#import "../common.typ": *

= Adjoint Functional Programming

== Lecture 1. Linear functional programming

- Origins of linearity is from linear logic
- from 1987 or before

#let isValue(x) = $#x "value"$

Language to be studied is called "snax"
	- Features
		- Substructural programming
		- Inference
			- Overloading? writing the same function that works both linearly and non-linearly
		- "Proof-theoretic compiler"
			- Everything up til the target C is either natural deduction, sequent calculus, etc.
	- Types
		- "if there's something you don't need in the language, don't put it in the language"
		- no empty type
		- unit type
			- 1
				- #tree(axi[], uni[$() : 1$])
				- #tree(axi[], uni[$isValue(())$])
			- +
				- ${e : A}{"inl" e : A + B}$
				- ${e : B}{"inr" e : A + B}$
				- #tree(axi[$isValue("v")$], uni[$isValue("inl" v)$])
				- #tree(axi[$isValue("v")$], uni[$isValue("inr" v)$])
			- $A + B :equiv +\{"inl":A, "inr": B\}$
			- label set $L eq.not emptyset$ , finite
				- finiteness is important
			- nat = $+\{ "zero" : 1 , "succ" : "nat" \}$
				- recursion is required to define an infinite amount of things
					- "equirecursive" types
						- nat is "equal" to the recursive definition in some sense
			- k
				- $\{(k \in l) e : A_e}{k(e) : +\{l : A_l\}_{l : L}}$
			- $ceil(0) = "zero" ()$

  - Computational rules

    Define negation of booleans

    $"not" (x : "bool") : "bool" \
    "not" x = "match" x "with" \
    | "true" u arrow.r "false" u\
    | "false" u arrow.r "true" u \
    $

    Function definitiosn like these operate at the meta level.
    Avoid introducing function types today.

    Linear functional programming means *every variable must be used exactly once.*

    *NOTE:* Above cannot be defined $"true" u arrow.r "false" ()$, since $u$ is not used.

    DOn't need a garbage collector for linear functional programming language

    #tree(
      axi[$e : +{l : A_l}_(l in L)$],
      axi[$x : A_l$],
      axi[$tack.r e_l : C(l in L)$],
      nary(3)[$"match" e "with" (l(x) arrow.r.double e_l)_(l in L) : C$],
    )

    Cannot write a function $"duplicate"(x) = (x , x)$ because of linear programming

    #tree(
      axi[$Delta tack.r e_1: A$],
      axi[$Gamma tack.r e_2 : B$],
      bin[$Delta,Gamma tack.r (e_1, e_2) : A times B$],
    )

    so introducing contexts

    $Gamma :equiv (dot) | Gamma , x : A$

    where the variable $x$ must be unique.

    #tree(axi[], uni[$x : A tack.r x : A$])

    *NOTE:* Left side is $x : A$, not $Gamma , x : A$ because those others would be unused, violating the single use rule.

    #tree(
      axi[$Delta , e : A times B$],
      axi[$Gamma , x : A, y : B tack.r e' : C$],
      nary(2)[$Delta , Gamma tack.r "match" e "with" (x , y) => e' : C$],
    )

    Have to split your variables to check $e$ and $e'$. This is not the way to implement it.

    To implement it, check $"FV"(e)$ and $"FV"(e')$. These correspond to $Delta$ and $Gamma$ respectively. This is expensive.

    - An easier way is to have each judgement give back another context containing the variables not use. Called a "subtractive" approach.

    - Additive approach returns the variables used, and then check to make sure variables don't exist on the output.

    GOing backk

    #tree(
      axi[],
      uni[$dot tack.r () : 1$],
    )

    // #tree(
    //   axi[$Delta tack.r e : +{l : A_l}_(l in L)$],
    //   axi[$Gamma $]
    // )

  Defining plus:

  - 
    $"plus" (x : "nat") (y : "nat") : "nat" \
    "plus" x " " y = "match" x "with" \
    | "zero" () arrow.r.double #text(fill: red)[$"zero" ()$] \
    | "succ" x' arrow.r.double "succ" ("plus" x' y)
    $

    This is problematic because it doesn't use up $y$ in the first branch.

  - 
    $"plus" (x : "nat") (y : "nat") : "nat" \
    "plus" x " " y = "match" x "with" \
    | "zero" () arrow.r.double y \
    | "succ" x' arrow.r.double "succ" ("plus" x' y)
    $

    This uses up all inputs exactly.

  #tree(
    axi[$Gamma tack.r e : A_k (k in L)$],
    uni[$Gamma tack.r k(e) : +{l : A_l}_(l in L)$],
  )