---
layout: post
title: "Lets Talk About Sets"
date: 2013-01-05 16:12
comments: true
external-url:
categories: basic, measures, sets
author: Ranjit Jhala
published: false
---
In the posts so far, we've seen how LiquidHaskell allows you to use SMT
solvers to specify and verify *numeric* invariants -- denominators
that are non-zero, integer indices that are within the range of an
array, vectors that have the right number of dimensions and so on.
However, SMT solvers are not limited to numbers, and in fact, support
rather expressive logics. In the next couple of posts, lets see how
LiquidHaskell uses SMT to talk about **sets of values**, for example,
the contents of a list, and to specify and verify properties about
those sets.
24: module ListSets where
Talking about Sets (In Logic)
=============================
First, we need a way to talk about sets in the refinement logic. We could
roll our own special Haskell type, but why not just use the `Set a` type
from `Data.Set`.
35: import Data.Set hiding (filter, split)
36: import Prelude hiding (reverse, filter)
The above, also instructs LiquidHaskell to import in the various
specifications defined for the `Data.Set` module that we have *predefined*
in [Data/Set.spec][setspec]
Lets look at the specifications.
The `embed` directive tells LiquidHaskell to model the **Haskell**
46: type constructor `Set` with the **SMT** type constructor `Set_Set`.
47:
48: module spec Data.Set where
49:
50: embed Set as Set_Set
First, we define the logical operators (i.e. `measure`s)
54: measure Set_sng :: a -> (Set a)
55: measure Set_cup :: (Set a) -> (Set a) -> (Set a)
56: measure Set_cap :: (Set a) -> (Set a) -> (Set a)
57: measure Set_dif :: (Set a) -> (Set a) -> (Set a)
Next, we define predicates on `Set`s
61: measure Set_emp :: (Set a) -> Prop
62: measure Set_mem :: a -> (Set a) -> Prop
63: measure Set_sub :: (Set a) -> (Set a) -> Prop
Interpreted Operations
----------------------
The above operators are *interpreted* by the SMT solver. That is, just
like the SMT solver *"knows that"*
2 + 2 == 4
the SMT solver *"knows that"*
(Set_sng 1) == (Set_cap (Set_sng 1) (Set_cup (Set_sng 2) (Set_sng 1)))
This is because, the above formulas belong to a decidable Theory of Sets
which can be reduced to McCarthy's more general [Theory of Arrays][mccarthy].
See [this recent paper][z3cal] if you want to learn more about how modern SMT
solvers *"know"* the above equality holds...
Talking about Sets (In Code)
============================
Of course, the above operators exist purely in the realm of the
refinement logic, beyond the grasp of the programmer.
To bring them down (or up, or left or right) within reach of Haskell code,
we can simply *assume* that the various public functions in `Data.Set` do
the *Right Thing* i.e. produce values that reflect the logical set operations:
First, the functions that create `Set` values
95: empty :: {v:(Set a) | (Set_emp v)}
96: singleton :: x:a -> {v:(Set a) | v = (Set_sng x)}
Next, the functions that operate on elements and `Set` values
100: insert :: Ord a => x:a -> xs:(Set a) -> {v:(Set a) | v = (Set_cup xs (Set_sng x))}
101: delete :: Ord a => x:a -> xs:(Set a) -> {v:(Set a) | v = (Set_dif xs (Set_sng x))}
Then, the binary `Set` operators
105: union :: Ord a => xs:(Set a) -> ys:(Set a) -> {v:(Set a) | v = (Set_cup xs ys)}
106: intersection :: Ord a => xs:(Set a) -> ys:(Set a) -> {v:(Set a) | v = (Set_cap xs ys)}
107: difference :: Ord a => xs:(Set a) -> ys:(Set a) -> {v:(Set a) | v = (Set_dif xs ys)}
And finally, the predicates on `Set` values:
111: isSubsetOf :: Ord a => xs:(Set a) -> ys:(Set a) -> {v:Bool | (Prop v) <=> (Set_sub xs ys)}
112: member :: Ord a => x:a -> xs:(Set a) -> {v:Bool | (Prop v) <=> (Set_mem x xs)}
**Note:** Oh quite. We shouldn't and needn't really *assume*, but should and
will *guarantee* that the functions from `Data.Set` implement the above types.
But thats a story for another day...
Proving Theorems With LiquidHaskell
===================================
OK, lets take our refined operators from `Data.Set` out for a spin!
One pleasant consequence of being able to precisely type the operators
from `Data.Set` is that we have a pleasant interface for using the SMT
solver to *prove theorems* about sets, while remaining firmly rooted in
Haskell.
First, lets write a simple function that asserts that its input is `True`
131: {-@ boolAssert :: {v: Bool | (Prop v)} -> {v:Bool | (Prop v)} @-}
132: {VV : (GHC.Types.Bool) | (? Prop([VV]))}
-> {VV : (GHC.Types.Bool) | (? Prop([VV]))}boolAssert True = {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV = True)}True
133: boolAssert False = [(GHC.Types.Char)] -> {VV : (GHC.Types.Bool) | false}error {VV : [(GHC.Types.Char)] | (len([VV]) >= 0)}"boolAssert: False? Never!"
Now, we can use `boolAssert` to write some simple properties. (Yes, these do
indeed look like QuickCheck properties -- but here, instead of generating
tests and executing the code, the type system is proving to us that the
properties will *always* evaluate to `True`)
Lets check that `intersection` is commutative ...
144: {-@ prop_cap_comm :: Set Int -> Set Int -> Bool @-}
145: prop_cap_comm :: Set Int -> Set Int -> Bool
146: (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int)) -> (GHC.Types.Bool)prop_cap_comm (Data.Set.Base.Set (GHC.Types.Int))x (Data.Set.Base.Set (GHC.Types.Int))y
147: = {VV : (GHC.Types.Bool) | (? Prop([VV]))}
-> {VV : (GHC.Types.Bool) | (? Prop([VV]))}boolAssert
148: ({VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)})
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}$ (GHC.Types.Bool)({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cap([xs;
ys]))}`intersection` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y) GHC.Classes.Eq (Data.Set.Base.Set GHC.Types.Int)== ({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cap([xs;
ys]))}`intersection` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x)
that `union` is associative ...
154: {-@ prop_cup_assoc :: Set Int -> Set Int -> Set Int -> Bool @-}
155: prop_cup_assoc :: Set Int -> Set Int -> Set Int -> Bool
156: (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int))
-> (GHC.Types.Bool)prop_cup_assoc (Data.Set.Base.Set (GHC.Types.Int))x (Data.Set.Base.Set (GHC.Types.Int))y (Data.Set.Base.Set (GHC.Types.Int))z
157: = {VV : (GHC.Types.Bool) | (? Prop([VV]))}
-> {VV : (GHC.Types.Bool) | (? Prop([VV]))}boolAssert
158: ({VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)})
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}$ (GHC.Types.Bool)({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` ({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = z)}z)) x:(Data.Set.Base.Set (GHC.Types.Int))
-> y:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (GHC.Types.Bool) | ((? Prop([VV])) <=> (x = y))}== (Data.Set.Base.Set (GHC.Types.Int))({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y) xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = z)}z
and that `union` distributes over `intersection`.
164: {-@ prop_cap_dist :: Set Int -> Set Int -> Set Int -> Bool @-}
165: prop_cap_dist :: Set Int -> Set Int -> Set Int -> Bool
166: (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int))
-> (GHC.Types.Bool)prop_cap_dist (Data.Set.Base.Set (GHC.Types.Int))x (Data.Set.Base.Set (GHC.Types.Int))y (Data.Set.Base.Set (GHC.Types.Int))z
167: = {VV : (GHC.Types.Bool) | (? Prop([VV]))}
-> {VV : (GHC.Types.Bool) | (? Prop([VV]))}boolAssert
168: ({VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)})
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}$ ({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cap([xs;
ys]))}`intersection` ({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = z)}z))
169: x:(Data.Set.Base.Set (GHC.Types.Int))
-> y:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (GHC.Types.Bool) | ((? Prop([VV])) <=> (x = y))}== (Data.Set.Base.Set (GHC.Types.Int))({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cap([xs;
ys]))}`intersection` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y) xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` ({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cap([xs;
ys]))}`intersection` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = z)}z)
Of course, while we're at it, lets make sure LiquidHaskell
doesn't prove anything thats *not* true ...
176: {-@ prop_cup_dif_bad :: Set Int -> Set Int -> Bool @-}
177: prop_cup_dif_bad :: Set Int -> Set Int -> Bool
178: (Data.Set.Base.Set (GHC.Types.Int))
-> (Data.Set.Base.Set (GHC.Types.Int)) -> (GHC.Types.Bool)prop_cup_dif_bad (Data.Set.Base.Set (GHC.Types.Int))x (Data.Set.Base.Set (GHC.Types.Int))y
179: = {VV : (GHC.Types.Bool) | (? Prop([VV]))}
-> {VV : (GHC.Types.Bool) | (? Prop([VV]))}boolAssert
180: ((GHC.Types.Bool)
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)})
-> (GHC.Types.Bool)
-> {VV : (GHC.Types.Bool) | (? Prop([VV])),(VV != False)}$ {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x x:(Data.Set.Base.Set (GHC.Types.Int))
-> y:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (GHC.Types.Bool) | ((? Prop([VV])) <=> (x = y))}== (Data.Set.Base.Set (GHC.Types.Int))({VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = x)}x xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_cup([xs;
ys]))}`union` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y) xs:(Data.Set.Base.Set (GHC.Types.Int))
-> ys:(Data.Set.Base.Set (GHC.Types.Int))
-> {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = Set_dif([xs;
ys]))}`difference` {VV : (Data.Set.Base.Set (GHC.Types.Int)) | (VV = y)}y
Hmm. You do know why the above doesn't hold, right? It would be nice to
get a *counterexample* wouldn't it. Well, for the moment, there is
QuickCheck!
Thus, the refined types offer a nice interface for interacting with the SMT
solver in order to prove theorems in LiquidHaskell. (BTW, The [SBV project][sbv]
describes another approach for using SMT solvers from Haskell, without the
indirection of refinement types.)
While the above is a nice warm up exercise to understanding how
LiquidHaskell reasons about sets, our overall goal is not to prove
theorems about set operators, but instead to specify and verify
properties of programs.
The Set of Values in a List
===========================
Lets see how we might reason about sets of values in regular Haskell programs.
Lets begin with Lists, the jack-of-all-data-types. Now, instead of just
talking about the **number of** elements in a list, we can write a measure
that describes the **set of** elements in a list:
A measure for the elements of a list, from [Data/Set.spec][setspec]
208:
209: measure listElts :: [a] -> (Set a)
210: listElts ([]) = {v | (? Set_emp(v))}
211: listElts (x:xs) = {v | v = (Set_cup (Set_sng x) (listElts xs)) }
That is, `(elts xs)` describes the set of elements contained in a list `xs`.
Next, to make the specifications concise, lets define a few predicate aliases:
219: {-@ predicate SameElts X Y = ((listElts X) = (listElts Y)) @-}
220: {-@ predicate SubElts X Y = (Set_sub (listElts X) (listElts Y)) @-}
221: {-@ predicate UnionElts X Y Z = ((listElts X) = (Set_cup (listElts Y) (listElts Z))) @-}
A Trivial Identity
------------------
OK, now lets write some code to check that the `elts` measure is sensible!
230: {-@ listId :: xs:[a] -> {v:[a]| (SameElts v xs)} @-}
231: forall a.
x1:[a]
-> {VV : [a] | (len([VV]) = len([x1])),
(listElts([VV]) = Set_cup([listElts([x1]); listElts([x1])])),
(listElts([VV]) = listElts([x1])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])])),
(len([VV]) >= 0)}listId [] = forall <p :: a -> a -> Bool>.
{VV : [{VV : a | false}]<p> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0)}[]
232: listId (x:xs) = {VV : a | (VV = x)}x forall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}: forall a.
x1:[a]
-> {VV : [a] | (len([VV]) = len([x1])),
(listElts([VV]) = Set_cup([listElts([x1]); listElts([x1])])),
(listElts([VV]) = listElts([x1])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])])),
(len([VV]) >= 0)}listId {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs
That is, LiquidHaskell checks that the set of elements of the output list
is the same as those in the input.
A Less Trivial Identity
-----------------------
Next, lets write a function to `reverse` a list. Of course, we'd like to
verify that `reverse` doesn't leave any elements behind; that is that the
output has the same set of values as the input list. This is somewhat more
interesting because of the *tail recursive* helper `go`. Do you understand
the type that is inferred for it? (Put your mouse over `go` to see the
inferred type.)
249: {-@ reverse :: xs:[a] -> {v:[a]| (SameElts v xs)} @-}
250: forall a. xs:[a] -> {VV : [a] | (listElts([VV]) = listElts([xs]))}reverse = x1:{VV : [{VV : a | false}]<\_ VV -> false> | (len([VV]) = 0)}
-> x2:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([x2])])),
(listElts([VV]) = Set_cup([listElts([x2]); listElts([x1])])),
(len([VV]) >= 0)}go {VV : [{VV : a | false}]<\_ VV -> false> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0),
(len([VV]) >= 0)}[]
251: where
252: acc:{VV : [a] | (len([VV]) >= 0)}
-> x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([acc]);
listElts([x1])])),
(listElts([VV]) = Set_cup([listElts([x1]); listElts([acc])])),
(len([VV]) >= 0)}go {VV : [a] | (len([VV]) >= 0)}acc [] = {VV : [a] | (VV = acc),(len([VV]) >= 0)}acc
253: go acc (y:ys) = acc:{VV : [a] | (len([VV]) >= 0)}
-> x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([acc]);
listElts([x1])])),
(listElts([VV]) = Set_cup([listElts([x1]); listElts([acc])])),
(len([VV]) >= 0)}go ({VV : a | (VV = y)}yforall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}:{VV : [a] | (VV = acc),(len([VV]) >= 0)}acc) {VV : [a] | (VV = ys),(len([VV]) >= 0)}ys
Appending Lists
---------------
Next, here's good old `append`, but now with a specification that states
that the output indeed includes the elements from both the input lists.
263: {-@ append :: xs:[a] -> ys:[a] -> {v:[a] | (UnionElts v xs ys) } @-}
264: forall a.
x1:[a]
-> ys:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([ys])])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([x1])])),
(len([VV]) >= 0)}append [] [a]ys = {VV : [a] | (VV = ys),(len([VV]) >= 0)}ys
265: append (x:xs) ys = {VV : a | (VV = x)}x forall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}: forall a.
x1:[a]
-> ys:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([ys])])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([x1])])),
(len([VV]) >= 0)}append {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs {VV : [a] | (VV = ys),(len([VV]) >= 0)}ys
Filtering Lists
---------------
Lets round off the list trilogy, with `filter`. Here, we can verify
that it returns a **subset of** the values of the input list.
275: {-@ filter :: (a -> Bool) -> xs:[a] -> {v:[a]| (SubElts v xs)} @-}
276:
277: forall a.
(a -> (GHC.Types.Bool))
-> x2:[a]
-> {VV : [a] | (? Set_sub([listElts([VV]); listElts([x2])])),
(listElts([x2]) = Set_cup([listElts([x2]); listElts([VV])])),
(len([VV]) >= 0)}filter a -> (GHC.Types.Bool)f [] = forall <p :: a -> a -> Bool>.
{VV : [{VV : a | false}]<p> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0)}[]
278: filter f (x:xs)
279: | a -> (GHC.Types.Bool)f {VV : a | (VV = x)}x = {VV : a | (VV = x)}x forall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}: forall a.
(a -> (GHC.Types.Bool))
-> x2:[a]
-> {VV : [a] | (? Set_sub([listElts([VV]); listElts([x2])])),
(listElts([x2]) = Set_cup([listElts([x2]); listElts([VV])])),
(len([VV]) >= 0)}filter a -> (GHC.Types.Bool)f {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs
280: | otherwise = forall a.
(a -> (GHC.Types.Bool))
-> x2:[a]
-> {VV : [a] | (? Set_sub([listElts([VV]); listElts([x2])])),
(listElts([x2]) = Set_cup([listElts([x2]); listElts([VV])])),
(len([VV]) >= 0)}filter a -> (GHC.Types.Bool)f {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs
Merge Sort
==========
Lets conclude this entry with one larger example: `mergeSort`.
We'd like to verify that, well, the list that is returned
contains the same set of elements as the input list.
And so we will!
But first, lets remind ourselves of how `mergeSort` works
1. `split` the input list into two halves,
2. `sort` each half, recursively,
3. `merge` the sorted halves to obtain the sorted list.
Split
-----
We can `split` a list into two, roughly equal parts like so:
305: forall a.
x1:[a]
-> ({VV : [a] | (? Set_sub([listElts([VV]); listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])])),
(len([VV]) >= 0)} , {VV : [a] | (? Set_sub([listElts([VV]);
listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(len([VV]) >= 0)})<\x1 VV -> (? Set_sub([listElts([VV]);
listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(len([VV]) >= 0)>split [] = forall a b <p2 :: a -> b -> Bool>. x1:a -> b<p2 x1> -> (a , b)<p2>({VV : [{VV : a | false}]<\_ VV -> false> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0),
(len([VV]) >= 0)}[], {VV : [{VV : a | false}]<\_ VV -> false> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0),
(len([VV]) >= 0)}[])
306: split (x:xs) = forall a b <p2 :: a -> b -> Bool>. x1:a -> b<p2 x1> -> (a , b)<p2>({VV : a | (VV = x)}xforall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}:{VV : [a] | (? Set_sub([listElts([VV]); listElts([xs])])),
(VV = zs),
(VV = zs),
(len([VV]) = len([zs])),
(listElts([VV]) = Set_cup([listElts([zs]); listElts([zs])])),
(listElts([VV]) = listElts([zs])),
(listElts([xs]) = Set_cup([listElts([xs]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([ys]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([ys]); listElts([VV])])),
(listElts([zs]) = Set_cup([listElts([zs]); listElts([VV])])),
(len([VV]) >= 0)}zs, {VV : [a] | (? Set_sub([listElts([VV]); listElts([xs])])),
(VV = ys),
(VV = ys),
(len([VV]) = len([ys])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([ys])])),
(listElts([VV]) = listElts([ys])),
(listElts([xs]) = Set_cup([listElts([xs]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([zs]); listElts([VV])])),
(listElts([ys]) = Set_cup([listElts([ys]); listElts([VV])])),
(len([VV]) >= 0)}ys)
307: where
308: ({VV : [a] | (? Set_sub([listElts([VV]); listElts([xs])])),
(VV = ys),
(len([VV]) = len([ys])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([ys])])),
(listElts([VV]) = listElts([ys])),
(listElts([xs]) = Set_cup([listElts([xs]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([zs]); listElts([VV])])),
(listElts([ys]) = Set_cup([listElts([ys]); listElts([VV])])),
(len([VV]) >= 0)}ys, {VV : [a] | (? Set_sub([listElts([VV]); listElts([xs])])),
(VV = zs),
(len([VV]) = len([zs])),
(listElts([VV]) = Set_cup([listElts([zs]); listElts([zs])])),
(listElts([VV]) = listElts([zs])),
(listElts([xs]) = Set_cup([listElts([xs]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([ys]); listElts([VV])])),
(listElts([xs]) = Set_cup([listElts([ys]); listElts([VV])])),
(listElts([zs]) = Set_cup([listElts([zs]); listElts([VV])])),
(len([VV]) >= 0)}zs) = forall a.
x1:[a]
-> ({VV : [a] | (? Set_sub([listElts([VV]); listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])])),
(len([VV]) >= 0)} , {VV : [a] | (? Set_sub([listElts([VV]);
listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(len([VV]) >= 0)})<\x1 VV -> (? Set_sub([listElts([VV]);
listElts([x1])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(listElts([x1]) = Set_cup([listElts([x1]);
listElts([VV])])),
(len([VV]) >= 0)>split {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs
LiquidHaskell verifies that the relevant property of `split` is
314: {-@{\ys zs -> (UnionElts xs ys zs)}> @-}
The funny syntax with angle brackets simply says that the output is a
a *pair* `(ys, zs)` whose union is the list of elements of the input.
(**Aside** yes, this is indeed a dependent tuple; we will revisit tuples
later to understand whats going on with the odd syntax.)
Merge
-----
Dually, we `merge` two (sorted) lists like so:
329: forall a.
(GHC.Classes.Ord a) =>
xs:[a]
-> x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([xs])])),
(listElts([VV]) = Set_cup([listElts([xs]); listElts([x1])])),
(len([VV]) >= 0)}merge [a]xs [] = {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs
330: merge [] ys = {VV : [a] | (len([VV]) >= 0)}ys
331: merge (x:xs) (yozz:ys)
332: | {VV : a | (VV = x)}x x:a
-> y:a -> {VV : (GHC.Types.Bool) | ((? Prop([VV])) <=> (x <= y))}<= {VV : a | (VV = yozz)}yozz = {VV : a | (VV = x)}x forall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}: forall a.
(GHC.Classes.Ord a) =>
xs:[a]
-> x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([xs])])),
(listElts([VV]) = Set_cup([listElts([xs]); listElts([x1])])),
(len([VV]) >= 0)}merge {VV : [a] | (VV = xs),(len([VV]) >= 0)}xs ({VV : a | (VV = yozz)}yozzforall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}:{VV : [a] | (VV = ys),(len([VV]) >= 0)}ys)
333: | otherwise = {VV : a | (VV = yozz)}yozz forall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}: forall a.
(GHC.Classes.Ord a) =>
xs:[a]
-> x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([xs])])),
(listElts([VV]) = Set_cup([listElts([xs]); listElts([x1])])),
(len([VV]) >= 0)}merge ({VV : a | (VV = x)}xforall <p :: a -> a -> Bool>.
y:a
-> ys:[a<p y>]<p>
-> {VV : [a]<p> | (len([VV]) = (1 + len([ys]))),
(listElts([VV]) = Set_cup([Set_sng([y]); listElts([ys])]))}:{VV : [a] | (VV = xs),(len([VV]) >= 0)}xs) {VV : [a] | (VV = ys),(len([VV]) >= 0)}ys
As you might expect, the elements of the returned list are the union of the
elements of the input, or as LiquidHaskell might say,
340: {-@ merge :: (Ord a) => x:[a] -> y:[a] -> {v:[a] | (UnionElts v x y)} @-}
Sort
----
Finally, we put all the pieces together by
349: {-@ mergeSort :: (Ord a) => xs:[a] -> {v:[a] | (SameElts v xs)} @-}
350: forall a.
(GHC.Classes.Ord a) =>
x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([x1])])),
(listElts([VV]) = listElts([x1])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])]))}mergeSort [] = forall <p :: a -> a -> Bool>.
{VV : [{VV : a | false}]<p> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0)}[]
351: mergeSort [x] = {VV : [{VV : a | false}]<\_ VV -> false> | (? Set_emp([listElts([VV])])),
(len([VV]) = 0),
(len([VV]) >= 0)}[{VV : a | (VV = x)}x]
352: mergeSort xs = x:[a]
-> y:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x]);
listElts([y])]))}merge (forall a.
(GHC.Classes.Ord a) =>
x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([x1])])),
(listElts([VV]) = listElts([x1])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])]))}mergeSort {VV : [a] | (VV = ys),
(VV = ys),
(len([VV]) = len([ys])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([ys])])),
(listElts([VV]) = listElts([ys])),
(listElts([ys]) = Set_cup([listElts([ys]); listElts([VV])])),
(len([VV]) >= 0)}ys) (forall a.
(GHC.Classes.Ord a) =>
x1:[a]
-> {VV : [a] | (listElts([VV]) = Set_cup([listElts([x1]);
listElts([x1])])),
(listElts([VV]) = listElts([x1])),
(listElts([x1]) = Set_cup([listElts([x1]); listElts([VV])]))}mergeSort {VV : [a] | (VV = zs),
(VV = zs),
(len([VV]) = len([zs])),
(listElts([VV]) = Set_cup([listElts([zs]); listElts([zs])])),
(listElts([VV]) = listElts([zs])),
(listElts([zs]) = Set_cup([listElts([zs]); listElts([VV])])),
(len([VV]) >= 0)}zs)
353: where
354: ({VV : [a] | (VV = ys),
(len([VV]) = len([ys])),
(listElts([VV]) = Set_cup([listElts([ys]); listElts([ys])])),
(listElts([VV]) = listElts([ys])),
(listElts([ys]) = Set_cup([listElts([ys]); listElts([VV])])),
(len([VV]) >= 0)}ys, {VV : [a] | (VV = zs),
(len([VV]) = len([zs])),
(listElts([VV]) = Set_cup([listElts([zs]); listElts([zs])])),
(listElts([VV]) = listElts([zs])),
(listElts([zs]) = Set_cup([listElts([zs]); listElts([VV])])),
(len([VV]) >= 0)}zs) = xs:[a]
-> ([a] , [a])<\ys VV -> (listElts([xs]) = Set_cup([listElts([ys]);
listElts([VV])]))>split {VV : [a] | (len([VV]) >= 0)}xs
The type given to `mergeSort`guarantees that the set of elements in the
output list is indeed the same as in the input list. Of course, it says
nothing about whether the list is *actually sorted*.
Well, Rome wasn't built in a day...
[sbv]: https://github.com/LeventErkok/sbv
[setspec]: https://github.com/ucsd-progsys/liquidhaskell/blob/master/include/Data/Set.spec
[mccarthy]: http://www-formal.stanford.edu/jmc/towards.ps
[z3cal]: http://research.microsoft.com/en-us/um/people/leonardo/fmcad09.pdf