logical relations

ahmed/notes.typ

@@ -7,6 +7,7 @@
 #let mapstostar = $op(arrow.r.long.bar)^*$
 #let safe = $"safe"$
 #let db(x) = $bracket.l.double #x bracket.r.double$
+#let TODO = text(fill: red)[*TODO*]
 
 = Mon. Jun 3 \@ 14:00
 
@@ -140,6 +141,8 @@ The property we're interested in is: $safe(e)$
 
 *Definition (safe).* $safe(e) :equiv forall e' . (e mapstostar e') arrow.r.double isValue(e') or exists e'' . (e' mapsto e'')$
 
+*Definition (Semantic Type Soundness).* $dot tack.r e : tau$ then $safe(e)$
+
 Common technique:
 
 - Focus on the values, when do values belong to the relation, when are they safe? 
@@ -155,7 +158,8 @@ The body also needs to be well-typed.
 
 $
 V db(tau_1 arrow.r tau_2) = { lambda (x : tau_1) . e | forall v in V db(tau_1) . subst(x, v, e) in Epsilon db(tau_2) } \
-Epsilon db(tau) = { e | forall e' . e mapstostar e' and "irreducible"(e') arrow.r.double e' in V db(tau)}
+Epsilon db(tau) = { e | forall e' . e mapstostar e' and "irreducible"(e') arrow.r.double e' in V db(tau)} \
+"irreducible"(e) :equiv cancel(exists) e' . e mapsto e'
 $
 
 #rect[
@@ -163,3 +167,84 @@ $
 Unsafe code blocks are an example of a something that may not be _syntactically well-formed_, but are still represented by logical relations because they behave the same at the boundary.
 ]
 
+The difference between semantic soundness with logical relations vs. progress and preservation is that you don't need to prove anything about intermediate states of running programs.
+
+Can't just work with closed terms because of this rule:
+
+#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  $]
+)
+
+(Slightly more general theorem)
+
+#rect(width: 100%)[
+  *Definition (Fundamental property of logical relations).* 
+  Theorem we are trying to prove
+  $
+    Gamma tack.r e : tau arrow.r.double Gamma tack.r.double e : tau
+  $
+  - "syntactically well typed means semantically well typed"
+]
+
+Focus on terms with open variables
+
+Use $gamma$ as a substitution from variables to values.
+
+#rect[$G db(Gamma)$]
+
+- $G db(dot) = { emptyset }$
+- $G #db[$Gamma , x : tau$] = { gamma[x mapsto v] | gamma in G db(Gamma) and v in V db(tau)}$
+
+  "sound inputs will give you sound outputs". This is the important case
+
+Now define semantic soundness:
+
+$
+Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau)$
+
+Prove:
+
+#set enum(numbering: "a.")
+
+1. $dot tack.r e : tau arrow.r.double e in Epsilon db(tau)$
+  - this cannot be proved by focusing on closed terms alone
+  - This is why the fundamental property must be used, in the case of $Gamma = dot$, then the substitution $gamma$ is empty and the result is $e in Epsilon db(tau)$
+2. $e in Epsilon db(tau) arrow.r.double safe(e)$
+
+#set enum(numbering: "1.")
+
+To prove A:
+
+_Proof._ By induction on typing derivations
+
+- 
+  Case example:
+
+  #tree(
+    axi[],
+    uni[$Gamma tack.r "true" : "bool"$]
+  )
+
+  Show $Gamma tack.r.double "true" : "bool"$. Suppose $gamma in G db(Gamma)$.
+  Show $gamma("true") in Epsilon db("bool") equiv "true" in Epsilon db("bool")$.
+  Suffices to show $"true" in V db("bool")$, which is true by definition.
+
+  False is proved similarly.
+
+- 
+  Case example:
+
+  #tree(
+    axi[$Gamma(x) = tau$],
+    uni[$Gamma tack.r x : tau$]
+  )
+
+  Show $Gamma tack.r.double x : tau$. #TODO
+
+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.