XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=591fd31d9421859776cb4220a92ecb8a3076f16e;hp=46ca442dfeba847526e8f6a5683ce51609aee528;hb=26574827b88c32f00baaf941c4eac1aaebac839d;hpb=458cadab1427b0fc0f7bc8689f1dddb18b2201e7
diff git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn
index 46ca442d..591fd31d 100644
 a/hints/assignment_7_hint_5.mdwn
+++ b/hints/assignment_7_hint_5.mdwn
@@ 1,7 +1,7 @@
* How shall we handle \[[∃x]]. As we said, GS&V really tell us how to interpret \[[∃xPx]], but what they say about this breaks naturally into two pieces, such that we can represent the update of `s` with \[[∃xPx]] as:
+* How shall we handle \[[∃x]]. As we said, GS&V really tell us how to interpret \[[∃xPx]], but what they say about this breaks naturally into two pieces, such that we can represent the update of our starting set `u` with \[[∃xPx]] as:
 s >>= \[[∃x]] >>= \[[Px]]
+ u >>=_{set} \[[∃x]] >>=_{set} \[[Px]]
What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
@@ 12,38 +12,45 @@
> The referent system of the remaining possibilities will be extended with a new peg, which is associated with `x`. And for each old possibility `i` in `s`, there will be just as many extensions `i[x/d]` in the new state `s'` and there are entities `d` which in the possible world of `i` have the property P.
 Deferring the "property P" part, this says:
+ Deferring the "property P" part, this corresponds to:
 s updated with \[[∃x]] ≡
 s >>= (fun (r, h) > List.map (fun d > newpeg_and_bind 'x' d) domain)
+ u updated with \[[∃x]] ≡
+ let extend_one = fun one_dpm >
+ fun truth_value >
+ if truth_value = false
+ then empty_set
+ else List.map (fun d > new_peg_and_assign 'x' d) domain
+ in bind_set u extend_one
+
+ where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
+
+ let new_peg_and_assign (var_to_bind : char) (d : entity) =
+ fun ((r, h) : assignment * store) >
+ (* first we calculate an unused index *)
+ let newindex = List.length h
+ (* next we store d at h[newindex], which is at the very end of h *)
+ (* the following line achieves that in a simple but inefficient way *)
+ in let h' = List.append h [d]
+ (* next we assign 'x' to location newindex *)
+ in let r' = fun v >
+ if v = var_to_bind then newindex else r v
+ (* the reason for returning true as an initial element should now be apparent *)
+ in (true, r',h')
 That is, for each pair `(r, h)` in `s`, we collect the result of extending `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities (here designated `domain`), and binding the variable `x` to the index of that peg.
+ What's going on here? For each `bool dpm` in `u` that wraps a `true`, we collect `dpm`s that are the result of extending their input `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities, and binding the variable `x` to the index of that peg. For `bool dpm`s in `u` that wrap `false`, we just discard them. We could if we wanted instead return `unit_set (unit_dpm false)`.
 A later step can then filter out all the possibilities in which the entity `d` we did that with doesn't have property P.
+ A later step can then filter out all the `dpm`s according to which the
+entity `d` we did that with doesn't have property P.
 So if we just call the function `(fun (r, h) > ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
+ So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
 s >>= \[[∃x]] >>= \[[Px]]
+ u >>= \[[∃x]] >>= \[[Px]]
or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
 bind_set (bind_set s \[[∃x]]) \[[Px]]
+ bind_set (bind_set u \[[∃x]]) \[[Px]]
* In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:

 > { i &elem; s  i does not subsist in s[φ] }

 where `i` *subsists* in s[φ]
if there are any `i'` that *extend* `i` in s[φ]
.

 Here's how we can represent that:

 bind_set s (fun (r, h) >
 let u = unit_set (r, h)
 in let descendents = u >>= \[[φ]]
 in if descendents = empty_set then u else empty_set



+* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).