#import "prooftree.typ": *
#import "@preview/showybox:2.0.1": showybox
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= Type theory crash course

== MLTT + Sets

Important features in MLTT:

#let Nat = $sans("Nat")$
#let Vect = $sans("Vect")$

- Dependent types and functions.
  e.g $ "concatenate": Pi_(m,n:"Nat") Vect(m) -> Vect(n) -> Vect(m+n) $
  - Function arrows always associate to the right.
- All functions are total.

Goal: to write well-typed programs. (implementing an algorithm and proving a mathematical statement are the same)

=== Judgements

$ "context" tack.r "conclusion" $

#let defeq = $equiv$

#table(
  columns: 2,
  stroke: gray,
  $Gamma$, [sequence of variable declarations (contexts are always well-formed)],
  $Gamma tack.r A$, [$A$ is well-formed *type* in context $Gamma$],
  $Gamma tack.r a : A$, [*term* $a$ is well-formed and of type $A$],
  $Gamma tack.r A defeq B$, [types $A$ and $B$ are *convertible*],
  $Gamma tack.r a defeq b : A$, [$a$ is convertible to $b$ in type $A$],
)

Example:

$ "isZero?"& : Nat -> "Bool" \
  "isZero?"& (n) :defeq "??"
$

At this point, looking for $(n: Nat) tack.r "isZero?"(n) : "Bool"$.

=== Inference rules

#prooftree(
  axiom($J_1$),
  axiom($J_2$),
  axiom($J_3$),
  rule(n: 3, $J$),
)

For example:

#prooftree(
  axiom($Gamma tack.r a defeq b : A$),
  rule($Gamma tack.r b defeq a : A$),
)

#let subst(name, replacement, expr) = $#expr [ #replacement \/ #name ]$

=== Interpreting types as sets

You can interpret types as sets, where $a : A$ is interpreted as $floor(a) : floor(A)$.

- Univalent mathematics can _not_ be interpreted as sets. There are extra axioms that breaks the interpretation.
- The judgement $a : A$ cannot be proved or disproved.
- For ex. 2 of natural numbers and 2 of integers can be converted but are inherently different values.

=== Convertibility

If $x : A$ and $A defeq B$, then $x : B$. We are thinking of these types as literally the same.

If $a defeq a'$ then $f @ a defeq f @ a'$.

=== Declaring types and terms

4 types of rules:

#let typeIntroTable(formation, introduction, elimination, computation) = table(
  columns: 2,
  stroke: 0in,
  [#text(fill: blue, [Formation])], [#formation],
  [#text(fill: blue, [Introduction])], [#introduction],
  [#text(fill: blue, [Elimination])], [#elimination],
  [#text(fill: blue, [Computation])], [#computation],
)

#typeIntroTable(
  [a way to construct a new type],
  [way to construct *canonical terms* of the type],
  [how to use a term of the new type to construct terms of other types],
  [what happens when one does Introduction followed by Elimination],
)

Example (context $Gamma$ are elided):

#typeIntroTable(
  [If $A$ and $B$ are types, then $A -> B$ is a type
    #prooftree(axiom($Gamma tack.r A$), axiom($Gamma tack.r B$), rule(n: 2, $Gamma tack.r A -> B$))],
  [If $(x : A) tack.r b : B$, then $tack.r lambda (x : A) . b(x) : A -> B$
    - $b$ is an expression that might involve $x$],
  [If we have a function $f : A -> B$, and $a : A$, then $f @ a : B$ (or $f(a) : B$)],
  [What is the result of the application? $(lambda(x : A) . b(x)) @ a defeq subst(a, x, b)$
    - Substitution $subst(a, x, b)$ is built-in],
)

Questions:

- *What does the lambda symbol mean?* Lambda is just notation. It could also be written $tack.r "lambda"((x:A), b(x)) : A -> B$.

Another example: the singleton type

#let unit = $bb(1)$
#let tt = $t t$
#let rec = $sans("rec")$

#typeIntroTable(
  [$unit$ is a type],
  [$tt : unit$],
  [If $x : unit$ and $C$ is a type and $c : C$, then $rec_unit (C, c, x) : C$],
  [$rec_unit (C, c, t) defeq c$
    - Interpretation in sets: a one-element set],
)

- Question: *How to construct this using lambda abstraction?*
  - (Structural rule: having $tack.r c : C$ means $x : unit tack.r c : C$, which by the lambda introduction rule gives us $lambda x.c : unit -> C$)

Booleans

#let Bool = $sans("Bool")$
#let tru = $sans("true")$
#let fls = $sans("false")$

#typeIntroTable(
  [$Bool$ is a type],
  [$tru : Bool$ and $fls : Bool$],
  [If $x : Bool$ and $C$ is a type and $c, c' : C$, then $rec_Bool (C, c, c', x) : C$],
  [$rec_Bool (C, c, c', tru) defeq c$ and $rec_Bool (C, c, c', fls) defeq c'$],
)

Empty type

#let empty = $bb(0)$

#typeIntroTable(
  [$empty$ is a type],
  [_(no introduction rule)_],
  [If $x : empty$ and $C$ is a type, then $rec_empty (C, x) : C$],
  [_(no computation rule)_],
)

Natural numbers

#let zero = $sans("zero")$
#let suc = $sans("suc")$

#typeIntroTable(
  [$Nat$ is a type],
  [- $zero : Nat$
  - If $n : Nat$, then $suc(n) : Nat$],
  [If $C$ is a type and $c_0:C$ and $c_s:C->C$ and $x: Nat$, then $rec_Nat (C,c_0,c_s,x) : C$],
  [- $rec_Nat (C, c_0, c_s, zero) defeq c_0$
  - $rec_Nat (C, c_0, c_s, suc(n)) defeq c_s @ (rec_Nat (C, c_0, c_s, n))$
  We can define computation rule on naturals using a universal property],
)