Following up on PR 1296, implement both APIs for search_sorted.

Context: #1296 (comment)

The more basic API, still named `search_sorted`, returns a `Post n`.
The idea is that the elements of `xs` are fence sections, and we find
the position between them (inclusive on either end) where `x` falls in
the ordering.

In terms of this, we now define `search_sorted_exact` (better name?),
which returns a `Maybe n`, which is the index of an element of `xs`
that equals `x` exactly, or `Nothing` if such does not exist.

Also reorder the prelude slightly to try to both maintain semantic
groupings and respect name resolution dependencies.

Author: axch

lib/prelude.dx

@@ -591,34 +591,6 @@ def i_to_n(x:Int) -> Maybe Nat =
     then Nothing
     else Just $ unsafe_i_to_n x
 
-'## Fencepost index sets
-
-struct Post(segment:Type) =
-  val : Nat
-
-instance Ix(Post segment) given (segment|Ix)
-  def size'() = size segment + 1
-  def ordinal(i) = i.val
-  def unsafe_from_ordinal(i) = Post(i)
-
-def left_post(i:n) -> Post n given (n|Ix) =
-  unsafe_from_ordinal(n=Post n, ordinal i)
-
-def right_post(i:n) -> Post n given (n|Ix) =
-  unsafe_from_ordinal(n=Post n, ordinal i + 1)
-
-interface NonEmpty(n|Ix)
-  first_ix : n
-
-def last_ix() ->> n given (n|NonEmpty) =
-  unsafe_from_ordinal(unsafe_i_to_n(n_to_i(size n) - 1))
-
-instance NonEmpty(Post n) given (n|Ix)
-  first_ix = unsafe_from_ordinal(n=Post n, 0)
-
-instance NonEmpty(())
-  first_ix = unsafe_from_ordinal(0)
-
 '### Monoid
 A [monoid](https://en.wikipedia.org/wiki/Monoid) is a things that have an associative binary operator and an identity element.
 This is a very useful and general calls of things.
@@ -901,6 +873,12 @@ instance Ix(Maybe a) given (a|Ix)
       False -> Just $ unsafe_from_ordinal o
       True  -> Nothing
 
+interface NonEmpty(n|Ix)
+  first_ix : n
+
+instance NonEmpty(())
+  first_ix = unsafe_from_ordinal(0)
+
 instance NonEmpty(Bool)
   first_ix = unsafe_from_ordinal 0
 
@@ -918,6 +896,40 @@ instance NonEmpty(Either(a,b)) given (a|NonEmpty, b|Ix)
 instance NonEmpty(Maybe a) given (a|Ix)
   first_ix = unsafe_from_ordinal 0
 
+'## Fencepost index sets
+
+struct Post(segment:Type) =
+  val : Nat
+
+instance Ix(Post segment) given (segment|Ix)
+  def size'() = size segment + 1
+  def ordinal(i) = i.val
+  def unsafe_from_ordinal(i) = Post(i)
+
+def left_post(i:n) -> Post n given (n|Ix) =
+  unsafe_from_ordinal(n=Post n, ordinal i)
+
+def right_post(i:n) -> Post n given (n|Ix) =
+  unsafe_from_ordinal(n=Post n, ordinal i + 1)
+
+def left_fence(p:Post n) -> Maybe n given (n|Ix) =
+  ix = ordinal p
+  if ix == 0
+    then Nothing
+    else Just $ unsafe_from_ordinal $ ix -| 1
+
+def right_fence(p:Post n) -> Maybe n given (n|Ix) =
+  ix = ordinal p
+  if ix == size n
+    then Nothing
+    else Just $ unsafe_from_ordinal ix
+
+def last_ix() ->> n given (n|NonEmpty) =
+  unsafe_from_ordinal(unsafe_i_to_n(n_to_i(size n) - 1))
+
+instance NonEmpty(Post n) given (n|Ix)
+  first_ix = unsafe_from_ordinal(n=Post n, 0)
+
 def scan(
     init:a,
     body:(n, a)->(a,b)
@@ -2016,27 +2028,45 @@ instance Arbitrary(Fin n) given (n)
 
 '### Searching
 
-'returns the highest index `i` such that `xs.i <= x`
+'Returns the bucket of `x` assuming boundaries `xs` as a `Post n`.
+The boundaries must already be sorted, and are inclusive on the left.
+
+'In other words, if there is an index `i` such that `xs.i <= x`,
+returns the `right_post` of the highest such index; otherwise returns
+`first_ix : Post n`, which is also the `left_post` of the minimum `i`.
 
-def search_sorted(xs:n=>a, x:a) -> Maybe n given (n|Ix, a|Ord) =
+'This is equivalent to the right-biased formulation: if an index `i`
+exists such that `x < xs.i`, returns the `left_post` of the least such
+`i`, otherwise returns `last_ix : Post n`, i.e., the `right_post` of
+the maximum `i`.
+
+def search_sorted(xs:n=>a, x:a) -> Post n given (n|Ix, a|Ord) =
   if size n == 0
-    then Nothing
+    then first_ix
     else if x < xs[from_ordinal 0]
-      then Nothing
+      then first_ix
       else
         low  <- with_state(0)
         high <- with_state(size n)
         _ <- iter
         numLeft = n_to_i (get high) - n_to_i (get low)
         if numLeft == 1
-          then Done $ Just $ from_ordinal $ get low
+          then Done $ right_post $ from_ordinal $ get low
           else
             centerIx = get low + unsafe_i_to_n (numLeft `idiv` 2)
             if x < xs[from_ordinal centerIx]
               then high := centerIx
               else low  := centerIx
             Continue
 
+'If `i` exists such that `xs.i == x`, returns `Just` of the largest
+such `i`, otherwise returns `Nothing`.
+
+def search_sorted_exact(xs:n=>a, x:a) -> Maybe n given (n|Ix, a|Ord) =
+  case left_fence(search_sorted(xs, x)) of
+    Just i -> if xs[i] == x then Just i else Nothing
+    Nothing -> Nothing
+
 '### min / max etc
 
 def min_by(f:(a)->o, x:a, y:a) -> a given (o|Ord, a) = select(f x < f y, x, y)
@@ -2318,8 +2348,7 @@ def lines(source:String) -> List String =
 -- cdf should include 0.0 but not 1.0
 def categorical_from_cdf(cdf: n=>Float, key: Key) -> n given (n|Ix) =
   r = rand key
-  case search_sorted(cdf, r) of
-    Just(i) -> i
+  from_just $ left_fence $ search_sorted(cdf, r)
 
 def normalize_pdf(xs: d=>Float) -> d=>Float given (d|Ix) = xs / sum xs
 

lib/set.dx

@@ -82,7 +82,7 @@ def set_intersect(
   UnsafeAsSet(nx, xs) = sx
   UnsafeAsSet(ny, ys) = sy
   -- This could be done in O(nx + ny) instead of O(nx log ny).
-  isInYs = \x. case search_sorted ys x of
+  isInYs = \x. case search_sorted_exact ys x of
     Just x -> True
     Nothing -> False
   AsList(n', intersection) = filter xs isInYs
@@ -100,7 +100,7 @@ struct Element(set:(Set a)) given (a|Ord) =
 -- type), but maybe it's easier to read if it's explicit.
 def member(x:a, set:(Set a)) -> Maybe (Element set) given (a|Ord) =
   UnsafeAsSet(_, elts) = set
-  case search_sorted elts x of
+  case search_sorted_exact elts x of
     Just n -> Just $ Element(ordinal n)
     Nothing -> Nothing
 

lib/stats.dx

@@ -166,7 +166,7 @@ instance OrderedDist(Binomial, Nat, Float)
     lpdf = for i:(Fin tp1). ln $ density d (ordinal i)
     cdf = cdf_for_categorical lpdf
     mi = search_sorted cdf q
-    ordinal $ from_just mi
+    ordinal $ from_just $ left_fence mi
 
 
 '### Exponential distribution

tests/sort-tests.dx

@@ -5,6 +5,12 @@ import sort
 :p is_sorted $ sort [9, 3, 7, 4, 6, 1, 9, 1, 9, -1, 10, 10, 100, 0]
 > True
 
+:p
+  xs = [1,2,4]
+  for i:(Fin 6).
+    search_sorted_exact(xs, ordinal i)
+> [Nothing, (Just 0), (Just 1), Nothing, (Just 2), Nothing]
+
 '### Lexical Sorting Tests
 
 :p "aaa" < "bbb"