lecture 1

.envrc

@@ -1 +0,0 @@
-use flake

.gitignore

@@ -1,2 +1,3 @@
 .direnv
 .*.aux
+*.pdf

Lecture1.typ

@@ -0,0 +1,168 @@
+#import "prooftree.typ": *
+#import "@preview/showybox:2.0.1": showybox
+#set page(width: 5.6in, height: 9in, margin: 0.4in)
+
+= 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],
+)
+

Test.v

@@ -4,4 +4,3 @@ Lemma myfirstlemma : 2 + 2 = 4.
 Proof.
   apply idpath.
 Defined.
-

prooftree.typ

@@ -0,0 +1,405 @@
+#let prooftree(
+  spacing: (
+    horizontal: 1em,
+    vertical: 0.5em,
+    lateral: 0.5em,
+  ),
+  label: (
+    // TODO: split offset into horizontal and vertical
+    offset: -0.1em,
+    side: left,
+    padding: 0.2em,
+  ),
+  line-stroke: 0.5pt,
+  ..rules
+) = context {
+  // Check parameters and compute normalized settings
+  let settings = {
+    // Check basic validity of `rules`.
+    if rules.pos().len() == 0 {
+      panic("The `rules` argument cannot be empty.")
+    }
+
+
+    // Check the types of the parameters.
+    assert(
+      type(spacing) == "dictionary",
+      message: "The value `" + repr(spacing) + "` of the `spacing` argument was expected"
+        + "to have type `dictionary` but instead had type `" + type(spacing) + "`."
+    )
+    assert(
+      type(label) == "dictionary",
+      message: "The value `" + repr(label) + "` of the `label` argument was expected"
+        + "to have type `dictionary` but instead had type `" + type(label) + "`."
+    )
+    assert(
+      type(line-stroke) == "length",
+      message: "The value `" + repr(line-stroke) + "` of the `line-stroke` argument was expected"
+        + "to have type `length` but instead had type `" + type(line-stroke) + "`."
+    )
+
+    // Check validity of `spacing`'s keys.
+    for (key, value) in spacing {
+      if key not in ("horizontal", "vertical", "lateral", "h", "v", "l") {
+        panic("The key `" + key + "` in the `spacing` argument `" + repr(spacing) + "` was not expected.")
+      }
+      if type(value) != "length" {
+        panic(
+          "The value `" + repr(value) + "` of the key `" + key + "` in the `spacing` argument `" + repr(spacing)
+          + "` was expected to have type `length` but instead had type `" + type(value) + "`."
+        )
+      }
+    }
+
+    // Check exclusivity of `spacing`'s keys.
+    let mutually_exclusive(key1, key2, keys) = {
+      assert(
+        key1 not in keys or key2 not in keys,
+        message: "The keys `" + key1 + "` and `" + key2 + "` in the `spacing` argument `"
+          + repr(spacing) + "` are mutually exclusive."
+      )
+    }
+    mutually_exclusive("horizontal", "h", spacing.keys())
+    mutually_exclusive("vertical", "v", spacing.keys())
+    mutually_exclusive("lateral", "l", spacing.keys())
+    
+    // Check validity of `label`'s keys.
+    let expected = ("offset": "length", "side": "alignment", "padding": "length")
+    for (key, value) in label {
+      if key not in expected {
+        panic("The key `" + key + "` in the `label` argument `" + repr(label) + "` was not expected.")
+      }
+      if type(value) != expected.at(key) {
+        panic(
+          "The value `" + repr(value) + "` of the key `" + key + "` in the `label` argument `" + repr(label)
+          + "` was expected to have type `" + type.at(key) + "` but instead had type `" + type(value) + "`."
+        )
+      }
+    }
+    if "side" in label {
+      assert(
+        label.side == left or label.side == right,
+        message: "The value for the key `side` in the argument `label` can only be either "
+          + "`left` (default) or `right`, but instead was `" + repr(label.side) + "`."
+      )
+    }
+
+    (
+      spacing: (
+        horizontal: spacing.at("horizontal", default: spacing.at("h", default: 1.5em)).to-absolute(),
+        vertical: spacing.at("vertical", default: spacing.at("v", default: 0.5em)).to-absolute(),
+        lateral: spacing.at("lateral", default: spacing.at("l", default: 0.5em)).to-absolute(),
+      ),
+      label: (
+        offset: label.at("offset", default: -0.1em).to-absolute(),
+        side: label.at("side", default: left),
+        padding: label.at("padding", default: 0.2em).to-absolute(),
+      ),
+      line-stroke: line-stroke.to-absolute(),
+    )
+  }
+
+  // Holds the current "pending" rules, i.e. those without a parent
+  let stack = ()
+  // Holds all the measures
+  let layouts = ()
+
+  // First pass: compute the layout of each rule given the one of its children
+  for (i, rule) in rules.pos().enumerate() {
+    let to_pop = rule.__prooftree_to_pop
+    let measure_func = rule.__prooftree_measure_func
+
+    assert(
+      to_pop <= stack.len(),
+      message: "The rule `" + repr(rule.__prooftree_raw) + "` was expecting at least "
+        + str(to_pop) + " rules in the stack, but only " + str(stack.len()) + " were present."
+    )
+
+    // Remove the children from the stack
+    let children = stack.slice(stack.len() - to_pop)
+    stack = stack.slice(0, stack.len() - to_pop)
+
+    // Compute the layout and push
+    let layout = measure_func(i, settings, children)
+    stack.push(layout)
+    layouts.push(layout)
+  }
+
+  assert(
+    stack.len() == 1,
+    message: "Some rule remained unmatched: " + str(stack.len()) + " roots were found but only 1 was expected."
+  )
+
+  let last = stack.pop()
+
+  let content = {
+    let offsets = range(rules.pos().len()).map(_ => (0pt, 0pt))
+
+    // Second pass: backward draw each rule and compute offset of children
+    for (i, rule) in rules.pos().enumerate().rev() {
+      let (dx, dy) = offsets.at(i)
+      let layout = layouts.at(i)
+
+      // Update the offsets of the children
+      for (j, cdx, cdy) in layout.at("children_offsets", default: ()) {
+        offsets.at(j) = (dx + cdx, dy + cdy)
+      }
+
+      // Draw at the correct offset
+      let draw_func = rule.__prooftree_draw_func
+      place(left + bottom, dx: dx, dy: -dy, draw_func(settings, layout))
+    }
+  }
+
+  block(width: last.width, height: last.height, content)
+}
+
+#let axiom(label: none, body) = {
+  // Check arguments
+  {
+    // Check the type of `label`.
+    assert(
+      type(label) in ("string", "content", "none"),
+      message: "The type of the `label` argument `" + repr(label) + "` was expected to be "
+      + "`none`, `string` or `content` but was instead `" + type(label) + "`."
+    )
+  }
+
+  // TODO: allow the label to be aligned on left, right or center (default and current).
+
+  (
+    __prooftree_raw: body,
+    __prooftree_to_pop: 0,
+    __prooftree_measure_func: (i, settings, children) => {
+      // Compute the size of the body
+      let body_size = measure(body)
+      let body_width = body_size.width.to-absolute()
+      let body_height = body_size.height.to-absolute()
+
+      // Compute width of the base (including space)
+      let base_width = body_width + 2 * settings.spacing.lateral
+
+      // Update layout if a label is present
+      let (width, height) = (base_width, body_height)
+      let base_side = 0pt
+      let (label_left, label_bottom) = (0pt, 0pt)
+      if label != none {
+        // Compute the size of the label
+        let label_size = measure(label)
+        let label_width = label_size.width
+        let label_height = label_size.height
+        
+        // Update width and offsets from the left
+        width = calc.max(base_width, label_width)
+        base_side = (width - base_width) / 2
+        label_left = (width - label_width) / 2
+
+        // Compute bottom offset and update height
+        label_bottom = height + 1.5 * settings.spacing.vertical
+        height = label_bottom + label_height
+      }
+
+      return (
+        index: i,
+        width: width,
+        height: height,
+        base_left: base_side,
+        base_right: base_side,
+        main_left: base_side,
+        main_right: base_side,
+
+        // Extra for draw
+        body_left: base_side + settings.spacing.lateral,
+        label_left: label_left,
+        label_bottom: label_bottom,
+      )
+    },
+    __prooftree_draw_func: (settings, l) => {
+      // Draw body
+      place(left + bottom, dx: l.body_left, body)
+      
+      // Draw label
+      if label != none {
+        place(left + bottom, dx: l.label_left, dy: -l.label_bottom, label)
+      }
+    }
+  )
+}
+
+#let rule(
+  n: 1,
+  label: none,
+  root
+) = {
+  // Check arguments
+  {
+    // Check validity of the `n` parameter
+    assert(
+      type(n) == "integer",
+      message: "The type of the `n` argument `" + repr(n) + "` was expected to be "
+      + "`integer` but was instead `" + type(n) + "`."
+    )
+
+    // Check the type of `label`.
+    assert(
+      type(label) in ("string", "dictionary", "content", "none"),
+      message: "The type of the `label` argument `" + repr(label) + "` was expected to be "
+      + "`none`, `string`, `content` or `dictionary` but was instead `" + type(label) + "`."
+    )
+    // If the type of `label` was string then it's good, otherwise we need to check its keys.
+    if type(label) == "dictionary" {
+      for (key, value) in label {
+        // TODO: maybe consider allowing `top`, `top-left` and `top-right` if `rule(n: 0)` gets changed.
+        if key not in ("left", "right") {
+          panic("The key `" + key + "` in the `label` argument `" + repr(label) + "` was not expected.")
+        }
+        if type(value) not in ("string", "content") {
+          panic(
+            "The value `" + repr(value) + "` of the key `" + key + "` in the `label` argument `" + repr(label)
+            + "` was expected to have type `string` or `content` but instead had type `" + type(value) + "`."
+          )
+        }
+      }
+    }
+  }
+
+  (
+    __prooftree_raw: root,
+    __prooftree_to_pop: n,
+    __prooftree_measure_func: (i, settings, children) => {
+      let width(it) = measure(it).width.to-absolute()
+      let height(it) = measure(it).height.to-absolute()
+
+      let label = label
+      if type(label) == "none" {
+        label = (left: none, right: none)
+      }
+      if type(label) in ("string", "content") {
+        label = (
+          left: if settings.label.side == left { label } else { none },
+          right: if settings.label.side == right { label } else { none }
+        )
+      }
+      label = (
+        left: label.at("left", default: none),
+        right: label.at("right", default: none),
+      )
+
+      // Size of root
+      let root_width = width(root)
+      let root_height = height(root)
+
+      // Width of base, which includes spacing as well
+      let base_width = 2 * settings.spacing.lateral + root_width
+
+      // Bottom offset of the line and children
+      let line_bottom = root_height + settings.spacing.vertical
+      let children_bottom = line_bottom + settings.spacing.vertical
+
+      // Left/right offset of bases of extreme children
+      let (child_base_left, child_base_right) = (0pt, 0pt)
+      if n != 0 {
+        child_base_left = children.first().base_left
+        child_base_right = children.last().base_right
+      }
+
+      // Width and height of children, and width of their combined bases
+      let children_width = children
+        .map(c => c.width)
+        .intersperse(settings.spacing.horizontal)
+        .sum()
+      let children_height = children.map(c => c.height).fold(0pt, calc.max)
+      let children_base_width = children_width - child_base_left - child_base_right
+
+      // Width of the line
+      let line_width = calc.max(children_base_width, base_width)
+      
+      // Left/right offsets of lateral children main
+      let (child_main_left, child_main_right) = (0pt, 0pt)
+      if n != 0 {
+        child_main_left = children.first().main_left
+        child_main_right = children.last().main_right
+      }
+
+      // Offset of bases from line start (same for left/right)
+      let base_from_line = (line_width - base_width) / 2
+      let children_base_from_line = (line_width - children_base_width) / 2
+
+      // Space for labels
+      let (label_left_width, label_right_width) = (0pt, 0pt)
+      let (label_left_height, label_right_height) = (0pt, 0pt)
+      if label.left != none {
+        label_left_width = width(label.left) + settings.label.padding
+        label_left_height = height(label.left)
+      }
+      if label.right != none {
+        label_right_width = width(label.right) + settings.label.padding
+        label_right_height = height(label.right)
+      }
+
+      // Left/right offsets of line = max of labels and children main
+      let line_left = calc.max(label_left_width, child_base_left - children_base_from_line)
+      let line_right = calc.max(label_right_width, child_base_right - children_base_from_line)
+
+      // Left/right offsets of base
+      let base_left = line_left + base_from_line
+      let base_right = line_right + base_from_line
+
+      // Left/right offsets of children
+      let children_left = line_left + children_base_from_line - child_base_left
+      let children_right = line_right + children_base_from_line - child_base_right
+
+      // Left/right offsets of main
+      let main_left = calc.min(line_left, children_left + child_main_left)
+      let main_right = calc.min(line_right, children_right + child_main_right)
+
+      // Full width and height
+      let width = line_left + line_width + line_right
+      let height = children_bottom + children_height
+
+      // Incrementally compute the relative offset of each child
+      let children_offsets = ()
+      for c in children {
+        children_offsets.push((c.index, children_left, children_bottom))
+        children_left += c.width + settings.spacing.horizontal
+      }
+
+      (
+        index: i,
+        width: width,
+        height: height,
+        base_left: base_left,
+        base_right: base_right,
+        main_left: main_left,
+        main_right: main_right,
+        children_offsets: children_offsets,
+
+        // Extra for draw
+        label: label,
+        root_left: base_left + settings.spacing.lateral,
+        line_left: line_left,
+        line_bottom: line_bottom,
+        line_width: line_width,
+        label_left: line_left - label_left_width,
+        label_right: line_left + line_width + settings.label.padding,
+        label_left_bottom: root_height + settings.spacing.vertical + settings.line-stroke / 2 - label_left_height / 2 - settings.label.offset,
+        label_right_bottom: root_height + settings.spacing.vertical + settings.line-stroke / 2 - label_right_height / 2 - settings.label.offset,
+      )
+    },
+    __prooftree_draw_func: (settings, l) => {
+      // Draw root content
+      place(left + bottom, dx: l.root_left, root)
+      
+      // Draw line
+      place(left + bottom, dx: l.line_left, dy: -l.line_bottom, line(length: l.line_width, stroke: settings.line-stroke))
+
+      // Draw labels
+      if l.label.left != none {
+        place(left + bottom, dx: l.label_left, dy: -l.label_left_bottom, l.label.left)
+      }
+      if l.label.right != none {
+        place(left + bottom, dx: l.label_right, dy: -l.label_right_bottom, l.label.right)
+      }
+    }
+  )
+}