ye

pfenning/notes.typ

@@ -1,3 +1,4 @@
+#import "@preview/fletcher:0.4.5" as fletcher: diagram, node, edge
 #import "../common.typ": *
 #import "@preview/prooftrees:0.1.0": *
 #import "@preview/algo:0.3.3": algo, i, d, comment, code
@@ -373,3 +374,109 @@ $with { l : A_l}_(l in L)$
   axi[$Gamma tack.r e : with { l : A_l}_(l in L) (forall l in L)$],
   uni[$Gamma tack.r e.k : A_k (k in L)$]
 )
+
+#pagebreak()
+
+=== Lecture 3
+
+- negation
+- Mixing linear & non-linear programs
+- Mode checking & inference
+
+#diagram((
+  node((0, -0.8), "Unrestricted"),
+  edge("-|>"),
+  node((0.8, 0), "Strict (at least once)"),
+  node((-0.8, 0), "Affine (at most once)"),
+  edge("-|>"),
+  node((0, 0.8), "Linear"),
+  edge((0, -0.8), (-0.8, 0), "-|>"),
+  edge((0.8, 0), (0, 0.8), "-|>"),
+))
+
+Cannot write map:
+
+```
+fail decl map (f : nat -> nat) (xs : list) : list
+defn map f xs = match xs with
+  | 'nil () => 'nil ()
+  | 'cons (x, xs) => 'cons (f x, map f xs)
+```
+
+This is because $f$ isn't used in the first line but used twice in the second line.
+
+We could write some iterator:
+
+```
+type iterator = &{'next : nat -> nat * iterator, 'done : 1}
+
+decl iterate (iter : iterator) (xs : list) : list
+defn iterate iter xs = match xs with
+  | 'nil () => (match iter.'done with | () => 'nil ())
+  | 'cons (x, xs) => (match iter.'next x with | (y, iter) => 'cons ('succ y, iterate iter xs))
+```
+
+This is a linear implementation of a function that adds 1 to everything in the list.
+
+=== Modes
+
+Take the entire language and parameterize by modes:
+
+$A, B_m ::= 1_m &| A_m times B_m | +{l : A^l_m}_(l in L) | arrow.b^k_m A_k (k >= m) \
+&| A_m arrow B_m | \&{l:A^l_m}_(l in L) | arrow.t^i_m A_i (i <= m)
+$
+
+These are implemented in the code:
+
+```
+type nat[k] = +{'zero : 1, 'succ : nat[k]}
+```
+
+Modes need to be _guarded_. For example:
+
+```
+type nat[k] = +{'zero : 1, 'succ : down[k] nat[k]}
+
+type list[m k] = +{
+  'nil : 1,
+  'cons : down[k] nat[k] * down[m] list[m k],
+}
+```
+
+The `m` is used because it comes first.
+
+$ (!A_L)_L eq.delta arrow.b^U_L arrow.t^U_L A_L$
+
+needs a partial ordering on U and L. Need to copy when it's in the U and then move it back into the L when you're done.
+
+```
+decl map (f : [mf] up[k] (nat[k] -> nat[k])) (xs : list[m k]) : list[r k]
+defn map f xs = match xs with
+  | 'nil () => 'nil ()
+  | 'cons (<x>, <xs>) => 'cons(<f.force x>, <map f xs>)
+```
+
+$Gamma$ can be multi-modal. This is how top-level declarations can be re-used.
+
+https://www.cs.cmu.edu/~fp/papers/tocl07.pdf
+
+Need to enforce independence, that $Gamma tack.r e : A_m$ means $Gamma >= m$
+
+Pointer:
+
+#let wrap(e) = $angle.l #e angle.r $
+
+#tree(
+  axi[$Gamma >= k$],
+  axi[$Gamma tack.r e : A_k$],
+  bin[$Gamma tack.r wrap(e) : arrow.b^k_m A_k$]
+)
+
+*NOTE:* The bottom would not be valid if $Gamma cancel(>=) k$
+
+#tree(
+  axi[$Delta >= m >= r$],
+  axi[$Delta tack.r e : arrow.b^k_m A_k$],
+  axi[$Gamma , x : A_k tack.r e' : C_r$],
+  nary(3)[$Delta Gamma tack.r "match" e with wrap(x) => e' : C_r$],
+)