then let obj = List.nth h (r 'x') in q obj
else false
in (truth_value', r, h)
- in bind_set u (fun one_dpm -> unit_set (bind_dpm one_dpm eliminator))
+ in set_bind u (fun one_dpm -> set_unit (dpm_bind one_dpm eliminator))
- The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `bind_set` operation. This works by taking each `dpm` in the set and returning a `unit_set` of a filtered `dpm`. The definition of `bind_set` takes care of collecting together all of the `unit_set`s that result for each different set element we started with.
+ The first seven lines here just perfom the operation we described: return a `bool dpm` computation that only yields `true` when its input `(r, h)` associates variable `x` with the right sort of entity. The last line performs the `set_bind` operation. This works by taking each `dpm` in the set and returning a `set_unit` of a filtered `dpm`. The definition of `set_bind` takes care of collecting together all of the `set_unit`s that result for each different set element we started with.
We can call the `(fun one_dpm -> ...)` part \[[Qx]] and then updating `u` with \[[Qx]] will be:
- bind_set u \[[Qx]]
+ set_bind u \[[Qx]]
or as it's written using Haskell's infix notation for bind:
* Now what do we do with predicates? As before, we suppose we have a function `q` that maps entities to `bool`s. We want to turn it into a function that maps `entity dpm`s to `bool dpm`s. Eventually we'll need to operate not just on single `dpm`s but on sets of them, but first things first. We'll begin by lifting `q` into a function that takes `entity dpm`s as arguments and returns `bool dpm`s:
- fun entity_dpm -> bind_dpm entity_dpm (fun e -> unit_dpm (q e))
+ fun entity_dpm -> dpm_bind entity_dpm (fun e -> dpm_unit (q e))
- Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `unit_set`:
+ Now we have to transform this into a function that again takes single `entity dpm`s as arguments, but now returns a `bool dpm set`. This is easily done with `set_unit`:
- fun entity_dpm -> unit_set (bind_dpm entity_dpm (fun e -> unit_dpm (q e)))
+ fun entity_dpm -> set_unit (dpm_bind entity_dpm (fun e -> dpm_unit (q e)))
Finally, we realize that we're going to have a set of `bool dpm`s to start with, and we need to monadically bind \[[Qx]] to them. We don't want any of the monadic values in the set that wrap `false` to become `true`; instead, we want to apply a filter that checks whether values that formerly wrapped `true` should still continue to do so.
let eliminator : bool -> bool dpm =
fun truth_value ->
if truth_value = false
- then unit_dpm false
- else bind_dpm entity_dpm (fun e -> unit_dpm (q e))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ then dpm_unit false
+ else dpm_bind entity_dpm (fun e -> dpm_unit (q e))
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
Applied to an `entity_dpm`, that yields a function that we can bind to a `bool dpm set` and that will transform the doubly-wrapped `bool` into a new `bool dpm set`.
in let entity_dpm = getx
in let eliminator = fun truth_value ->
if truth_value = false
- then unit_dpm false
- else bind_dpm entity_dpm (fun e -> unit_dpm (q e))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ then dpm_unit false
+ else dpm_bind entity_dpm (fun e -> dpm_unit (q e))
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
<!--
or, simplifying:
in (obj, r, h)
in let eliminator = fun truth_value ->
if truth_value
- then bind_dpm getx (fun e -> unit_dpm (q e))
- else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ then dpm_bind getx (fun e -> dpm_unit (q e))
+ else dpm_unit false
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
-->
- If we simplify and unpack the definition of `bind_dpm`, that's equivalent to:
+ If we simplify and unpack the definition of `dpm_bind`, that's equivalent to:
let getx = fun (r, h) ->
let obj = List.nth h (r 'x')
if truth_value
then (fun (r, h) ->
let (a, r', h') = getx (r, h)
- in let u' = (fun e -> unit_dpm (q e)) a
+ in let u' = (fun e -> dpm_unit (q e)) a
in u' (r', h')
- ) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ ) else dpm_unit false
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
which can be further simplified to:
then (fun (r, h) ->
let obj = List.nth h (r 'x')
let (a, r', h') = (obj, r, h)
- in let u' = (fun e -> unit_dpm (q e)) a
+ in let u' = (fun e -> dpm_unit (q e)) a
in u' (r', h')
- ) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ ) else dpm_unit false
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
let eliminator = fun truth_value ->
if truth_value
then (fun (r, h) ->
let obj = List.nth h (r 'x')
- in let u' = unit_dpm (q obj)
+ in let u' = dpm_unit (q obj)
in u' (r, h)
- ) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ ) else dpm_unit false
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
-->
let eliminator = fun truth_value ->
then (fun (r, h) ->
let obj = List.nth h (r 'x')
in (q obj, r, h)
- ) else unit_dpm false
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ ) else dpm_unit false
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
This is a function that takes a `bool dpm` as input and returns a `bool dpm set` as output.
then let obj = List.nth h (r 'x') in q obj
else false
in (truth_value', r, h))
- in fun one_dpm -> unit_set (bind_dpm one_dpm eliminator)
+ in fun one_dpm -> set_unit (dpm_bind one_dpm eliminator)
Can you persuade yourself that these are equivalent?)
* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We monadically bind this operaration to whatever `bool dpm set` we already have on hand:
- bind_set u \[[Qx]]
+ set_bind u \[[Qx]]
or: