update

Makefile

@@ -1,4 +1,9 @@
-deploy:
+SOURCES := $(shell find . -name "*.typ")
+
+oplss.pdf: oplss.typ $(SOURCES)
+	typst compile $<
+
+deploy: oplss.pdf
 	rsync -azr \
 		--exclude .git \
 		./ root@veil:/home/blogDeploy/public/public-files/oplss-2024

ahmed/day1.agda

@@ -24,4 +24,5 @@ data evaluationContext : Set where
   dot : evaluationContext
   EIf_Then_Else_ : evaluationContext → Term → Term → evaluationContext
   EAppLeft : evaluationContext → Term → evaluationContext
-  EAppRight : Value → evaluationContext → evaluationContext
+  EAppRight : Value → evaluationContext → evaluationContext
+

ahmed/notes.typ

@@ -1,11 +1,11 @@
 #import "../common.typ": *
 #import "@preview/prooftrees:0.1.0": *
-#show: doc => conf(doc)
+#show: doc => conf("Logical Relations", doc)
 
 #let ifthenelse(e, e1, e2) = $"if" #e "then" #e1 "else" #e2$
 #let safe = $"safe"$
 
-= Mon. Jun 3 \@ 14:00
+= Lecture 1, Mon. Jun 3 \@ 14:00
 
 == Logical relations
 
@@ -195,8 +195,7 @@ Use $gamma$ as a substitution from variables to values.
 
 Now define semantic soundness:
 
-$
-Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau)$
+$ Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau) $
 
 Prove:
 
@@ -242,4 +241,135 @@ To prove B:
 _Proof._ Suppose $e'$ s.t. $e mapstostar e'$ .
 
 - Case : $not "irreducible" (e')$ then $exists e'' . e' mapsto e''$
-- Case : $"irreducible" (e')$ then $isValue(e')$ trivially.
+- Case : $"irreducible" (e')$ then $isValue(e')$ trivially.
+
+#pagebreak()
+
+= Lecture 2, Tue. Jun 4 \@ 11:00
+
+#set enum(numbering: "A.")
+
+Recall the lemmas we're proving:
+
+1. $dot tack.r e : tau arrow.r.double e in Epsilon db(tau)$
+2. $e in Epsilon db(tau) arrow.r.double safe(e)$
+
+#set enum(numbering: "1.")
+
+To prove lemma A, we must prove the fundamental property: $Gamma tack.r e:tau arrow.r.double Gamma tack.r.double e:tau$. "Syntactically well-typed implies semantic well-typed"
+
+$ Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau) $
+
+=== Proving A case for $lambda$
+
+Case:
+
+#tree(
+  axi[$Gamma , x : tau tack.r e : tau_2$],
+  uni[$Gamma tack.r lambda (x : tau_1) . e : tau_1 arrow.r tau_2  $]
+)
+
+*Show:* $Gamma tack.r.double lambda (x : tau_1) . e : tau_1 arrow tau_2$.
+
+Suppose $gamma in G db(Gamma)$. Then we ned to *show*: 
+
+$ &gamma (lambda (x : tau_1). e) in Epsilon db(tau_1 arrow tau_2) \
+equiv &lambda (x:tau_1).gamma(e) in Epsilon db(tau_1 arrow tau_2)
+$
+
+Suffices to show: $lambda (x:tau_1) gamma(e) in V db(tau_1 arrow tau_2)$.
+
+Suppose $v in V db(tau_1)$. Then we need to show: 
+
+$ &subst(x, v, gamma(e)) in Epsilon db(tau_2) \
+equiv & gamma[x mapsto v] (e) in Epsilon db(tau_2)$
+
+By the induction hypothesis, $Gamma,x:tau_1 tack.r e:tau_2$, so $Gamma,x:tau_1 tack.r.double e:tau_2$.
+
+Because $gamma[x mapsto v] in G db(#[$Gamma, x : tau_1$])$, we can instantiate $Gamma,x:tau_1$ with $gamma[x mapsto v]$. (basically just translating it over to $gamma$)
+
+Therefore, $gamma[x mapsto v](e) in Epsilon db(tau_2)$
+
+#TODO Do the app case at home.
+
+== Adding recursive types
+
+$ Omega :equiv (lambda x . x " " x) (lambda x . x " " x) $
+
+Cannot be type-checked in STLC, but adding recursive types allows this to be typechecked.
+
+$ tau ::= 1 | "bool" | tau_1 arrow tau_2 | alpha | mu alpha . tau $
+
+For example:
+
+- $"tree" = mu alpha . 1 + ("int" times alpha times alpha)$
+- $"intlist" = mu alpha . 1 + ("int" times alpha)$
+
+$ e ::= ... | "fold"(e) | "unfold"(e) $
+
+#tree(
+  axi[$Gamma tack.r e : subst(alpha, mu alpha . tau, tau)$],
+  uni[$Gamma tack.r "fold"(e) : mu alpha . tau $],
+)
+
+#tree(
+  axi[$Gamma tack.r e : mu alpha . tau $],
+  uni[$Gamma tack.r "unfold"(e) : subst(alpha, mu alpha . tau, tau)$],
+)
+
+*NOTE*: Fold usually needs to be annotated with $mu alpha . tau$ in order to guide type checking.
+
+== Type-checking $Omega$
+
+$ Omega :equiv (lambda (x : ?) . x " " x) (lambda (x : ?) . x " " x) $
+
+With inductive types, we can give the second $x$ some type $alpha$.
+Then the first $x$ that calls it would be $alpha arrow tau$ for some $tau$.
+
+$ "SA" : (mu alpha . alpha arrow tau) arrow tau \
+"SA" :equiv lambda (x : mu alpha . alpha arrow tau) . ("unfold" x) x$
+
+Then re-define $Omega$ to be:
+
+$ Omega' :equiv "SA" ("fold" "SA") $
+
+== Proving soundness for lambda calculus with recursive types with logical relations
+
+$ v ::= ... | "fold"(v) $
+
+*NOTE:* Unfold will not be a value.
+
+$ E ::= ... | "fold" E $
+
+New reduction rule:
+
+$ "unfold"("fold" v) mapsto v $
+
+$V db(mu alpha . tau) = { "fold"(v) | forall v in V db(subst(alpha, mu alpha. tau, tau))  }$
+
+This is *not* a well-founded relation, because $V$ would depend on itself. This was an open problem (ATTAPL exercise) for a long time.
+
+=== Step Indexing
+
+This shows that this is well founded.
+
+"If i were to run my program for $k$ steps, then I would not be able to tell that $V$ isn't well-founded"
+
+#rect[$V_k db(tau)$]
+
+$V_k db("bool") = {"true" , "false"}$
+
+$V_k db(mu alpha . tau) = {"fold" v | v in V_(k-1) db(subst(alpha , mu alpha. tau , tau))}$
+
+- Doing $"unfold"("fold" x) mapsto x$ "consumes" a step of $k$, so that's why $v$ already consumes a step.
+
+$V_k db(tau_1 arrow tau_2) = {lambda (x : tau_1) . e | forall (j lt.eq k, v in V_j db(tau_1)) arrow.r.double subst(x, v, e) in Epsilon_j db(tau_2)}$
+
+- Think of this as a timeline, starting with $k$ steps left
+- $j+1$ is the number of steps left when you get to the application of the lambda
+- Use 1 step to do the application, resulting in $j$ steps
+- Eventually reach 0 steps left.
+
+$Epsilon_k db(tau) = { e | forall (j < k, e') arrow.r.double e mapsto e' and "irreducible"(e') arrow.r.double e' in V_(k-j) db(tau)}$
+
+What $k$ doesn't matter because the entire relations are universally quantified over all $k$s.

common.typ

@@ -7,9 +7,11 @@
 #let subst(x, v, e) = $#e [#v\/#x]$
 #let TODO = text(fill: red)[*TODO*]
 
-#let conf(doc) = {
+#let conf(title, doc) = {
   show link: underline
-  set page(width: 5in, height: 8in, margin: 0.4in)
+  set page(width: 6in, height: 9in, margin: 0.3in)
+
+  text(size: 30pt)[#title]
   doc
 }
 

pfenning/notes.typ

@@ -141,6 +141,8 @@ Language to be studied is called "snax"
     uni[$Gamma tack.r k(e) : +{l : A_l}_(l in L)$],
   )
 
+#pagebreak()
+
 == Lecture 2
 
 Type system