - IO as a State monad with RealWorld type 269 | ```OCaml 270 | type 'a io = realworld -> ('a * realworld) 271 | ``` 272 | ```OCaml 273 | type 'a io = realworld state 274 | ``` 275 | ### Interactive output 276 | ```OCaml 277 | val put_str : string -> unit io 278 | ``` 279 | ```OCaml 280 | val put_str : string -> unit 281 | ``` 282 | - Main function of type unit io 283 | ```ocaml 284 | (* Monad implementation for type io *) 285 | module IOMonad:MonadBind with type 'a m = 'a io = struct 286 | type 'a m = 'a io 287 | let return x = IO (fun () -> x) 288 | let (>>=) m f = IO (fun () -> 289 | let (IO m') = m in 290 | let (IO m'') = f (m' ()) in 291 | m'' () 292 | ) 293 | end 294 | 295 | (* main *) 296 | module IOMain = struct 297 | 298 | let (let*) = IOMonad.(>>=) 299 | 300 | let main = 301 | let* _ = put_str "Hello" in 302 | put_str "world!" 303 | end 304 | ``` 305 | -------------------------------------------------------------------------------- /chapter21/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter21/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter21/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter22/README.md: -------------------------------------------------------------------------------- 1 | # Monads Categorically 2 | ## Utilities used by code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | module type Representable = sig 9 | type rep (* Representing type 'a' *) 10 | type 'a t 11 | include Functor with type 'a t := 'a t 12 | val tabulate : (rep -> 'a) -> 'a t 13 | val index : 'a t -> (rep -> 'a) 14 | end 15 | module type MonadJoin = sig 16 | type 'a t 17 | include Functor with type 'a t := 'a t 18 | val join : 'a t t -> 'a t 19 | val return : 'a -> 'a t 20 | end 21 | module type Adjunction = sig 22 | type 'a f 23 | type 'a r 24 | include Functor with type 'a t = 'a f 25 | include Representable with type 'a t = 'a r 26 | val unit : 'a -> 'a f r 27 | val counit : 'a r f -> 'a 28 | end 29 | let (<.>) f g x = f (g x) 30 | let uncurry f (a, b) = f a b 31 | ``` 32 | ## Introduction 33 | - In CT, a monad is an endofunctor T equipped with a pair of natural transformations mu and eta 34 | - mu is the NT from the square functor to T 35 | - mu :: T² -> T 36 | - muₐ :: T(Tₐ) -> Tₐ 37 | - eta is the NT between Identity functor I and T 38 | - eta :: I -> T 39 | - etaₐ :: a -> Tₐ 40 | - Kleisli arrow between a and b is a morphism a -> T b 41 | - Composition of two such arrows can be implemented using mu 42 | - f :: a -> T b 43 | - g :: b -> T c 44 | - Implementing composition of f and g using mu 45 | ```ocaml 46 | module Kleisli(M : MonadJoin) = struct 47 | (* compose *) 48 | let (<.>) f g x = f (g x) 49 | 50 | let (>=>) f g = M.join <.> M.fmap g <.> f 51 | end 52 | ``` 53 | - In components 54 | ```ocaml 55 | module Kleisli(M : MonadJoin) = struct 56 | 57 | let (>=>) f g a = M.join (M.fmap g (f a)) 58 | end 59 | ``` 60 | - To make Kleisli arrows form a category, we want their composition to be associative and eta at a, to be the identity kleisli arrow at a 61 | - In terms of monoid laws 62 | - mu is multiplication 63 | - eta is unit 64 | - Monoidal categories 65 | ```ocaml 66 | module type Monoid = sig 67 | type m 68 | val mappend : m -> m -> m 69 | val mempty : m 70 | end 71 | ``` 72 | - definition of mappend 73 | ```OCaml 74 | val mappend : m -> (m -> m) 75 | ``` 76 | - Alternate definition of mappend 77 | ```OCaml 78 | val mu : (m, m) -> m 79 | ``` 80 | - Alt definition of mempty 81 | ```OCaml 82 | val eta : unit -> m 83 | ``` 84 | - Associativity in monoids 85 | ```OCaml 86 | mu (x, mu(y, z)) = mu (mu (x, y), z) 87 | ``` 88 | - Towards point-free 89 | - LHS 90 | ```OCaml 91 | (compose mu (bimap id mu))(x, (y, z)) 92 | ``` 93 | - RHS 94 | ```OCaml 95 | (compose mu (bimap mu id))((x, y), z) 96 | ``` 97 | - We want to be able to express function equality in point-free notation like this, but it isn't possible just yet 98 | ```OCaml 99 | compose mu (bimap id mu) = compose mu (bimap mu id) 100 | ``` 101 | - Associator - establish Isomorphism between two pairs 102 | ```ocaml 103 | let alpha ((x, y), z) = (x, (y, z)) 104 | ``` 105 | - With the help of the associator, we can write this point-free 106 | ```OCaml 107 | compose mu (compose (bimap id mu) alpha) = compose mu (bimap mu id) 108 | ``` 109 | - Moving unit laws to point-free notation. This is the unit law without point-free 110 | ```OCaml 111 | mu (eta (), x) = x 112 | mu (x, eta ()) = x 113 | ``` 114 | - Using bimap 115 | ```OCaml 116 | (compose mu (bimap eta id))((), x) = lambda((), x) 117 | (compose mu (bimap id eta))(x, ()) = rho(x, ()) 118 | ``` 119 | - lambda - left unitor and rho - right unitor 120 | ```ocaml 121 | let lambda ((), x) = x 122 | ``` 123 | ```ocaml 124 | let rho (x, ()) = x 125 | ``` 126 | - Point free versions 127 | ```OCaml 128 | mu . bimap id eta = rho 129 | mu . bimap eta id = lambda 130 | ``` 131 | - Associativity and unit laws for cartesian product are only valid upto isomorphism. 132 | - A monoidal category is a category C equipped with a bifunctor called the tensor product `tensor :: C x C -> C` and a distinct object i called the unit object together with three natural isomorphisms - associator, left and right unitors 133 | alpha_{abc} :: (a `tensor` b) `tensor` c -> a `tensor` (b `tensor` c) 134 | lambdaₐ :: i `tensor` a -> a 135 | rhoₐ :: a `tensor` i -> a 136 | ### Monoid in a Monoidal Category 137 | - Define monoid in monoidal category 138 | - object m 139 | - Use tensor product to form higher powers of m 140 | - To form a monoid: 141 | - U :: m `tensor` m 142 | - N :: i -> m 143 | - tensor product has to be a bifunctor 144 | ### Monads as Monoids 145 | - Monads are just monoids in the category of endofunctors 146 | ### Monads from Adjunctions 147 | - Adjunction is a pair of functors going back and forth between C and D 148 | - Endofunctors : compose R L and compose L R 149 | - L z = z x s 150 | - R b = s => b 151 | ```ocaml 152 | type ('s, 'a) state = State of ('s -> 'a * 's) 153 | ``` 154 | ```ocaml 155 | type ('s, 'a) prod = Prod of 'a * 's 156 | ``` 157 | ```ocaml 158 | type ('s, 'a) reader = Reader of ('s -> 'a) 159 | ``` 160 | ```ocaml 161 | module AdjunctionState(S:sig type s end)(F:Functor with type 'a t = (S.s, 'a) prod)(R:Representable with type 'a t = (S.s, 'a) reader):Adjunction = struct 162 | type 'a f = (S.s, 'a) prod 163 | type 'a r = (S.s, 'a) reader 164 | include F 165 | include R 166 | let unit a = Reader (fun s -> Prod (a, s)) 167 | let counit (Prod (Reader f, s)) = f s 168 | end 169 | ``` 170 | - Composition of reader after product is the state functor 171 | ```ocaml 172 | type ('s, 'a) state = State of ('s -> 'a * 's) 173 | ``` 174 | - Run_state 175 | ```ocaml 176 | let run_state (State f) s = f s 177 | ``` 178 | - Definiing join 179 | ```OCaml 180 | val ssa : ('s, ('s, 'a) state) state 181 | val run_state ssa : 's -> (('s, 'a) state, 's) 182 | ``` 183 | - join 184 | ```ocaml 185 | let join : ('s, ('s, 'a) state) state -> ('s, 'a) state = fun ssa -> 186 | State (uncurry run_state <.> run_state ssa) 187 | ``` 188 | - Not every adjunction gives rise to a monad but every monad can be factorized into a composition of two adjoint functors 189 | -------------------------------------------------------------------------------- /chapter22/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter22/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter22/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter23/README.md: -------------------------------------------------------------------------------- 1 | # Comonads 2 | ## Utilities used by code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | let id a = a 9 | let compose f g x = f (g x) 10 | type ('s, 'a) prod = Prod of 'a * 's 11 | type ('s, 'a) reader = Reader of ('s -> 'a) 12 | ``` 13 | ## Introduction 14 | - Monad - composing kleisli arrows 15 | ```OCaml 16 | 'a -> 'b m 17 | ``` 18 | - Comonad 19 | ```OCaml 20 | 'a w -> 'b 21 | ``` 22 | - CoKleisli arrow - analog of the fish operator 23 | ```ocaml 24 | module type CoKleisli = sig 25 | type 'a w 26 | val (=>=) : ('a w -> 'b) -> ('b w -> 'c) -> ('a w -> 'c) 27 | end 28 | ``` 29 | - CoKleisli - identity arrow - dual of return 30 | ```OCaml 31 | val extract : 'a w -> 'a 32 | ``` 33 | - Comonad type 34 | ```ocaml 35 | module type Comonad = sig 36 | type 'a w 37 | include Functor with type 'a t := 'a w 38 | val extract : 'a w -> 'a 39 | val (=>=) : ('a w -> 'b) -> ('b w -> 'c) -> ('a w -> 'c) 40 | end 41 | ``` 42 | - Reader monad dissection 43 | ```OCaml 44 | 'a * 'e -> 'b 45 | ``` 46 | - With currying 47 | ```OCaml 48 | 'a -> ('e -> 'b) 49 | ``` 50 | - They already have the co-Kleisli arrows 51 | ```ocaml 52 | type ('e, 'a) product = P of 'e * 'a 53 | ``` 54 | - Implementing composition for product 55 | ```ocaml 56 | let (=>=) f g = fun (P (e, a)) -> 57 | let b = f (P (e, a)) in 58 | let c = g (P (e, b)) in 59 | c 60 | ``` 61 | - Implementing extract for product 62 | ```ocaml 63 | let extract (P (e, a)) = a 64 | ``` 65 | - Product comonad can be used to perform exactly the same computations as the reader monad. 66 | - Reader functor is the right adjoint of the product functor 67 | ### Dissecting the composition 68 | ```OCaml 69 | module CoKleisliImpl = struct 70 | type 'a w 71 | let (=>=) : ('a w -> 'b) -> ('b w -> 'c) -> ('a w -> 'c) = fun f g -> 72 | g ... 73 | end 74 | ``` 75 | - Dual of bind is extend 76 | ```OCaml 77 | val extend : ('a w -> 'b) -> 'a w -> 'b w 78 | ``` 79 | - Implementing composition using extend 80 | ```ocaml 81 | module type CoKleisliExtend = sig 82 | type 'a w 83 | val extend : ('a w -> 'b) -> 'a w -> 'b w 84 | end 85 | module CoKleisliImpl(C : CoKleisliExtend) = struct 86 | type 'a w = 'a C.w 87 | let (=>=) : ('a w -> 'b) -> ('b w -> 'c) -> ('a w -> 'c) = fun f g -> 88 | compose g (C.extend f) 89 | end 90 | ``` 91 | - Duplicate 92 | ```OCaml 93 | val duplicate : 'a w -> 'a w w 94 | ``` 95 | - Three equivalent definitions of co-monad - Co-Kleisli arrows, extends or duplicate 96 | ```ocaml 97 | module type ComonadBase = sig 98 | type 'a w 99 | include Functor with type 'a t = 'a w 100 | val extract : 'a w -> 'a 101 | end 102 | 103 | module type ComonadDuplicate = sig 104 | type 'a w 105 | val duplicate : 'a w -> 'a w w 106 | end 107 | 108 | module type ComonadExtend = sig 109 | type 'a w 110 | val extend : ('a w -> 'b) -> 'a w -> 'b w 111 | end 112 | 113 | module type Comonad = sig 114 | type 'a w 115 | include ComonadBase with type 'a w := 'a w 116 | include ComonadExtend with type 'a w := 'a w 117 | include ComonadDuplicate with type 'a w := 'a w 118 | end 119 | 120 | (* Construct a full comonad instance using one of the following modules *) 121 | module ComonadImplViaExtend: functor(C:ComonadBase)(D:ComonadDuplicate with type 'a w = 'a C.w) -> Comonad with type 'a w = 'a C.w = functor(C:ComonadBase)(D:ComonadDuplicate with type 'a w = 'a C.w) -> struct 122 | include C 123 | include D 124 | let extend f wa = (C.fmap f) (D.duplicate wa) 125 | end 126 | 127 | module ComonadImplViaDuplicate: functor (C:ComonadBase)(E:ComonadExtend with type 'a w = 'a C.w) -> Comonad with type 'a w = 'a C.w = functor(C:ComonadBase)(E:ComonadExtend with type 'a w = 'a C.w) -> struct 128 | include C 129 | include E 130 | let duplicate (wa : 'a w):'a w w = E.extend id wa 131 | end 132 | ``` 133 | ### Stream comonad 134 | ```ocaml 135 | type 'a stream = Cons of 'a * 'a stream Lazy.t;; 136 | ``` 137 | - Functor instance 138 | ```ocaml 139 | module StreamFunctor : Functor with type 'a t = 'a stream = struct 140 | type 'a t = 'a stream 141 | let rec fmap f = function 142 | | Cons (x, xs) -> Cons (f x, Lazy.from_val (fmap f (Lazy.force xs))) 143 | end 144 | ``` 145 | - Get the first element of stream - extract 146 | ```ocaml 147 | let extract (Cons (x, _)) = x 148 | ``` 149 | - Duplicate produces stream of streams 150 | ```ocaml 151 | let rec duplicate (Cons (x, xs)) = Cons (Cons (x, xs), Lazy.from_val (duplicate (Lazy.force xs))) 152 | ``` 153 | - Complete comonad instance 154 | ```ocaml 155 | (* Implement Extract *) 156 | module StreamComonadBase(F:Functor with type 'a t = 'a stream):ComonadBase with type 'a w = 'a stream = struct 157 | type 'a w = 'a stream 158 | include F 159 | let extract (Cons (x, _)) = x 160 | end 161 | 162 | (* Implement duplicate *) 163 | module StreamComonadDuplicate : ComonadDuplicate with type 'a w = 'a stream = struct 164 | type 'a w = 'a stream 165 | let rec duplicate (Cons (x, xs)) = Cons (Cons (x, xs), Lazy.from_val (duplicate (Lazy.force xs))) 166 | end 167 | 168 | (* Generate full Comonad Instance *) 169 | module StreamComonad : Comonad with type 'a w = 'a stream = ComonadImplViaExtend(StreamComonadBase(StreamFunctor))(StreamComonadDuplicate) 170 | ``` 171 | - Analog of advance 172 | ```ocaml 173 | let tail (Cons (_, xs)) = Lazy.force xs 174 | ``` 175 | - sum 176 | ```ocaml 177 | let rec sum_s n (Cons (x, xs)) = 178 | if n <= 0 then 0 else x + sum_s (n - 1) (Lazy.force xs) 179 | ``` 180 | - average 181 | ```ocaml 182 | let average n stm = Float.(of_int (sum_s n stm) /. of_int n) 183 | ``` 184 | - movingAverage 185 | ```ocaml 186 | let moving_average n = StreamComonad.extend (average n) 187 | ``` 188 | ### Comonad Categorically 189 | - NT reversed for comonad. E : T -> I and D : T -> T^2 190 | - components of these transformations correspond to extract and duplicate 191 | - Monad can be derived from adjunction - R `compose` L - Monad 192 | - L `compose` R - Comonad 193 | - counit of the adjunction - E : L `compose` R -> I - extract 194 | - D : L `compose` R `compose` L `compose` R - duplicate 195 | - monad is a monoid 196 | - Is Comonad a `comonoid` 197 | - `monoid` - an object in the monoidal category 198 | - U : m * m -> m 199 | - N : i -> m 200 | - Comonoid - Reversing the above morphisms 201 | - D : m -> m * m 202 | - E : m -> i 203 | ```ocaml 204 | module type Comonoid = sig 205 | type m 206 | val split : m -> m * m 207 | val destroy : m -> unit 208 | end 209 | ``` 210 | - destroy 211 | ```ocaml 212 | let destroy _ = () 213 | ``` 214 | - split 215 | ```OCaml 216 | let split x = (f x, g x) 217 | ``` 218 | - Comonoid laws dual to monoid laws 219 | ```OCaml 220 | (* lambda is the left unitor and rho is the right unitor *) 221 | (* <.> is used as compose below *) 222 | lambda <.> (bimap destroy id) <.> split = id 223 | rho <.> (bimap id destroy) <.> split = id 224 | ``` 225 | - Plugging in the definitions 226 | ```OCaml 227 | lambda (bimap destroy id (split x)) 228 | = lambda (bimap destroy id (f x, g x)) 229 | = lambda ((), g x) 230 | = g x 231 | ``` 232 | - So we can conclude that g = id and f = id 233 | - split becomes 234 | ```ocaml 235 | let split x = (x, x) 236 | ``` 237 | - Every object is a trivial comonoid 238 | - Monad is a monoid in the category of endofunctors 239 | - Comonad is a Comonoid in the category of endofunctors 240 | ### Store Comonad 241 | - Dual of a state monad - store comonad 242 | - L z = z * s 243 | - R a = s => a 244 | - For costate(Store) comonad 245 | - Comonad - L `compose` R 246 | - L (R a) = (s => a) * s 247 | - Prod as L and Reader as R 248 | ```ocaml 249 | type ('s, 'a) store = Store of (('s -> 'a) * 's) 250 | ``` 251 | - counit of the adjunction Ea : ((s => a) * s) -> a 252 | ```ocaml 253 | let counit (Prod (Reader f, s)) = f s 254 | ``` 255 | - extract function 256 | ```ocaml 257 | let extract (Store (f, s)) = f s 258 | ``` 259 | - unit of adjunction 260 | ```ocaml 261 | let unit a = Reader (fun s -> Prod (a, s)) 262 | ``` 263 | - Partial construction of Store 264 | ```ocaml 265 | let make_store f = fun s -> Store (f, s) 266 | ``` 267 | - duplicate - D : L `compose` R -> L `compose` R `compose` L `compose` R -> L `compose` Eta `compose` R 268 | ```ocaml 269 | let duplicate (Store (f, s)) = Store (make_store f, s) 270 | ``` 271 | - complete defintion 272 | ```ocaml 273 | module StoreComonadBase(S: sig type s end)(F: Functor with type 'a t = (S.s, 'a) store):ComonadBase with type 'a w = (S.s, 'a) store = struct 274 | type 'a w = (S.s, 'a) store 275 | include F 276 | let extract (Store (f, s)) = f s 277 | end 278 | 279 | module StoreComonadDuplicate(S: sig type s end) : ComonadDuplicate with type 'a w = (S.s, 'a) store = struct 280 | type 'a w = (S.s, 'a) store 281 | let duplicate (Store (f, s)) = Store (make_store f, s) 282 | end 283 | 284 | (* Generate Full comonad *) 285 | module StoreComonad(S : sig type s end)(F:Functor with type 'a t = (S.s, 'a) store) : Comonad with type 'a w = (S.s, 'a) store = ComonadImplViaExtend(StoreComonadBase(S)(F))(StoreComonadDuplicate(S)) 286 | ``` 287 | - Reader of the store - generalized container of `a`s that are keyed using elements of type s 288 | - Second argument of the store - current position in the stream 289 | - duplicate - creates an infinite stream indexed by an element of type s 290 | - Basis for Lens 291 | ```OCaml 292 | 'a -> ('s, 'a) store 293 | ``` 294 | - Equivalent to 295 | ```OCaml 296 | val get : 'a -> 's 297 | val set : 'a -> 's -> 'a 298 | ``` 299 | -------------------------------------------------------------------------------- /chapter23/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter23/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter23/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter24/README.md: -------------------------------------------------------------------------------- 1 | # F-Algebras 2 | ### Utilities used by code below 3 | ```ocaml 4 | module type Algebra = functor (F : sig type 'a f end) -> sig 5 | type 'a algebra = 'a F.f -> 'a 6 | end 7 | module Algebra : Algebra = functor (F : sig type 'a f end) -> struct 8 | type 'a algebra = 'a F.f -> 'a 9 | end 10 | module type Functor = sig 11 | type 'a t 12 | val fmap : ('a -> 'b) -> 'a t -> 'b t 13 | end 14 | ``` 15 | ### Introduction 16 | - Monoid - set, single object category, object in monoidal category 17 | - F Algebra 18 | ```ocaml 19 | module type Algebra = functor(F : sig type 'a f end) -> sig 20 | type 'a algebra = 'a F.f -> 'a 21 | end 22 | ``` 23 | - MonoidF functor 24 | ```ocaml 25 | type 'a mon_f = MEmpty | Mappend of ('a * 'a) 26 | ``` 27 | - RingF functor 28 | ```ocaml 29 | type 'a ring_f = RZero 30 | | ROne 31 | | RAdd of ('a * 'a) 32 | | RMul of ('a * 'a) 33 | | RNeg of 'a 34 | ``` 35 | - evalZ function 36 | ```ocaml 37 | module Ring = struct 38 | 39 | module RingAlg = Algebra(struct type 'a f = 'a ring_f end) 40 | 41 | let eval_z : 'a RingAlg.algebra = function 42 | | RZero -> 0 43 | | ROne -> 1 44 | | RAdd (m, n) -> m + n 45 | | RMul (m, n) -> m * n 46 | | RNeg n -> -n 47 | end 48 | ``` 49 | - Recursion 50 | ```ocaml 51 | type expr = 52 | RZero 53 | | ROne 54 | | RAdd of (expr * expr) 55 | | RMul of (expr * expr) 56 | | RNeg of expr 57 | ``` 58 | - Ring Evaluator with a recursive definition 59 | ```ocaml 60 | let rec eval_z : expr -> int = function 61 | | RZero -> 0 62 | | ROne -> 1 63 | | RAdd (e1, e2) -> eval_z e1 + eval_z e2 64 | | RMul (e1, e2) -> eval_z e1 * eval_z e2 65 | | RNeg e -> - eval_z e 66 | ``` 67 | - Depth-one tree 68 | ```ocaml 69 | type 'a ring_f1 = ('a ring_f) ring_f 70 | ``` 71 | - Depth two tree 72 | ```ocaml 73 | type 'a ring_f2 = (('a ring_f) ring_f) ring_f 74 | ``` 75 | - D2 via D1 76 | ```ocaml 77 | type 'a ring_f2 = 'a ring_f ring_f1 78 | ``` 79 | - Applying an endofunctor infinitely many times produces a fixed point 80 | ```ocaml 81 | module Fix(F : Functor) = struct 82 | type 'a fix = Fix of (('a fix) F.t) 83 | end 84 | ``` 85 | - Constructor name - Fix is a convention 86 | ```ocaml 87 | module Fix(F : Functor) = struct 88 | type 'a fix = In of (('a fix) F.t) 89 | end 90 | ``` 91 | - Fix as a function 92 | ```ocaml 93 | module Fix(F : Functor) = struct 94 | type 'a fix = Fix of (('a fix) F.t) 95 | let fix : 'a fix F.t -> 'a fix = fun f -> Fix f 96 | end 97 | ``` 98 | - unfix 99 | ```ocaml 100 | module Fix(F : Functor) = struct 101 | type 'a fix = Fix of (('a fix) F.t) 102 | let unfix : 'a fix -> 'a fix F.t = fun (Fix f) -> f 103 | end 104 | ``` 105 | ### Category of F-Algebras 106 | - F Algebras form a category 107 | - carrier object : a 108 | - morphism : f : F a -> a 109 | - Objects in that category are a pair (a, f) 110 | - Morphisms in the F-algebra category : (a, f) -> (b, g) 111 | - m: a -> b 112 | - Homomorphism of F-algebras 113 | - F m : F a -> F b 114 | - g : F b -> b 115 | - g compose F m = m compose f 116 | - Initial Algebra 117 | - carrier i 118 | - j :: F i -> i 119 | - Lambek's theorem : j is an isomorphism 120 | - There is a unique homomorphism m from initial object to any other F-algebra 121 | - j : F i -> i ; m : j -> i ; F m : F i -> F a ; f : F a -> a 122 | - A new algebra 123 | - Carrier : F i 124 | - morphism : F j : F (F i) -> F i 125 | - (i, j) is the initial algebra. Unique homomorphism 'm' must connect initial algebra (i, j) with (F i, F j) 126 | - j : F i -> i ; m : i -> F i ; F m : F i -> F (F i); F j : F (F i) -> F i 127 | - A new algebra 128 | - F j : F (F i) -> F i 129 | - j : F i -> i 130 | - j is a homomorphism of algebras (F i, F j) to (i, j) 131 | - j compose m is a homomorphism of two algebras (i, j) and (F i, F j) 132 | - i compose m = id_i 133 | - m compose j = id_Fi 134 | - i is the inverse of m and m is the inverse of j 135 | - j is an isomorphism between F i and i 136 | - j is the constructor Fix 137 | - i is the Fix f 138 | - m is the inverse unFix 139 | ### Natural Numbers 140 | - zero : 1 -> N ; succ : N -> N 141 | - 1 + N -> N 142 | - As functor 143 | ```ocaml 144 | type 'a nat_f = ZeroF | SuccF of 'a 145 | ``` 146 | - Fixed point (Initial algebra) 147 | ```ocaml 148 | type nat = Zero | Succ of nat 149 | ``` 150 | - Peano representation for natural numbers 151 | ### Catamorphisms 152 | - Initial algebra - Fix f 153 | - Evaluator is the constructor Fix 154 | - Unique morphism m : Initial algebra to any other algebra 155 | - Algebra : carrier a and evaluator is alg 156 | - Fix : f (Fix f) -> Fix f 157 | - m : Fix f -> a 158 | - fmap m (f (Fix f)) -> f a 159 | - alg : f a -> a 160 | - m is the evaluator of the fixed point 161 | - m evaluates the whole expression tree 162 | - m = alg . fmap m . unfix 163 | - cata in OCaml 164 | ```ocaml 165 | module Cata(F : Functor) = struct 166 | type 'a fix = Fix of 'a fix F.t 167 | let fix : 'a fix F.t -> 'a fix = fun f -> Fix f 168 | let unfix : 'a fix -> 'a fix F.t = fun (Fix f) -> f 169 | let rec cata : ('a F.t -> 'a) -> 'a fix -> 'a = fun alg fixf -> alg (F.fmap (cata alg) (unfix fixf)) 170 | end 171 | ``` 172 | - Functor for natural numbers 173 | ```ocaml 174 | type 'a nat_f = ZeroF | SuccF of 'a 175 | ``` 176 | - Carrier Type : (int, int) 177 | - Algebra 178 | ```ocaml 179 | let rec fib = function 180 | | ZeroF -> (1, 1) 181 | | SuccF (m, n) -> (n, m + n) 182 | ``` 183 | - Algebra for NatF defines the recurrence relation and the catmorphism just evaluates the n-th element of that sequence 184 | ### Folds 185 | ```ocaml 186 | type ('e, 'a) list_f = NilF | ConsF of ('e * 'a) 187 | ``` 188 | - Replacing 'a with the result of recursion - 'e list 189 | ```ocaml 190 | type 'e list' = Nil | Cons of ('e * 'e list') 191 | ``` 192 | - Algebra for a list functor picks a particular carrier type and defines a function that does pattern matching on the two constructors 193 | ```ocaml 194 | let len_alg = function 195 | | ConsF (e, n) -> n + 1 196 | | NilF -> 0 197 | ``` 198 | - Traditional list length 199 | ```ocaml 200 | let length xs = List.fold_right (fun e n -> n + 1) xs 0 201 | ``` 202 | - Two arguments to fold_r are the two components of the algebra 203 | ```ocaml 204 | let sum_alg = function 205 | | ConsF (e, s) -> e +. s 206 | | NilF -> 0.0 207 | ``` 208 | - sum using foldr 209 | ```ocaml 210 | let sum xs = List.fold_right (fun e s -> e +. s) xs 0.0 211 | ``` 212 | ### CoAlgebras 213 | - Direction of the morphism is reversed 214 | - a -> F a 215 | - Coalgebras for a given functor also form a category 216 | - Homomorphisms preserve the coalgebraic structure 217 | - The terminal object (t, u) is the final coalgebra 218 | - For every alg (a, f), there is a unique homomorphism m such that 219 | - u : t -> F t; m : a -> t; F m : F a -> F t; f : a -> F a 220 | - Terminal coalgebra is a fixed point of the functor 221 | - morphism : u : t -> F t is an isomorphism 222 | - Terminal coalgebra -> recipe for generating infinite data structures 223 | - Cata is used to evaluate initial algebra 224 | - Ana is used to evaluate final coalgebra 225 | ```ocaml 226 | module Ana(F:Functor) = struct 227 | 228 | type 'a fix = Fix of 'a fix F.t 229 | 230 | let rec ana : ('a -> 'a F.t) -> 'a -> 'a fix = fun coalg a -> Fix (F.fmap (ana coalg) (coalg a)) 231 | end 232 | ``` 233 | - Stream as an example 234 | ```ocaml 235 | type ('e, 'a) stream_f = StreamF of ('e * 'a) 236 | 237 | module Stream_Functor(E : sig type e end) : Functor with type 'a t = (E.e, 'a) stream_f = struct 238 | type 'a t = (E.e, 'a) stream_f 239 | let fmap f = function 240 | | StreamF (e, a) -> StreamF (e, f a) 241 | end 242 | ``` 243 | - Fixed point 244 | ```ocaml 245 | type 'e stream = Stream of ('e * 'e stream) 246 | ``` 247 | - Generating Sieve of eratosthenes 248 | ```ocaml 249 | (* OCaml library `gen` provides useful helpers for 250 | potentially infinite iterators. You can install it 251 | with `opam install gen`. To use it in the toplevel, 252 | you need to `#require "gen"` *) 253 | let era : int Gen.t -> (int, int Gen.t) stream_f = 254 | fun ilist -> 255 | let notdiv = fun p n -> (mod) n p != 0 in 256 | let p = Gen.get_exn ilist in 257 | StreamF (p, Gen.filter (notdiv p) ilist) 258 | ``` 259 | - Primes 260 | ```ocaml 261 | module Stream_Int = Stream_Functor(struct type e = int end) 262 | module Ana_Stream = Ana(Stream_Int) 263 | 264 | (* The fixpoint translated to OCaml is eager in its evaluation. 265 | Hence, executing the following function will cause overflow. 266 | So, wrapping it inside a lazy *) 267 | let primes = lazy (Ana_Stream.ana era (Gen.init (fun i -> i + 2))) 268 | ``` 269 | - to_list_c 270 | ```ocaml 271 | module List_C(E : sig type e end) = struct 272 | module Stream_F: Functor with type 'a t = (E.e, 'a) stream_f = Stream_Functor(E) 273 | module Cata_Stream = Cata(Stream_F) 274 | 275 | let to_list_c : E.e list Cata_Stream.fix -> E.e list = 276 | fun s_fix -> 277 | Cata_Stream.cata (fun (StreamF (e, a)) -> e :: a) s_fix 278 | 279 | end 280 | ``` 281 | - Fixed point is the initial algebra and final coalgebra 282 | - Endofunctor may have many fixed points 283 | - Initial algebra is the least fixed point 284 | - Final Coalgebra is the greatest fixed point 285 | - unfold 286 | ```OCaml 287 | (* Gen.t is used to represent infinite data structures like haskell's lazy list *) 288 | val unfold : ('b -> ('a * 'b) option) -> 'b -> 'a Gen.t 289 | ``` 290 | - A lens can be represented as a pair of getter and setter. 291 | - set 292 | ```OCaml 293 | val set : 'a -> 's -> 'a 294 | val get : 'a -> 's 295 | ``` 296 | - a is a product type; s is the field type 297 | ```OCaml 298 | (a, (s, s -> a)) 299 | ``` 300 | ```OCaml 301 | 'a -> ('s, 'a) store 302 | ``` 303 | - Functor 304 | ```ocaml 305 | (* Store is the comonad version of State *) 306 | type ('s, 'a) store = Store of ('s -> 'a) * 's 307 | ``` 308 | - Lens is a coalgebra for functor with carrier type a 309 | - Lens is a coalgebra that is compatible with comonad structure 310 | -------------------------------------------------------------------------------- /chapter24/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter24/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter24/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter25/README.md: -------------------------------------------------------------------------------- 1 | # Algebra for monads 2 | ### Utilities used by code below 3 | ```ocaml 4 | let compose f g x = f (g x) 5 | let ( <.> ) = compose 6 | module type Functor = sig 7 | type 'a t 8 | val fmap : ('a -> 'b) -> 'a t -> 'b t 9 | end 10 | module type MonadJoin = sig 11 | type 'a t 12 | include Functor with type 'a t := 'a t 13 | val join : 'a t t -> 'a t 14 | val return : 'a -> 'a t 15 | end 16 | module type Comonad = sig 17 | type 'a w 18 | include Functor with type 'a t := 'a w 19 | val extract : 'a w -> 'a 20 | val duplicate : 'a w -> 'a w w 21 | end 22 | ``` 23 | ### Introduction 24 | - Endofunctors - defines expressions 25 | - Algebras - Evaluate them 26 | - Monads - Allows us to form and manipulate expressions 27 | - Algebras + Monads 28 | - Relation between monads and adjunctions 29 | - Every adjunction defines a monad and a comonad 30 | - Monad is an endofunctor m equipped with two natural transformations that satisfy some coherence conditions 31 | - N_a : a -> m a 32 | - U_a : m (m a) -> m a 33 | - alg : m a -> a 34 | - First coherence condition 35 | - alg compose N_a = id_a 36 | - Second coherence condition 37 | - alg compose U_a = alg compose (m alg) 38 | ```OCaml 39 | compose alg return = id 40 | compose alg join = compose alg (fmap alg) 41 | ``` 42 | - Algebra for a list endofunctor 43 | - type a 44 | - function that produces a from [a] 45 | - foldr can be used to express that algebra 46 | ```OCaml 47 | val fold_right : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b 48 | ``` 49 | - List Algebra - foldr f z 50 | ```OCaml 51 | (* List module in the OCaml standard library accepts list before z *) 52 | List.fold_right f [x] z = f x z 53 | ``` 54 | - List Algebra is compatible with the monad if 55 | ```OCaml 56 | f x z = x 57 | ``` 58 | - z is the right unit 59 | ### T-algebras 60 | - T - Monads 61 | - Algebras compatible with monads - T algebras 62 | - T algebras for a Monad T in category C form a Eilenberg-Moore category C^T 63 | - Morphisms in that category are homomorphisms of algebras 64 | - Forgetful functor U^T from C^T to C ; maps (a, f) -> a 65 | - Maps homomorphism of T algebras to a corresponding morphism between carrier objects in C 66 | - Free functor(left adjoint to U^T) - F^T 67 | - maps an object a in C to C^T 68 | - Left adjoint functor aka Free functor (F^T) - goes from C to C^T 69 | - Right adjoint functor aka Forgetful functor (U^T) - goes from C^T to C 70 | - F^T maps object a in C to an Object(Free algebra) in C^T. This object is T a 71 | - morphisms in C^T are T (T a) -> T a 72 | - To show that the Free algebra is a T algebra 73 | - compose alg (N_a at T a) = id at T a 74 | - compose alg U_a = compose alg (T alg) 75 | - These are monadic laws 76 | - Every adjunction defines a monad 77 | - Adjunction between F^T and U^T also forms a monad and that monad is T. 78 | - T was used to construct C^T 79 | - Since we can form the category C^T for any monad and this category C^T forms a nice adjunction which leads to the formation of T monad, we can say that every monad can be formed from an adjunction 80 | - Adjunction between F^T and U^T 81 | - To prove: F^T is the left adjoint of U^T 82 | - unit and counit for this adjunction 83 | - Triangular identities are satisfied 84 | - Monad generated by this adjunction is the original monad T 85 | - unit of this adjunction 86 | - N :: I -> U^T `compose` F^T 87 | - Free functor -> Free algebra -> (T a, U_a) 88 | - Forgetful functor -> reduces the free algebra to T a 89 | - This is just a mapping from a to T a 90 | - unit of monad T provides this mapping 91 | - counit of this adjunction 92 | - E :: F^T `compose` U^T -> I 93 | - T-algebra (a, f) 94 | - Free algebra created bt F^T is (T a, U_a) 95 | - Forgetful functor F^T drops the evaluator 96 | - U^T `compose` F^T = T 97 | - counit of the adjunction produces monadic multiplication 98 | - U = R `compose` E `compose` L 99 | ### Kleisli Category 100 | - Kleisli category 101 | - Constructed from C and a monad T - C_T 102 | - Objects of C_T are objects of C but morphisms are different 103 | - f_K from a to b in C_T corresponds to a morphism f from a to T b in C 104 | - f_K is the kleisli arrow from a to b 105 | - Composition of morphisms in the Kleisli category is defined in terms of monadic composition of Kleisli arrows 106 | - In C_T 107 | - f_K : a -> b 108 | - g_K : b -> c 109 | - In C 110 | - f : a -> T b 111 | - g : b -> T c 112 | - Composition 113 | - h_K = g_K `compose` f_K 114 | - is a kleisli arrow in C 115 | - h : a -> T c 116 | - h = U `compose` T g `compose` f 117 | ```ocaml 118 | module Kleisli_Composition(T : MonadJoin) = struct 119 | let h g f = T.join <.> T.fmap g <.> f 120 | end 121 | ``` 122 | - Functor F from C to C_T 123 | ```OCaml 124 | module C_to_CT(T : Monad) = struct 125 | let on_objects = T.return <.> f 126 | end 127 | ``` 128 | - Functor G from C_T to C 129 | - on_objects : a -> T a 130 | - on_morphisms : f_K which corresponds to f : a -> T b in C 131 | - It produces a morphism T a -> T b in C 132 | - U at T b `compose` T f (Takes T a and produces T b) 133 | - Two functors F and G form an adjunction 134 | - G `compose` F forms the monad T 135 | - THere is a whole category of adjunctions Adj(C, T) that result in monad T on C 136 | - Kleisli adjunction is the initial object in this category 137 | - EM adjunction is the terminal object in this category 138 | ### Coalgebras for Comonads 139 | - coa : a -> W a 140 | - E(extract) and D(duplicate) are the nat trans defining comonad. 141 | ### Lenses 142 | - A lens can be written as a coalgebra 143 | - coalg_s :: a -> ('s, 'a) store 144 | ```ocaml 145 | type ('s, 'a) store = Store of ('s -> 'a) * 's 146 | ``` 147 | - Coalgebra as a pair of functions 148 | - set : 'a -> 's -> 'a 149 | - get : 'a -> 's 150 | - coalg_s a = Store ((set a), (get a)) 151 | - Store is a comonad 152 | ```ocaml 153 | module Store_comonad(S: sig type s end)(F : Functor with type 'a t = (S.s, 'a) store) : Comonad with type 'a w = (S.s, 'a) store = struct 154 | type 'a w = (S.s, 'a) store 155 | include F 156 | let extract : 'a w -> 'a = fun (Store (f, s)) -> f s 157 | let duplicate : 'a w -> ('a w) w = fun (Store (f, s)) -> Store ((fun s -> Store (f, s)), s) 158 | end 159 | ``` 160 | - When is a lens a coalgebra? 161 | ```ocaml 162 | module Store_Functor(S : sig type s end) : Functor with type 'a t = (S.s, 'a) store = struct 163 | type 'a w = (S.s, 'a) store 164 | type 'a t = 'a w 165 | 166 | let fmap g (Store (f, s)) = Store ((compose g f), s) 167 | end 168 | ``` 169 | - coalg_s a = Store ((set a), (get a)) 170 | - fmap coalg to result of coalg 171 | ```OCaml 172 | (* Assume <.> acts as compose *) 173 | Store ((coalg_store <.> set a), (get a)) 174 | ``` 175 | - Applying duplicate to result of coalg 176 | ```OCaml 177 | Store ((fun s -> Store (set a, s)), (get a)) 178 | ``` 179 | - Coalg must be equal to arbitrary s 180 | ```OCaml 181 | (* Pseudo OCaml *) 182 | let coalg_store (set a s) = Store ((set a), s) 183 | ``` 184 | - Expaning coalg 185 | ```OCaml 186 | (* Expaning coalg_store *) 187 | Store ((set (set a s)), (get (set a s))) = Store ((set a), s) 188 | ``` 189 | - Using the lens laws#1 190 | ```OCaml 191 | set (set a s) = set a 192 | ``` 193 | - Using lens law#2 194 | ```OCaml 195 | get (set a s) = s 196 | ``` 197 | - A well-behaved lens is a comonad coalgebra for the Store functor 198 | -------------------------------------------------------------------------------- /chapter25/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter25/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter25/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter26/README.md: -------------------------------------------------------------------------------- 1 | # Ends and Coends 2 | ### Utilities used by code below 3 | ### Introduction 4 | - Morphism : a -> b 5 | - a and b are related 6 | - Existence of the morphism is a proof of this relation 7 | - poset category 8 | - morphism is a relation 9 | - There may be mant proofs of the same relation between two objects 10 | - These proofs form a set called hom-set 11 | - Vary the objects, we get a mapping from pairs of objects to sets of "proofs" 12 | - Hom Functor : C(-, =) :: C^op x C -> Set 13 | - A relation may also involve two different categories C and D 14 | - p :: D^op x C -> Set 15 | - A profunctor from C to D 16 | ```ocaml 17 | module type Functor = sig 18 | type 'a t 19 | val fmap : ('a -> 'b) -> 'a t -> 'b t 20 | end 21 | module type Profunctor = sig 22 | type ('a,'b) p 23 | val dimap : ('c -> 'a) -> ('b -> 'd) -> ('a, 'b) p -> ('c, 'd) p 24 | end 25 | let id a = a 26 | ``` 27 | - Functoriality of the profunctor tells us that if we have a proof that a is related to b then we get the proof that c is related to d 28 | - as long as there is a morphism from c to a 29 | - another from b to d 30 | ```OCaml 31 | let dimap f id (P (b, b)) : ('a, 'b) p 32 | ``` 33 | - p (a, a) to p (a, b) 34 | ```OCaml 35 | let dimap id f (P (a, a)) : ('a, 'b) p 36 | ``` 37 | ### Dinatural transformations 38 | - Profunctors are functors. 39 | - NT between them in the standard way 40 | - Dinatural transformation between two profunctor p and q which are members of the functor category [C^op x C, Set] is a family of morphisms 41 | - f : a -> b 42 | - alpha_a : p a a -> q a a 43 | - p b a -> p f id_a -> p a a 44 | - p b a -> p id_b f -> p b b 45 | - p a a -> alpha_a -> q a a 46 | - p b b -> alpha_a -> q b b 47 | - q a a -> q id_a f -> q a b 48 | - q b b -> q f id_b -> q a b 49 | - This commuting condition is strictly weaker than naturality condition 50 | - Natural transformation alpha in [C^op x C, Set] is indexed by a pair of objects 51 | - Dinatural transformation is indexed by only one object 52 | ### Ends 53 | - Calculus of category theory 54 | - Calculus of ends and coends borrows ideas and notation from calculus 55 | - coend - infinite sum or integral 56 | - end - infinite product 57 | - End is the generalization of a limit 58 | ```OCaml 59 | (* There is no compose operator in OCaml *) 60 | compose (dimap id f) (alpha) = compose (dimap f id) alpha 61 | ``` 62 | - End formula 63 | ```OCaml 64 | 'a. ('a, 'a) p 65 | ``` 66 | - Wedge condition 67 | ```OCaml 68 | compose (dimap f id) pi = compose (dimap id f) pi 69 | ``` 70 | - Profunctor requirement 71 | ```ocaml 72 | module type Polymorphic_Projection = functor(P : Profunctor) -> sig 73 | type rank2_p = { p : 'c. ('c, 'c) P.p } 74 | val pi : rank2_p -> ('a, 'b) P.p 75 | end 76 | ``` 77 | - pi is the polymorphic projection 78 | ```ocaml 79 | module Pi(P : Profunctor) = struct 80 | type rank2_p = { p : 'a. ('a, 'a) P.p } 81 | let pi : rank2_p -> ('c, 'c) P.p = fun e -> e.p 82 | end 83 | ``` 84 | - Generalization of a constant functor to a constant profunctor 85 | - maps all pairs of objects to a single object c 86 | - maps all pairs of morphisms to identity morphism 87 | - A wedge is a dinatural transformation from that functor to the profunctor p 88 | - Dinatural hexagon to wedge diamond 89 | ### Ends as Equalizers 90 | - Commutation conditions as equalizer. 91 | ```ocaml 92 | module EndsEqualizer(P : Profunctor) = struct 93 | let lambda : ('a, 'a) P.p -> ('a -> 'b) -> ('a, 'b) P.p = fun paa f -> P.dimap id f paa 94 | let rho : ('b, 'b) P.p -> ('a -> 'b) -> ('a, 'b) P.p = fun pbb f -> P.dimap f id pbb 95 | end 96 | ``` 97 | - prod p 98 | ```ocaml 99 | module type ProdP = sig 100 | type ('a, 'b) p 101 | type ('a, 'b) prod_p = ('a -> 'b) -> ('a, 'b) p 102 | end 103 | ``` 104 | - diaprod 105 | ```ocaml 106 | module type DiaProd = sig 107 | type ('a, 'b) p 108 | type 'a diaprod = DiaProd of ('a, 'a) p 109 | end 110 | ``` 111 | - mappings from this prod 112 | ```ocaml 113 | module EndsWithDiaProd(P : Profunctor)(D : DiaProd with type ('a, 'b) p = ('a, 'b) P.p)(PP : ProdP with type ('a, 'b) p = ('a, 'b) P.p) = struct 114 | module E = EndsEqualizer(P) 115 | let lambdaP : 'a D.diaprod -> ('a, 'b) PP.prod_p = fun (DiaProd paa) -> E.lambda paa 116 | let rhoP : 'b D.diaprod -> ('a, 'b) PP.prod_p = fun (DiaProd pbb) -> E.rho pbb 117 | end 118 | ``` 119 | ### Natural Transformations as Ends 120 | ```ocaml 121 | (* Higher rank types can be introduced via records *) 122 | module NT_as_Ends(F : Functor)(G : Functor) = struct 123 | type set_of_nt = { nt : 'a. 'a F.t -> 'a G.t} 124 | end 125 | ``` 126 | - Naturality follows from parametricity 127 | - Coend 128 | ```ocaml 129 | module Coend(P : Profunctor) = struct 130 | type coend = Coend of {c : 'a. ('a, 'a) P.p } 131 | end 132 | ``` 133 | - Coequalizer 134 | - cowedge conditions can be summarized 135 | ```ocaml 136 | module type SumP = sig 137 | type a 138 | type b 139 | type ('a, 'b) p 140 | val f : b -> a 141 | val pab : (a, b) p 142 | end 143 | ``` 144 | ```ocaml 145 | module type DiagSum = sig 146 | type a 147 | type ('a, 'b) p 148 | val paa : (a, a) p 149 | end 150 | 151 | module CoEndImpl(P : Profunctor) = struct 152 | type a 153 | type b 154 | module type Sum_P = SumP with type ('a, 'b) p = ('a, 'b) P.p and type a = a and type b = b 155 | let lambda (module S : Sum_P) = 156 | (module struct type a = S.b type ('a, 'b) p = ('a, 'b) P.p let paa = P.dimap S.f id S.pab end : DiagSum) 157 | let rho (module S : Sum_P) = 158 | (module struct type a = S.a type ('a, 'b) p = ('a, 'b) P.p let paa = P.dimap id S.f S.pab end : DiagSum) 159 | end 160 | ``` 161 | - DiagSum 162 | ```ocaml 163 | module type DiagSum = sig 164 | type a 165 | type ('a, 'b) p 166 | val paa : (a, a) p 167 | end 168 | ``` 169 | - Profunctor Composition 170 | ```ocaml 171 | module type Procompose = sig 172 | type ('a, 'b) p 173 | type ('a, 'b) q 174 | type (_, _) procompose = 175 | | Procompose : (('a, 'c) q -> ('c, 'b) p) -> ('a, 'b) procompose 176 | end 177 | ``` 178 | -------------------------------------------------------------------------------- /chapter26/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter26/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter26/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter27/README.md: -------------------------------------------------------------------------------- 1 | # Kan Extensions 2 | ### Utilities used by code below 3 | ```ocaml 4 | module type Functor = sig 5 | type 'a t 6 | val fmap : ('a -> 'b) -> 'a t -> 'b t 7 | end 8 | module type Monoid = sig 9 | type m 10 | val mempty : m 11 | val mappend : m -> m -> m 12 | end 13 | module ListMonoid(T1 : sig type a end) : (Monoid with type m = T1.a list) = struct 14 | type m = T1.a list 15 | let mempty = [] 16 | let mappend xs ys = List.append xs ys 17 | end 18 | ``` 19 | ### Kan 20 | ```ocaml 21 | module type Ran = sig 22 | type 'a k 23 | type 'a d 24 | type 'a ran = Ran of { r : 'i. ('a -> 'i k) -> 'i d } 25 | end 26 | ``` 27 | ```OCaml 28 | val f : string -> int tree 29 | ``` 30 | ```OCaml 31 | (* Higher rank polymorphic functions can be achieved using records *) 32 | { r : 'i. (a -> 'i k) -> 'i } 33 | ``` 34 | ```ocaml 35 | module type Lst = sig 36 | type a 37 | type m 38 | module M : Monoid with type m = m 39 | type lst = (a -> M.m) -> M.m 40 | val f : lst 41 | end 42 | ``` 43 | ```ocaml 44 | let fold_map (type i) (module M : Monoid with type m = i) xs f = List.fold_left (fun acc -> fun a -> M.mappend acc (f a)) M.mempty xs 45 | 46 | let to_lst (type x) (xs : x list) = 47 | let module LM : Monoid with type m = x list = ListMonoid(struct type a = x end) in 48 | (module struct 49 | type a = x 50 | type m = x list 51 | module M = LM 52 | type lst = (a -> LM.m) -> LM.m 53 | let f = fun g -> fold_map (module LM) xs g 54 | end : Lst with type a = x) 55 | 56 | let from_lst (type x) (module LstImpl : Lst with type a = x and type m = x list) = 57 | LstImpl.f (fun x -> [x]) 58 | ``` 59 | ```ocaml 60 | module type Lan = sig 61 | type 'a k 62 | type 'a d 63 | type a 64 | type i 65 | val fk : i k -> a 66 | val di : i d 67 | end 68 | ``` 69 | ```ocaml 70 | module type Exp = sig 71 | type a 72 | type b 73 | type 'a d = I of 'a 74 | type 'a k = ('a * a) 75 | include Lan with type 'a k := a * 'a and type 'a d := 'a d and type a := b 76 | end 77 | ``` 78 | ```ocaml 79 | let to_exp (type a') (type b') = fun f -> 80 | (module struct 81 | type a = a' 82 | type b = b' 83 | type 'a d = I of 'a 84 | type 'a k = ('a * a) 85 | type i = unit 86 | let fk = fun (a, _) -> f a 87 | let di = I () 88 | end : Exp with type a = a' and type b = b') 89 | 90 | let from_exp (type a') (type b') (module E : Exp with type a = a' and type b = b') = fun a -> 91 | let (I i) = E.di in 92 | E.fk (a, i) 93 | ``` 94 | ### Free Functor 95 | ```ocaml 96 | module type FreeF = sig 97 | type 'a f 98 | type a 99 | type i 100 | val h : i -> a 101 | val fi : i -> i f 102 | end 103 | ``` 104 | ```ocaml 105 | module FreeFunctor(F : sig type 'a f end) : Functor = struct 106 | module type F = FreeF with type 'a f = 'a F.f 107 | type 'a t = (module F with type a = 'a) 108 | 109 | let fmap (type a') (type b') (f : a' -> b') = fun (module FF : F with type a = a') -> (module struct 110 | type 'a f = 'a F.f 111 | type a = b' 112 | type i = FF.i 113 | let h = fun i -> f (FF.h i) 114 | let fi = FF.fi 115 | end : F with type a = b') 116 | 117 | end 118 | ``` 119 | ```ocaml 120 | module type FreeF_Alt = sig 121 | type a 122 | type 'a f 123 | val freeF : (a -> 'i) -> 'i f 124 | end 125 | ``` 126 | ```ocaml 127 | module FreeFunctorAlt(F : sig type 'a f end) : Functor = struct 128 | module type F = FreeF_Alt with type 'a f = 'a F.f 129 | type 'a t = (module F with type a = 'a) 130 | 131 | let fmap (type a') (type b') f = fun (module FF : F with type a = a') -> 132 | (module struct 133 | type a = b' 134 | type 'a f = 'a F.f 135 | let freeF = fun bi -> 136 | FF.freeF (fun a -> bi (f a)) 137 | end : F with type a = b') 138 | end 139 | ``` 140 | -------------------------------------------------------------------------------- /chapter27/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter27/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter27/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter3/README.md: -------------------------------------------------------------------------------- 1 | # Categories Great and Small 2 | ## No Objects 3 | * Categories with no objects 4 | * No morphisms 5 | ### Monoid 6 | * Definition 7 | 1. Requires the operation to be associative. 8 | 2. There must be a special element that behaves like a unit. 9 | ```ocaml 10 | module type Monoid = sig 11 | type a 12 | val mempty : a 13 | val mappend : a -> a -> a 14 | end 15 | ``` 16 | * String Instance of Monoid 17 | ```ocaml 18 | module StringMonoid:Monoid = struct 19 | type a = string 20 | let mempty = "" 21 | let mappend = (^) 22 | end 23 | ``` 24 | * Function equality without specifying its arguments is described as "point-free". 25 | ### Monoid as Category 26 | * Monoid is a single object category. 27 | -------------------------------------------------------------------------------- /chapter3/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter3/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter3/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter30/README.md: -------------------------------------------------------------------------------- 1 | # Lawvere Theories 2 | ```ocaml 3 | type ('a, 'b) either = Left of 'a | Right of 'b 4 | type two = (unit, unit) either 5 | ``` 6 | ```OCaml 7 | val raise : unit -> 'a 8 | ``` 9 | ```ocaml 10 | type 'a option = (unit, 'a) either 11 | ``` 12 | ```ocaml 13 | type 'a option = None | Some of 'a 14 | ``` 15 | -------------------------------------------------------------------------------- /chapter30/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter30/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps 4 | (:x README.md) 5 | prelude.ml) 6 | (action 7 | (progn 8 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 9 | (diff? %{x} %{x}.corrected)))) 10 | -------------------------------------------------------------------------------- /chapter30/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "gen";; 2 | 3 | let () = Printexc.record_backtrace false 4 | -------------------------------------------------------------------------------- /chapter4/README.md: -------------------------------------------------------------------------------- 1 | # Kleisli Categories 2 | ### Writer in Haskell 3 | ```ocaml 4 | type 'a writer = 'a * string 5 | ``` 6 | * Morphisms from an arbitrary type to Writer type 7 | ```OCaml 8 | 'a -> 'b writer 9 | ``` 10 | * Kleisli for Writer 11 | ```ocaml 12 | module type Kleisli = 13 | sig 14 | type a 15 | type b 16 | type c 17 | val ( >=> ) : (a -> b writer) -> (b -> c writer) -> a -> c writer 18 | end 19 | ``` 20 | * Pure for Writer 21 | ```ocaml 22 | # let pure x = (x, "");; 23 | val pure : 'a -> 'a * string = 24 | ``` 25 | * upCase for Writer 26 | ```ocaml 27 | # let up_case : (string -> string writer) = fun s -> (String.uppercase s, "up_case ") 28 | val up_case : string -> string writer = 29 | ``` 30 | * toWords for Writer 31 | ```ocaml 32 | # let to_words : (string -> string list writer) = fun s -> (String.split s ~on:' ', "to_words ") 33 | val to_words : string -> string list writer = 34 | ``` 35 | * Example Kleisli application 36 | ```ocaml 37 | module KleisiExample(K : Kleisli with type a = string and type b = string and type c = string list) = struct 38 | let up_case_and_to_words : string -> string list writer = K.(>=>) up_case to_words 39 | end 40 | ``` 41 | -------------------------------------------------------------------------------- /chapter4/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter4/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter4/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter5/README.md: -------------------------------------------------------------------------------- 1 | # Products and Coproducts 2 | ### Utilities needed to compile the code below 3 | ```ocaml 4 | # let compose f g x = f (g x) 5 | val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = 6 | # let id x = x 7 | val id : 'a -> 'a = 8 | ``` 9 | ## Universal Construction 10 | * Universal Construction 11 | - Defining objects in terms of their relationships. 12 | * Initial Object 13 | - The object that has one and only arrow to any object in the category. 14 | - This object is unique upto isomorphism. 15 | * Absurd definition 16 | ```OCaml 17 | type void (* Uninhabited type *) 18 | val absurd : void -> 'a 19 | ``` 20 | * Terminal Object 21 | - One and only morphism coming to it from any object in the category. 22 | - Unique upto isomorphism. 23 | ```ocaml 24 | # let unit x = () 25 | val unit : 'a -> unit = 26 | ``` 27 | ```ocaml 28 | # let yes _ = true 29 | val yes : 'a -> bool = 30 | ``` 31 | ```ocaml 32 | # let no _ = false 33 | val no : 'a -> bool = 34 | ``` 35 | * For any Category C, we can reverse the arrows and define opposite category. 36 | * Constructions in the opposite category are prefixed with "co". 37 | ## Isomorphisms 38 | * Propositional equality, intensional equality, extensional equality and equality as a path in homotopy type theory. 39 | * Isomorphism is an even weaker notion of equivalence. 40 | * Isomorphism - Object that look the same and have an one to one mapping between them(Invertible morphism) 41 | * Pseudo Ocaml for expression function equality 42 | ```OCaml 43 | compose f g = id 44 | compose g f = id 45 | ``` 46 | * Initial and Terminal objects are *unique upto unique isomorphism*. 47 | * This uniqueness upto unique isomorphism is the basis for all universal construction. 48 | ## Products 49 | ```ocaml 50 | # let fst (a,b) = a 51 | val fst : 'a * 'b -> 'a = 52 | ``` 53 | ```ocaml 54 | # let snd (a, b) = b 55 | val snd : 'a * 'b -> 'b = 56 | ``` 57 | ```ocaml 58 | let fst (a,_) = a 59 | let snd (_, b) = b 60 | ``` 61 | * Object *c* and two morphisms *p* and *q* connecting it to *a* and *b* 62 | ```ocaml 63 | module type Chapter5_Product = sig 64 | type a 65 | type c 66 | type b 67 | val p : c -> a 68 | val q : c -> b 69 | end 70 | ``` 71 | * Example with *Int* and *Bool* 72 | ```ocaml 73 | module Chapter5_Product_Example: Chapter5_Product with type a = int and type b = bool and type c = int = struct 74 | type a = int 75 | type b = bool 76 | type c = int 77 | let p x = x 78 | let q _ = true 79 | end 80 | ``` 81 | * Example with *c* as (int, int, bool) 82 | ```ocaml 83 | module Chapter5_Product_Example2: Chapter5_Product = struct 84 | type a = int 85 | type b = bool 86 | type c = int * int * bool 87 | let p (x,_,_) = x 88 | let q (_,_,b) = b 89 | end 90 | ``` 91 | * P' and Q' from *p* and *q* using *m* 92 | ```OCaml 93 | let p' = compose Chapter5_Product_Example.p m 94 | let q' = compose Chapter5_Product_Example.q m 95 | ``` 96 | * m as a function returning pair (int, bool) 97 | ```ocaml 98 | let m (x:int) = (x, true) 99 | ``` 100 | ```ocaml 101 | let p x = fst (m x) 102 | let q x = snd (m x) 103 | ``` 104 | * With, m as a function taking (int, int, bool) 105 | ```ocaml 106 | let m ((x,_,b): int * int * bool) = (x, b) 107 | ``` 108 | * Pseudo OCaml showing function equivalence 109 | ```OCaml 110 | fst = compose p m' 111 | snd = compose q m' 112 | ``` 113 | * m' example 114 | ```ocaml 115 | # let m' ((x, b): int * bool) = (x, x, b) 116 | val m' : int * bool -> int * int * bool = 117 | ``` 118 | * m' another example 119 | ```ocaml 120 | # let m' ((x, b): int * bool) = (x, 42, b) 121 | val m' : int * bool -> int * int * bool = 122 | ``` 123 | * Projection example 124 | ```ocaml 125 | module type Chapter5_product_projection_example = functor (Product : Chapter5_Product) -> 126 | sig 127 | val m : Product.c -> Product.a * Product.b 128 | end 129 | module ProjectionImpl(Product:Chapter5_Product) = struct 130 | let m c = (Product.p c, Product.q c) 131 | end 132 | ``` 133 | * factorizer example 134 | ```ocaml 135 | module type Factorizer = functor (Product: Chapter5_Product) -> 136 | sig 137 | val factorizer : (Product.c -> Product.a) -> (Product.c -> Product.b) -> (Product.c -> Product.a * Product.b) 138 | end 139 | module FactorizerImpl(Product:Chapter5_Product) = struct 140 | let factorizer ca cb = (Product.p ca, Product.q cb) 141 | end 142 | ``` 143 | * CoProduct 144 | ```ocaml 145 | module type CoProduct = sig 146 | type a 147 | type b 148 | type c 149 | val i : a -> c 150 | val j : b -> c 151 | end 152 | ``` 153 | * Pseudo OCaml showing function equivalence 154 | ```OCaml 155 | i' == compose m i 156 | j' == compose m j 157 | ``` 158 | * Coproduct is the disjoint union of two sets. 159 | * example 160 | ```ocaml 161 | type contact = 162 | | PhoneNum of int 163 | | EmailAddr of string 164 | ``` 165 | * example function 166 | ```ocaml 167 | # let helpdesk = PhoneNum 2222222 168 | val helpdesk : contact = PhoneNum 2222222 169 | ``` 170 | * Either type 171 | ```ocaml 172 | type ('a, 'b) either = 173 | | Left of 'a 174 | | Right of 'b 175 | ``` 176 | * Factorizer 177 | ```ocaml 178 | let factorizer i j = function 179 | | Left a -> i a 180 | | Right b -> j b 181 | ``` 182 | * Definition of terminal object can be obtained by reversing the arrows of an initial object. 183 | * Definition of coproduct can be obtained by reversing the arrows of product 184 | * Pseudo OCaml 185 | ```OCaml 186 | p = compose fst m 187 | q = compose snd m 188 | ``` 189 | ```OCaml 190 | p () = fst (m ()) 191 | q () = snd (m ()) 192 | ``` 193 | * Functions are asymmetric. Should be defined for every element of its domain but doesn't have to cover the whole codomain. 194 | * Functions that tightly fill their codomain are called *surjective* or *onto*. 195 | * Functions are allowed to map many elements from the domain to a single element in the codomain. Collapsing functions are called *injective* or *one-to-one* 196 | * Functions that are neither embedding nor collapsing called *bijections*. They are symmetric and invertible. Example: Isomorphic functions. 197 | -------------------------------------------------------------------------------- /chapter5/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter5/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter5/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter6/README.md: -------------------------------------------------------------------------------- 1 | # Simple Algebraic Data Types 2 | ### Product Types 3 | * Implemented using pairs. 4 | * Not commutative. 5 | * Commmutative upto isomorphism. 6 | ```ocaml 7 | # let swap (a, b) = (b, a) 8 | val swap : 'a * 'b -> 'b * 'a = 9 | ``` 10 | * Nesting pairs are isomorphic.(Pseudo OCaml) 11 | ```OCaml 12 | (('a * 'b) * 'c) 13 | ``` 14 | ```OCaml 15 | ('a * ('b * 'c)) 16 | ``` 17 | * Types are different but elements are in one-to-one correspondence. 18 | ```ocaml 19 | # let alpha ((a, b), c) = (a, (b, c)) 20 | val alpha : ('a * 'b) * 'c -> 'a * ('b * 'c) = 21 | ``` 22 | * Invertible function Alpha. 23 | ```ocaml 24 | # let alpha_inv (a, (b, c)) = ((a, b), c) 25 | val alpha_inv : 'a * ('b * 'c) -> ('a * 'b) * 'c = 26 | ``` 27 | * Unit type is the unit of the product 28 | * Adding unit to a type 'a is isomorphic to 'a. 29 | ```OCaml 30 | 'a * unit 31 | ``` 32 | * Isomorphism example 33 | ```ocaml 34 | # let rho (a, ()) = a 35 | val rho : 'a * unit -> 'a = 36 | ``` 37 | ```ocaml 38 | # let rho_inv a = (a, ()) 39 | val rho_inv : 'a -> 'a * unit = 40 | ``` 41 | * _Set_ is a monoidal category. (A category that is also a monoid) 42 | * Pair as ADT 43 | ```ocaml 44 | type ('a, 'b) pair = P of 'a * 'b 45 | ``` 46 | * Example construction 47 | ```ocaml 48 | # let stmt = P ("This statement is", false) 49 | val stmt : (string, bool) pair = P ("This statement is", false) 50 | ``` 51 | * Type and Data constructors with same name. In Ocaml, data constructors start with an uppercase, though. 52 | ```ocaml 53 | type ('a, 'b) pair = Pair of ('a * 'b) 54 | ``` 55 | * Pair and (,) are interchangeable. 56 | ```ocaml 57 | # let stmt = ("This statement is", false) 58 | val stmt : string * bool = ("This statement is", false) 59 | ``` 60 | * Named products. 61 | ```ocaml 62 | type stmt = Stmt of string * int 63 | ``` 64 | ## Records 65 | * Problem in working with unnamed tuples 66 | ```ocaml 67 | # let starts_with_symbol (name, symbol, _) = String.is_prefix name ~prefix:symbol 68 | val starts_with_symbol : string * string * 'a -> bool = 69 | ``` 70 | * Element as a Record 71 | ```ocaml 72 | type element = 73 | { name: string; 74 | symbol: string; 75 | atomic_number: int; 76 | } 77 | ``` 78 | * The two representations are isomorphic. 79 | ```ocaml 80 | # let tuple_to_elem (name, symbol, atomic_number) = {name; symbol; atomic_number} 81 | val tuple_to_elem : string * string * int -> element = 82 | ``` 83 | ```ocaml 84 | # let elem_to_tuple {name; symbol; atomic_number} = (name, symbol, atomic_number) 85 | val elem_to_tuple : element -> string * string * int = 86 | ``` 87 | ```ocaml 88 | # let atomic_number {atomic_number} = atomic_number 89 | val atomic_number : element -> int = 90 | ``` 91 | * Using record syntax 92 | ```ocaml 93 | # let starts_with_symbol {name;symbol;_} = String.is_prefix name ~prefix:symbol 94 | val starts_with_symbol : element -> bool = 95 | ``` 96 | * Infix application only available for special characters 97 | ```ocaml 98 | (* OCaml only allows special characters in the infix operator. So, the above function name cannot be applied be infix. *) 99 | ``` 100 | ## Sum Types 101 | * Either type (Similar to OCaml's builtin Result type) 102 | ```ocaml 103 | type ('a, 'b) either = Left of 'a | Right of 'b 104 | ``` 105 | * Sum Types are commutative upto isomorphism. 106 | ```ocaml 107 | type ('a, 'b, 'c) one_of_three = Sinistrial of 'a | Medial of 'b | Dextral of 'c 108 | ``` 109 | * *Set* is a symmetric monoidal category with respect to coproduct. 110 | * Role of the binary operation is played by the disjoint sum(Either). 111 | * Role of the initial operation is played by the initial object(Void). 112 | * Pseudo OCaml 113 | ```OCaml 114 | 'a void either 115 | ``` 116 | * Sum type example 117 | ```ocaml 118 | type color = Red | Green | Blue 119 | ``` 120 | * Even simpler example 121 | ```ocaml 122 | type bool = True | False 123 | ``` 124 | * Maybe type (Similar to OCaml's builtin Option type) 125 | ```ocaml 126 | type 'a maybe = Nothing | Just of 'a 127 | ``` 128 | * Nothing type 129 | ```ocaml 130 | type nothing_type = Nothing 131 | ``` 132 | * Just type 133 | ```ocaml 134 | type 'a just_type = Just of 'a 135 | ``` 136 | * Maybe using Either 137 | ```ocaml 138 | type 'a maybe = (unit, 'a) either 139 | ``` 140 | * List type 141 | ```ocaml 142 | type 'a list = Nil | Cons of 'a * 'a list 143 | ``` 144 | * Maybe Tail 145 | ```ocaml 146 | type 'a maybe = Nothing | Just of 'a 147 | let maybe_tail = function | Nil -> Nothing | Cons (_, xs) -> Just xs 148 | ``` 149 | ## Algebra of Types 150 | * Pseudo Ocaml representation of types 151 | ```OCaml 152 | 'a * ('b, 'c) either 153 | ``` 154 | ```OCaml 155 | ('a * 'b, 'c * 'd) either 156 | ``` 157 | * Isomorphic example 158 | ```ocaml 159 | # let prod_to_sum (x, e) = match e with 160 | | Left y -> Left (x, y) 161 | | Right z -> Right (x, z) 162 | val prod_to_sum : 'a * ('b, 'c) either -> ('a * 'b, 'a * 'c) either = 163 | ``` 164 | ```ocaml 165 | # let sum_to_prod = function | Left (x, y) -> (x, Left y) | Right (x, z) -> (x, Right z) 166 | val sum_to_prod : ('a * 'b, 'a * 'c) either -> 'a * ('b, 'c) either = 167 | ``` 168 | * Sample value 169 | ```ocaml 170 | # let prod1 = (2, Left "Hi!") 171 | val prod1 : int * (string, 'a) either = (2, Left "Hi!") 172 | ``` 173 | * Defining list again 174 | ```ocaml 175 | type 'a list = Nil | Cons of 'a * 'a list 176 | ``` 177 | -------------------------------------------------------------------------------- /chapter6/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter6/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter6/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter7/README.md: -------------------------------------------------------------------------------- 1 | # Functors 2 | - Functors are functions on objects. 3 | - Functors map both objects and morphisms. 4 | - Functors preserve connections when it maps morphisms. 5 | - Functors must preserve the structure of a category. 6 | - Collapsing Functor 7 | - Maps every object in the source category to one selected object in the target category. 8 | - Maps every morphism to identity morphism. 9 | 10 | ### Utitlities 11 | ```ocaml 12 | let compose f g x = f (g x) 13 | ``` 14 | ### Maybe Functor 15 | - Maybe functor 16 | ```ocaml 17 | type 'a option = None | Some of 'a 18 | ``` 19 | - Example function 20 | ```ocaml 21 | module type AtoB = sig 22 | type a 23 | type b 24 | val f : a -> b 25 | end 26 | ``` 27 | - Function from 'a option to 'b option 28 | ```ocaml 29 | # let f' f = function | None -> None | Some x -> Some (f x) 30 | val f' : ('a -> 'b) -> 'a option -> 'b option = 31 | ``` 32 | - Morphism mapping as `fmap` 33 | ```ocaml 34 | module type Maybe_Functor = sig 35 | type a 36 | type b 37 | val fmap : (a -> b) -> (a option -> b option) 38 | end 39 | ``` 40 | - fmap as a function with two arguments. 41 | ```ocaml 42 | module type Maybe_Functor = sig 43 | type a 44 | type b 45 | val fmap : (a -> b) -> a option -> b option 46 | end 47 | ``` 48 | - Maybe Functor Example implementation. 49 | ```ocaml 50 | # let fmap f = function 51 | | None -> None 52 | | Some x -> Some (f x) 53 | val fmap : ('a -> 'b) -> 'a option -> 'b option = 54 | ``` 55 | - Id example 56 | ```ocaml 57 | # let id x = x 58 | val id : 'a -> 'a = 59 | ``` 60 | ### Utilities needed to compile the code(Can skip this section) 61 | ```ocaml 62 | module type Functor = sig 63 | type 'a t 64 | val fmap : ('a -> 'b) -> 'a t -> 'b t 65 | end 66 | (* Functor for Option type *) 67 | module OptionF : (Functor with type 'a t = 'a option) = struct 68 | type 'a t = 'a option 69 | let fmap f = function | None -> None | Some x -> Some (f x) 70 | end 71 | ``` 72 | ### Test functor laws 73 | - Test Id law (Syntactically correct OCaml but will not be compiled by mdx) 74 | ```OCaml 75 | module Test_Functor_Id(F: Functor) = struct 76 | open F 77 | let test_id x = assert ((fmap id x) = x) 78 | end 79 | ``` 80 | - Test Compose law (Syntactically correct OCaml but will not be compiled by mdx) 81 | ```OCaml 82 | module Test_Functor_Compose(F: Functor) = struct 83 | open F 84 | 85 | (* Compose *) 86 | let <.> f g x = f (g x) 87 | 88 | let test_compose f g x = assert (fmap (f <.> g) x = fmap f (fmap g x)) 89 | end 90 | ``` 91 | ### Typeclasses 92 | - OCaml doesn't have typeclasses. 93 | - But it has functor modules. 94 | ```ocaml 95 | module type Eq = sig 96 | type a 97 | val (==) : a -> a -> bool 98 | end 99 | ``` 100 | - Point data type 101 | ```ocaml 102 | type point = Pt of float * float 103 | ``` 104 | - Eq instance for Point 105 | ```ocaml 106 | module Point_Eq(E:Eq with type a = float) = struct 107 | type a = point 108 | let (==) (Pt (p1x, p1y)) (Pt (p2x, p2y)) = E.(p1x == p2x) && E.(p2x == p2y) 109 | end 110 | ``` 111 | - Functor for OCaml 112 | ```ocaml 113 | module type Functor = sig 114 | type 'a t 115 | val fmap : ('a -> 'b) -> 'a t -> 'b t 116 | end 117 | ``` 118 | - Functor instance for Option 119 | ```ocaml 120 | module Option_Functor:(Functor with type 'a t = 'a option) = struct 121 | type 'a t = 'a option 122 | let fmap f = function 123 | | None -> None 124 | | Some x -> Some (f x) 125 | end 126 | ``` 127 | - List Functor 128 | ```ocaml 129 | type 'a list = Nil | Cons of 'a * 'a list 130 | ``` 131 | - Fmap for list 132 | ```ocaml 133 | module type List_Functor_Type = sig 134 | type 'a t = 'a list 135 | val fmap : ('a -> 'b) -> 'a list -> 'b list 136 | end 137 | ``` 138 | - Fmap impl for list 139 | ```ocaml 140 | # let rec fmap f = function 141 | | Nil -> Nil 142 | | Cons (x, xs) -> Cons (f x, fmap f xs) 143 | val fmap : ('a -> 'b) -> 'a list -> 'b list = 144 | ``` 145 | - Functor instance for List 146 | ```ocaml 147 | module List_Functor : (Functor with type 'a t = 'a list) = struct 148 | type 'a t = 'a list 149 | let rec fmap f = function 150 | | Nil -> Nil 151 | | Cons (x, xs) -> Cons (f x, fmap f xs) 152 | end 153 | ``` 154 | ### Reader Functor 155 | - Function type 156 | ```ocaml 157 | type ('a, 'b) t = 'a -> 'b 158 | ``` 159 | - Partially Applied Function Type 160 | ```ocaml 161 | module type T = sig 162 | type t 163 | end 164 | module Partially_Applied_FunctionType(T : T) = struct 165 | type 'b t = T.t -> 'b 166 | end 167 | ``` 168 | - fmap for Reader 169 | ```ocaml 170 | module type Reader_Fmap_Example = sig 171 | val fmap : ('a -> 'b) -> ('r -> 'a) -> 'r -> 'b 172 | end 173 | ``` 174 | - Functor Instance for Reader 175 | ```ocaml 176 | module Reader_Functor(T: T):Functor = struct 177 | type 'a t = T.t -> 'a 178 | let fmap f ra = fun r -> f (ra r) 179 | end 180 | ``` 181 | - Reader Functor implementation - Even simpler 182 | ```ocaml 183 | # let fmap: ('a -> 'b) -> ('r -> 'a) -> ('r -> 'b) = compose 184 | val fmap : ('a -> 'b) -> ('r -> 'a) -> 'r -> 'b = 185 | ``` 186 | 187 | ### Functors as Containers 188 | - Infinite list 189 | ```ocaml 190 | # let nats = Caml.Stream.from (fun i -> Some (i + 1)) 191 | val nats : int Stream.t = 192 | ``` 193 | - Functors can be considered as a container of value(s) of the type over which it is parameterized or as containing a recipe for generating those values. 194 | - It doesn't matter if we are able to access the values inside the functor. All that matters is if we are able to manipulate those values using functions and making sure that these manipulations compose correctly. 195 | - Const 196 | ```ocaml 197 | type ('c, 'a) const = Const of 'c 198 | ``` 199 | - Const fmap signature example 200 | ```ocaml 201 | module type Const_Functor_Example = sig 202 | val fmap : ('a -> 'b) -> ('c, 'a) const -> ('c, 'b) const 203 | end 204 | ``` 205 | - Const functor instance. 206 | ```ocaml 207 | module Const_Functor(T : T) : Functor = struct 208 | type 'a t = (T.t, 'a) const 209 | let fmap f (Const c) = Const c (* or even let fmap _ c = c *) 210 | end 211 | ``` 212 | ### Functor Composition 213 | - A composition of two functors when acting on objects is just the composition of their respective object mappings. 214 | - Same for morphisms. 215 | ```ocaml 216 | # let maybe_tail = function 217 | | [] -> None 218 | | _ :: xs -> Some xs 219 | val maybe_tail : 'a list -> 'a list option = 220 | ``` 221 | - Using fmap of respective functors 222 | ```ocaml 223 | let square x = x * x 224 | let mis = Some (Cons (1, Cons (2, Cons (3, Nil)))) 225 | let mis2 = Option_Functor.fmap (List_Functor.fmap square) mis 226 | ``` 227 | - Composing fmap of list and option functors 228 | ```ocaml 229 | let fmapO = Option_Functor.fmap 230 | let fmapL = List_Functor.fmap 231 | let fmapC f l = (compose fmapO fmapL) f l 232 | let mis2 = fmapC (square) mis 233 | ``` 234 | - Viewing fmap as a function of one argument 235 | ```ocaml 236 | module type Fmap_Alt_Sig_Example = sig 237 | type 'a t 238 | val fmap : ('a -> 'b) -> ('a t -> 'b t) 239 | end 240 | ``` 241 | - square signature 242 | ```ocaml 243 | module type Square_Signature = sig 244 | val square : int -> int 245 | end 246 | ``` 247 | - Return type signature (Syntactically correct Ocaml but not compiled) 248 | ```OCaml 249 | int list -> int list 250 | ``` 251 | - First fmap takes above signature and then returns a function. 252 | ```OCaml 253 | int list option -> int list option 254 | ``` 255 | - Functors form a category. 256 | - Objects are categories. Morphisms are functors. 257 | - *Cat* category of all small categories. 258 | -------------------------------------------------------------------------------- /chapter7/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter7/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter7/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter8/README.md: -------------------------------------------------------------------------------- 1 | # Functoriality 2 | ### Utilities needed for the code below 3 | ```ocaml 4 | # let id : 'a -> 'a = fun x -> x 5 | val id : 'a -> 'a = 6 | # let compose f g x = f (g x) 7 | val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = 8 | ``` 9 | ```ocaml 10 | (* Functor definition given in previous chapter *) 11 | module type Functor = sig 12 | type 'a t 13 | val fmap : ('a -> 'b) -> 'a t -> 'b t 14 | end 15 | ``` 16 | ### Bifunctors 17 | - Functors are morphisms in Cat. 18 | - Bifunctors - Functors of two arguments. 19 | - Bifunctor maps every pair of objects, one from category C and one from category D, to category E. 20 | - Bifunctors map morphism from C and morphism from D to a morphism in E. 21 | - Pair of objects is an object in the category C x D and a pair of morphism is just a morphism in the category C x D. 22 | - Bifunctors - Functors in both arguments. 23 | ```ocaml 24 | (** You can represent bifunctor defintion in two forms and implement just and derive the other from it. *) 25 | module type BifunctorCore = sig 26 | type ('a, 'b) t 27 | val bimap : ('a -> 'c) -> ('b -> 'd) -> ('a, 'b) t -> ('c, 'd) t 28 | end 29 | 30 | module type BifunctorExt = sig 31 | type ('a, 'b) t 32 | val first : ('a -> 'c) -> ('a, 'b) t -> ('c, 'b) t 33 | val second : ('b -> 'd) -> ('a, 'b) t -> ('a, 'd) t 34 | end 35 | 36 | module BifunctorCore_Using_Ext(M : BifunctorExt):BifunctorCore = struct 37 | type ('a, 'b) t = ('a, 'b) M.t 38 | let bimap g h x = M.first g (M.second h x) 39 | end 40 | 41 | module BifunctorExt_Using_Core(M : BifunctorCore):BifunctorExt = struct 42 | type ('a, 'b) t = ('a, 'b) M.t 43 | let first g x = M.bimap g id x 44 | let second h x = M.bimap id h x 45 | end 46 | ``` 47 | - Example of bifunctor - Product type 48 | ```ocaml 49 | module Bifunctor_Product : BifunctorCore = struct 50 | type ('a, 'b) t = 'a * 'b 51 | let bimap f g (l, r) = (f l, g r) 52 | end 53 | ``` 54 | - bimap signature for product type. 55 | ```OCaml 56 | val bimap : ('a -> 'c) -> ('b -> 'd) -> ('a, 'b) Bifunctor_Product.t -> ('c, 'd) Bifunctor_Product.t 57 | ``` 58 | - Coproduct 59 | ```ocaml 60 | type ('a, 'b) either = Left of 'a | Right of 'b 61 | module Bifunctor_Either : BifunctorCore = struct 62 | type ('a, 'b) t = ('a, 'b) either 63 | let bimap f g = function | Left a -> Left (f a) | Right b -> Right (g b) 64 | end 65 | ``` 66 | - Monoidal category defines 67 | - a binary operator(must be a bifunctor) acting on objects. 68 | - a unit object 69 | - Set is a monoidal category with respect to cartesian product. 70 | - Set is a monoidal category with respect to disjoint union. 71 | ### Functoral ADT 72 | - ADTs are made of sum and product types. 73 | - Sum and Product types are functorial. 74 | - Functors compose. 75 | - Identity type 76 | ```ocaml 77 | type 'a id = Id of 'a 78 | ``` 79 | - Functor for Identity. 80 | ```ocaml 81 | module Identity_Functor : Functor = struct 82 | type 'a t = 'a id 83 | let fmap f (Id a) = Id (f a) 84 | end 85 | ``` 86 | - Maybe type 87 | ```ocaml 88 | type 'a option = None | Some of 'a 89 | ``` 90 | - None part can be represented using Const () functor. 91 | - Just part can be represented using Id functor. 92 | - Maybe type using Either 93 | ```ocaml 94 | (** OCaml doesn't have a built in Const type *) 95 | type ('a, 'b) const = Const of 'a;; 96 | (** OCaml doesn't have a built in either type *) 97 | type ('a, 'b) either = Left of 'a | Right of 'b 98 | (** Either type *) 99 | type 'a option = ((unit, 'a) const, 'a id) either 100 | ``` 101 | - Composition of functors is a functor. 102 | - In OCaml, abstracting over type constructors requires some extra work. 103 | ```ocaml 104 | (** OCaml doesn't support higher kinded types. So, we have to use module functors to emulate the behavior higher kinded types. There's less verbose options using type defunctionalization but it's more advanced and obscures the flow of this book *) 105 | module type BiComp = functor(BF: sig type ('a, 'b) t end)(FU : sig type 'a t end)(GU: sig type 'b t end) -> sig 106 | type ('a, 'b) bicomp = BiComp of ('a FU.t, 'b GU.t) BF.t 107 | end 108 | ``` 109 | - Bifunctor instance 110 | ```ocaml 111 | module BiCompBifunctor(BF: BifunctorCore)(FU:Functor)(GU:Functor):BifunctorCore = struct 112 | type ('a, 'b) t = BiComp of ('a FU.t, 'b GU.t) BF.t 113 | let bimap f g (BiComp x) = BiComp (BF.bimap (FU.fmap f) (GU.fmap g) x) 114 | end 115 | ``` 116 | - type of x in the definition (Pseudo OCaml) 117 | ```OCaml 118 | type ('a FU.t, 'b GU.t) BF.t 119 | ``` 120 | - Types of f1 and f2 (Pseudo OCaml) 121 | ```OCaml 122 | val f1 : a -> a' 123 | val f2 : b -> b' 124 | ``` 125 | - then final result of type bf (Pseudo OCaml) 126 | ```OCaml 127 | val bimap : (a FU.t -> a' FU.t) -> (b GU.t -> b' GU.t) -> (a FU.t, b GU.t) -> (a' FU.t, b' GU.t) 128 | ``` 129 | - Functors 130 | ```ocaml 131 | (** Deriving a functor in OCaml is not available as a language extension. You could try experimental library like ocsigen to derive functors.*) 132 | type 'a tree = Leaf of 'a | Node of 'a tree * 'a tree 133 | ``` 134 | - Functor for tree 135 | ```ocaml 136 | module TreeFunctor: Functor = struct 137 | type 'a t = 'a tree 138 | let rec fmap f = function 139 | | Leaf a -> Leaf (f a) 140 | | Node (l, r) -> Node (fmap f l, fmap f r) 141 | end 142 | ``` 143 | - Writer type 144 | ```ocaml 145 | type 'a writer = 'a * string 146 | ``` 147 | - Kleisli Composition (Using Writer) 148 | ```ocaml 149 | module KleisliComposition = struct 150 | let (>=>) : ('a -> 'b writer) -> ('b -> 'c writer) -> ('a -> 'c writer) = fun m1 m2 -> 151 | fun x -> 152 | let (y, s1) = m1 x in 153 | let (z, s2) = m2 y in 154 | (z, StringLabels.concat ~sep:"" [s1; s2]) 155 | end 156 | ``` 157 | - Kleisli Identity (Using writer) 158 | ```ocaml 159 | module KleisliIdentity = struct 160 | let return : 'a -> 'a writer = fun a -> (a, "") 161 | end 162 | ``` 163 | - Kleisli Fmap implementation - (Using Writer) 164 | ```ocaml 165 | module KleisliFunctor : Functor = struct 166 | type 'a t = 'a writer 167 | let fmap f = KleisliComposition.(>=>) id (fun x -> KleisliIdentity.return (f x)) 168 | end 169 | ``` 170 | - Covariant Functors 171 | ```ocaml 172 | module PartialArrow(T : sig type r end) = struct 173 | type 'a t = T.r -> 'a 174 | end 175 | ``` 176 | - Reader type 177 | ```ocaml 178 | type ('r, 'a) reader = 'r -> 'a 179 | ``` 180 | - Functor for Reader type 181 | ```ocaml 182 | module ReaderFunctor(In: sig type r end): Functor = struct 183 | type 'a t = (In.r, 'a) reader 184 | let fmap f g = compose f g 185 | end 186 | ``` 187 | - Reader flipped - Op 188 | ```ocaml 189 | type ('r, 'a) op = 'a -> 'r 190 | ``` 191 | - Fmap for Op 192 | ```OCaml 193 | val fmap : 'a 'b. ('a -> 'b) -> ('a -> 'r) -> ('b -> 'r) 194 | ``` 195 | - For every category C, there is a dual category C^op(Same objects as C but arrows reversed) 196 | - Mapping of categories that inverts the direction of morphisms is called contravariant functor. 197 | - Contravariant functor is a regular functor from the opposite category. 198 | ```ocaml 199 | module type Contravariant = sig 200 | type 'a t 201 | val contramap : ('b -> 'a) -> 'a t -> 'b t 202 | end 203 | ``` 204 | - Contravariant instance for op 205 | ```ocaml 206 | module OpContravariant(In : sig type r end) : Contravariant = struct 207 | type 'a t = (In.r, 'a) op 208 | let contramap f g = compose g f 209 | end 210 | ``` 211 | - Flip 212 | ```ocaml 213 | # let flip f b a = f a b 214 | val flip : ('a -> 'b -> 'c) -> 'b -> 'a -> 'c = 215 | ``` 216 | - Contramap and flip 217 | ```ocaml 218 | # let contramap : ('b -> 'a) -> ('r, 'a) op -> ('r, 'b) op = fun f g -> flip compose f g 219 | val contramap : ('b -> 'a) -> ('r, 'a) op -> ('r, 'b) op = 220 | ``` 221 | ### Profunctors 222 | - Function arrow is contravariant in its first argument and covariant in its second argument. 223 | - If the target category is *Set*, then we can describe this as Profunctor. 224 | ```ocaml 225 | (* Profunctor definition *) 226 | module type Profunctor = sig 227 | type ('a,'b) p 228 | val dimap : ('a -> 'b) -> ('c -> 'd) -> ('b, 'c) p -> ('a, 'd) p 229 | end 230 | 231 | (* Profunctor alternate definition *) 232 | module type ProfunctorExt = sig 233 | type ('a, 'b) p 234 | val lmap : ('a -> 'b) -> ('b, 'c) p -> ('a, 'c) p 235 | val rmap : ('b -> 'c) -> ('a, 'b) p -> ('a, 'c) p 236 | end 237 | 238 | (* Profunctor dimap defined using lmap and rmap *) 239 | module Profunctor_Using_Ext(PF: ProfunctorExt):Profunctor = struct 240 | type ('a, 'b) p = ('a, 'b) PF.p 241 | let dimap f g = compose (PF.lmap f) (PF.rmap g) 242 | end 243 | 244 | (** Profunctor lmap and rmap defined using dimap *) 245 | module ProfunctorExt_Using_Dimap(PF: Profunctor): ProfunctorExt = struct 246 | type ('a, 'b) p = ('a, 'b) PF.p 247 | let lmap f = PF.dimap f id 248 | let rmap g = PF.dimap id g 249 | end 250 | ``` 251 | - Profunctor Implementation for function type 252 | ```ocaml 253 | module ProfunctorArrow : Profunctor = struct 254 | type ('a, 'b) p = 'a -> 'b 255 | let dimap f g p = compose g (compose p f) 256 | end 257 | module ProfunctorExtArrow : ProfunctorExt = struct 258 | type ('a, 'b) p = 'a -> 'b 259 | let lmap f p = (flip compose) f p 260 | let rmap = compose 261 | end 262 | ``` 263 | - Profunctor : C^op x D -> Set 264 | - Hom-Functor: C^op x C -> Hom-Set (a, b) 265 | - Hom-Functor is a special case of Profunctor. 266 | -------------------------------------------------------------------------------- /chapter8/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter8/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter8/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /chapter9/README.md: -------------------------------------------------------------------------------- 1 | # Function Types 2 | ### Utilities 3 | ```ocaml 4 | (* Utilities needed for the rest of the scripts in the document to run. *) 5 | type ('a, 'b) either = Left of 'a | Right of 'b 6 | ``` 7 | - Function type is a set of morphisms between two sets. 8 | - A set of morphisms between any two objects in a category is called a hom-set. 9 | - In the category *Set*, every hom-set is itself an object in the same category, because it's also a set. 10 | - In other categories, hom-set is external to a category - external hom-sets 11 | - Internal hom-set - an object constructed in a category that represent the hom-sets. 12 | ### Universal Construction 13 | - Constructing an internal hom-set(function type) from scratch. 14 | - A pattern that involves three objects - function type we are constructing, argument type and result type. 15 | - Function application/evaluation connects these three types. 16 | - Pick an object z and a. 17 | - Product of them z x a is an object. 18 | - Arrow g from that object to b. 19 | - g maps every such object(z x a) to b. 20 | - *There is no function type, if there is no product type* 21 | ### Currying 22 | - Curring is built into the syntax of OCaml as well.(Pseudo OCaml) 23 | ```OCaml 24 | 'a -> ('b -> 'c) 25 | ``` 26 | - Unparenthesized signature 27 | ```OCaml 28 | 'a -> 'b -> 'c 29 | ``` 30 | - Multi argument functions 31 | ```ocaml 32 | # let catstr s s' = String.concat ~sep:"" [s;s'] 33 | val catstr : string -> string -> string = 34 | ``` 35 | - Same function written using one argument functions 36 | ```ocaml 37 | # let catstr = fun s -> fun s' -> String.concat ~sep:"" [s;s'] 38 | val catstr : string -> string -> string = 39 | ``` 40 | - Greet 41 | ```ocaml 42 | # let greet = catstr "Hello" 43 | val greet : string -> string = 44 | ``` 45 | - A Function of two variables 46 | ```OCaml 47 | 'a * 'b -> 'a 48 | ``` 49 | - curry 50 | ```ocaml 51 | # let curry f a b = f (a, b) 52 | val curry : ('a * 'b -> 'c) -> 'a -> 'b -> 'c = 53 | ``` 54 | - uncurry 55 | ```ocaml 56 | # let uncurry f p = f (fst p) (snd p) 57 | val uncurry : ('a -> 'b -> 'c) -> 'a * 'b -> 'c = 58 | ``` 59 | - Factorizer 60 | ```ocaml 61 | # let factorizer g = fun a -> (fun b -> g (a, b)) 62 | val factorizer : ('a * 'b -> 'c) -> 'a -> 'b -> 'c = 63 | ``` 64 | ### Exponentials 65 | - Function object or internal hom object between a and b, is often called the exponential denoted by b^a 66 | ### Cartesian Closed Categories 67 | - Larger family of categories called *Cartesian closed* 68 | - *Set* is just one example of such a category. 69 | - Cartesian closed category must contain 70 | - The terminal object 71 | - A product of any pair of objects 72 | - An exponential for any pair of objects. 73 | - Terminal object is a product of zero arity. 74 | - CCC provides models for the simply typed lambda calculus. 75 | - Dual of product and terminal object is the coproduct and initial object. 76 | - *Bicartesian closed category* - CCC that also supports coproduct and initial object and in which product can be distributed over coproduct 77 | ### Exponentials and ADT 78 | - Function types as exponentials fits very well into the scheme of ADTs. 79 | - Algebra vs BiCartesian Closed Category 80 | - Zero -> Initial object 81 | - One -> Terminal object 82 | - product -> product 83 | - sums -> coproduct 84 | - exponential -> exponentials (internal hom object) 85 | - Singleton set is the terminal object in *Set* 86 | - Exponentials of sums 87 | ```ocaml 88 | module type Exponential_Of_Sums_Example = sig 89 | val f : (int, float) either -> string 90 | end 91 | ``` 92 | - Implementation 93 | ```ocaml 94 | module Exp_Sum_Impl : Exponential_Of_Sums_Example = struct 95 | let f = function 96 | | Left n -> if n < 0 then "Negative int" else "Positive int" 97 | | Right x -> if Float.compare x 0.4 < 0 then "Negative double" else "Positive double" 98 | end 99 | ``` 100 | ### Curry-Howard Isomorphism 101 | - Correspondence between logic and ADT 102 | - Void - false 103 | - unit -> true 104 | - Product -> AND 105 | - Sum type -> OR 106 | - Function type -> logical implication 107 | - Every type can be interpreted as a proposition. 108 | - Writing programs is same as proving theorems. 109 | - eval example (pseudo OCaml) 110 | ```OCaml 111 | val eval : (('a -> 'b), 'a) -> 'b 112 | ``` 113 | - eval implementation 114 | ```ocaml 115 | # let eval (f,a) = f a 116 | val eval : ('a -> 'b) * 'a -> 'b = 117 | ``` 118 | - Mapping a V b => a to types (pseudo OCaml) 119 | ```OCaml 120 | ('a, 'b) either -> 'a 121 | ``` 122 | - Absurd (pseudo OCaml) 123 | ```OCaml 124 | val absurd : void -> 'a 125 | ``` 126 | -------------------------------------------------------------------------------- /chapter9/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (with-stdout-to dune.gen 3 | (run ocaml-mdx rule --direction=to-ml README.md --prelude=prelude.ml))) 4 | 5 | (alias 6 | (name runtest) 7 | (deps README.md) 8 | (action (diff dune.inc dune.gen))) 9 | 10 | (include dune.inc) -------------------------------------------------------------------------------- /chapter9/dune.inc: -------------------------------------------------------------------------------- 1 | (alias 2 | (name runtest) 3 | (deps (:x README.md) 4 | prelude.ml) 5 | (action (progn 6 | (run ocaml-mdx test --prelude=prelude.ml --direction=to-ml %{x}) 7 | (diff? %{x} %{x}.corrected) 8 | ))) 9 | -------------------------------------------------------------------------------- /chapter9/prelude.ml: -------------------------------------------------------------------------------- 1 | #require "core,core.top,ppx_jane";; 2 | 3 | open Base 4 | 5 | let () = Printexc.record_backtrace false 6 | -------------------------------------------------------------------------------- /dune: -------------------------------------------------------------------------------- 1 | (alias 2 | (name generate) 3 | (deps chapter1.ml)) 4 | 5 | (rule 6 | (targets chapter1/chapter.ml) 7 | (deps chapter1/chapter1.md) 8 | (action 9 | (run ocaml-mdx ))) -------------------------------------------------------------------------------- /dune-project: -------------------------------------------------------------------------------- 1 | (lang dune 1.8) 2 | -------------------------------------------------------------------------------- /ocaml-ctfp.install: -------------------------------------------------------------------------------- 1 | lib: [ 2 | "_build/install/default/lib/ocaml-ctfp/META" {"META"} 3 | "_build/install/default/lib/ocaml-ctfp/dune-package" {"dune-package"} 4 | "_build/install/default/lib/ocaml-ctfp/opam" {"opam"} 5 | ] 6 | doc: [ 7 | "_build/install/default/doc/ocaml-ctfp/LICENSE" 8 | "_build/install/default/doc/ocaml-ctfp/README.org" 9 | ] 10 | -------------------------------------------------------------------------------- /ocaml-ctfp.opam: -------------------------------------------------------------------------------- 1 | opam-version: "2.0" 2 | version: "0.0.1" 3 | synopsis: "OCaml code snippets for Category Theory For Programmers" 4 | maintainer: "Arulselvan Madhavan " 5 | authors: "Arulselvan Madhavan " 6 | license: "GPL 3.0" 7 | homepage: "https://github.com/ArulselvanMadhavan/ocaml-ctfp" 8 | bug-reports: "https://github.com/ArulselvanMadhavan/ocaml-ctfp/issues" 9 | depends: [ 10 | "ocaml" {>= "4.07.1"} 11 | "dune" {build} 12 | "patdiff" #For running diff based checks in tests 13 | "core" 14 | ] 15 | build: [ 16 | # ["dune" "subst"] {pinned} 17 | ["dune" "build" "-p" name "-j" jobs] 18 | ["dune" "runtest" "-p" name] {with-test} 19 | ] 20 | install: [make "install"] 21 | dev-repo: "git+https://" 22 | url { 23 | src: "git+file:///Users/arul.madhavan/dev/ocaml-projects/ocaml-ctfp#master" 24 | } 25 | -------------------------------------------------------------------------------- /playground/.merlin: -------------------------------------------------------------------------------- 1 | EXCLUDE_QUERY_DIR 2 | B _build/default/.playground.eobjs/byte 3 | S . 4 | FLG -w @a-4-29-40-41-42-44-45-48-58-59-60-40 -strict-sequence -strict-formats -short-paths -keep-locs 5 | -------------------------------------------------------------------------------- /playground/dune: -------------------------------------------------------------------------------- 1 | (rule 2 | (targets playground.ml) 3 | (deps playground.md) 4 | (action 5 | (progn 6 | (with-stdout-to %{targets} (run ocaml-mdx pp %{deps}))))) 7 | 8 | (executable 9 | (name playground)) 10 | 11 | (rule 12 | (with-stdout-to playground.md.out (run ./playground.exe))) 13 | 14 | (alias 15 | (name runtest) 16 | (deps (:x playground.md.out) (:y playground.md.out.expected)) 17 | (action (progn 18 | (run ocaml-mdx test --direction=infer-timestamp %{x}) 19 | (diff %{x} %{y})))) -------------------------------------------------------------------------------- /playground/dune-project: -------------------------------------------------------------------------------- 1 | (lang dune 1.8) 2 | -------------------------------------------------------------------------------- /playground/playground.md: -------------------------------------------------------------------------------- 1 | * mdx Playground 2 | ```ocaml 3 | print_endline "42" 4 | ``` 5 | -------------------------------------------------------------------------------- /playground/playground.md.out.expected: -------------------------------------------------------------------------------- 1 | 42 2 | --------------------------------------------------------------------------------