Update FirstClassFunctions_template from new source material

FirstClassFunctions_template.v

@@ -1,21 +1,26 @@
-Require Import Frap.
+Require Import Frap Program.
 
 
 (** * Some data fodder for us to compute with later *)
 
+(* Records are a handy way to define datatypes in terms of the named fields that
+ * each value must contain. *)
 Record programming_language := {
   Name : string;
   PurelyFunctional : bool;
   AppearedInYear : nat
 }.
 
+(* Here's a quick example of a set of programming languages, which we will use
+ * below in some example computations. *)
+
 Definition pascal := {|
   Name := "Pascal";
   PurelyFunctional := false;
   AppearedInYear := 1970
 |}.
 
-Definition c := {|
+Definition clang := {|
   Name := "C";
   PurelyFunctional := false;
   AppearedInYear := 1972
@@ -39,11 +44,16 @@ Definition ocaml := {|
   AppearedInYear := 1996
 |}.
 
-Definition languages := [pascal; c; gallina; haskell; ocaml].
+Definition languages := [pascal; clang; gallina; haskell; ocaml].
 
 
 (** * Classic list functions *)
 
+(* The trio of "map/filter/reduce" are commonly presented as workhorse
+ * *higher-order functions* for lists.  That is, they are functions that take
+ * functions as arguments. *)
+
+(* [map] runs a function on every position of a list to make a new list. *)
 Fixpoint map {A B} (f : A -> B) (ls : list A) : list B :=
   match ls with
   | nil => nil
@@ -51,7 +61,9 @@ Fixpoint map {A B} (f : A -> B) (ls : list A) : list B :=
   end.
 
 Compute map (fun n => n + 2) [1; 3; 8].
+(* Note the use of an *anonymous function* above via [fun]. *)
 
+(* [filter] keeps only those elements of a list that pass a Boolean test. *)
 Fixpoint filter {A} (f : A -> bool) (ls : list A) : list A :=
   match ls with
   | nil => nil
@@ -59,7 +71,12 @@ Fixpoint filter {A} (f : A -> bool) (ls : list A) : list A :=
   end.
 
 Compute filter (fun n => if n <=? 3 then true else false) [1; 3; 8].
+(* The [if ... then true else false] bit might seem wasteful.  Actually, the
+ * [<=?] operator has a fancy type that needs converting to [bool].  We'll get
+ * more specific about such types in a future class. *)
 
+(* [fold_left], a relative of "reduce," repeatedly applies a function to all
+ * elements of a list. *)
 Fixpoint fold_left {A B} (f : B -> A -> B) (ls : list A) (acc : B) : B :=
   match ls with
   | nil => acc
@@ -68,6 +85,7 @@ Fixpoint fold_left {A B} (f : B -> A -> B) (ls : list A) (acc : B) : B :=
 
 Compute fold_left max [1; 3; 8] 0.
 
+(* Another way to see [fold_left] in action: *)
 Theorem fold_left3 : forall {A B} (f : B -> A -> B) (x y z : A) (acc : B),
     fold_left f [x; y; z] acc = f (f (f acc x) y) z.
 Proof.
@@ -75,13 +93,22 @@ Proof.
   equality.
 Qed.
 
+(* Let's use these classics to implement a few simple "database queries" on the
+ * list of programming languages.  Note that each field name from
+ * [programming_language] is itself a first-class function, for projecting that
+ * field from any record! *)
+
 Compute map Name languages.
+(* names of languages *)
 
 Compute map Name (filter PurelyFunctional languages).
+(* names of purely functional languages *)
 
 Compute fold_left max (map AppearedInYear languages) 0.
+(* maximum year in which a language appeared *)
 
 Compute fold_left max (map AppearedInYear (filter PurelyFunctional languages)) 0.
+(* maximum year in which a purely functional language appeared *)
 
 (* To avoid confusing things, we'll revert to the standard library's (identical)
  * versions of these functions for the remainder. *)
@@ -90,21 +117,8 @@ Reset map.
 
 (** * Sorting, parameterized in a comparison operation *)
 
-Fixpoint insert {A} (le : A -> A -> bool) (new : A) (ls : list A) : list A :=
-  match ls with
-  | [] => [new]
-  | x :: ls' =>
-    if le new x then
-      new :: ls
-    else
-      x :: insert le new ls'
-  end.
-
-Fixpoint insertion_sort {A} (le : A -> A -> bool) (ls : list A) : list A :=
-  match ls with
-  | [] => []
-  | x :: ls' => insert le x (insertion_sort le ls')
-  end.
+Fixpoint insertion_sort {A} (le : A -> A -> bool) (ls : list A) : list A.
+Admitted.
 
 Fixpoint sorted {A} (le : A -> A -> bool) (ls : list A) : bool :=
   match ls with
@@ -116,17 +130,22 @@ Fixpoint sorted {A} (le : A -> A -> bool) (ls : list A) : bool :=
     end
   end.
 
-Theorem insertion_sort_sorted : forall {A} (le : A -> A -> bool) ls,
-    sorted le (insertion_sort le ls) = true.
+(* Main theorem: [insertion_sort] produces only sorted lists. *)
+Theorem insertion_sort_sorted : forall {A} (le : A -> A -> bool),
+    forall ls,
+      sorted le (insertion_sort le ls) = true.
 Proof.
 Admitted.
 
+(* Let's do a quick example of using [insertion_sort] with a concrete
+ * comparator. *)
+
 Definition not_introduced_later (l1 l2 : programming_language) : bool :=
   if AppearedInYear l1 <=? AppearedInYear l2 then true else false.
 
 Compute insertion_sort
         not_introduced_later
-        [gallina; pascal; c; ocaml; haskell].
+        [gallina; pascal; clang; ocaml; haskell].
 
 Corollary insertion_sort_languages : forall langs,
     sorted not_introduced_later (insertion_sort not_introduced_later langs) = true.
@@ -134,6 +153,272 @@ Proof.
 Admitted.
 
 
+(** * A language of functions and its interpreter *)
+
+Inductive dyn :=
+| Bool (b : bool)
+| Number (n : nat)
+| List (ds : list dyn).
+
+Definition dmap (f : dyn -> dyn) (x : dyn) : dyn :=
+  match x with
+  | List ds => List (map f ds)
+  | _ => x
+  end.
+
+Definition dfilter (f : dyn -> dyn) (x : dyn) : dyn :=
+  match x with
+    List ds => List (filter (fun arg => match f arg with
+                                        | Bool b => b
+                                        | _ => false
+                                        end) ds)
+  | _ => x
+  end.
+
+Definition disZero (x : dyn) : dyn :=
+  match x with
+  | Number 0 => Bool true
+  | Number _ => Bool false
+  | _ => x
+  end.
+
+Definition dnot (x : dyn) : dyn :=
+  match x with
+  | Bool b => Bool (negb b)
+  | x => x
+  end.
+
+Inductive xform :=
+| Identity
+| Compose (xf1 xf2 : xform)
+| Map (xf1 : xform)
+| Filter (xf1 : xform)
+
+| ConstantBool (b : bool)
+| ConstantNumber (n : nat)
+| IsZero
+| Not.
+
+Fixpoint transform (xf : xform) : dyn -> dyn :=
+  match xf with
+  | Identity => id
+  | Compose f1 f2 => compose (transform f1) (transform f2)
+  | Map f => dmap (transform f)
+  | Filter f => dfilter (transform f)
+
+  | ConstantBool b => fun _ => Bool b
+  | ConstantNumber n => fun _ => Number n
+  | IsZero => disZero
+  | Not => dnot
+  end.
+
+Compute transform (Map IsZero) (List [Number 0; Number 1; Number 2; Number 0; Number 3]).
+Compute transform (Filter IsZero) (List [Number 0; Number 1; Number 2; Number 0; Number 3]).
+
+Fixpoint optimize (xf : xform) : xform.
+Admitted.
+
+Theorem optimize_ok : forall xf x, transform (optimize xf) x = transform xf x.
+Proof.
+Admitted.
+
+(** ** Are these really optimizations?  Can they ever grow a term's size? *)
+
+Fixpoint size (xf : xform) : nat :=
+  match xf with
+  | Identity
+  | Not
+  | IsZero
+  | ConstantBool _
+  | ConstantNumber _ => 1
+
+  | Compose xf1 xf2 => 1 + size xf1 + size xf2
+  | Map xf
+  | Filter xf => 1 + size xf
+  end.
+
+Theorem optimize_optimizes : forall xf, size (optimize xf) <= size xf.
+Proof.
+Admitted.
+
+(** ** More interestingly, the same is true of the action of these
+       transformations on concrete values! *)
+
+Fixpoint sum (ls : list nat) : nat :=
+  match ls with
+  | nil => 0
+  | x :: ls' => x + sum ls'
+  end.
+
+Fixpoint dsize (d : dyn) : nat :=
+  match d with
+  | Bool _
+  | Number _ => 1
+  | List ds => 1 + sum (map dsize ds)
+  end.
+
+Theorem neverGrow : forall xf x,
+    dsize (transform xf x) <= dsize x.
+Proof.
+Admitted.
+
+
+(** * Combinators for syntax-tree transformations *)
+
+(* Let's reprise the imperative language from the end of Interpreters. *)
+
+Inductive arith : Set :=
+| Const (n : nat)
+| Var (x : var)
+| Plus (e1 e2 : arith)
+| Minus (e1 e2 : arith)
+| Times (e1 e2 : arith).
+
+Definition valuation := fmap var nat.
+
+Fixpoint interp (e : arith) (v : valuation) : nat :=
+  match e with
+  | Const n => n
+  | Var x =>
+    match v $? x with
+    | None => 0
+    | Some n => n
+    end
+  | Plus e1 e2 => interp e1 v + interp e2 v
+  | Minus e1 e2 => interp e1 v - interp e2 v
+  | Times e1 e2 => interp e1 v * interp e2 v
+  end.
+
+Inductive cmd :=
+| Skip
+| Assign (x : var) (e : arith)
+| Sequence (c1 c2 : cmd)
+| Repeat (e : arith) (body : cmd).
+
+Fixpoint selfCompose {A} (f : A -> A) (n : nat) : A -> A :=
+  match n with
+  | O => fun x => x
+  | S n' => fun x => selfCompose f n' (f x)
+  end.
+
+Lemma selfCompose_extensional : forall {A} (f g : A -> A) n x,
+  (forall y, f y = g y)
+  -> selfCompose f n x = selfCompose g n x.
+Proof.
+  induct n; simplify; try equality.
+
+  rewrite H.
+  apply IHn.
+  trivial.
+Qed.
+
+Fixpoint exec (c : cmd) (v : valuation) : valuation :=
+  match c with
+  | Skip => v
+  | Assign x e => v $+ (x, interp e v)
+  | Sequence c1 c2 => exec c2 (exec c1 v)
+  | Repeat e body => selfCompose (exec body) (interp e v) v
+  end.
+
+Fixpoint seqself (c : cmd) (n : nat) : cmd :=
+  match n with
+  | O => Skip
+  | S n' => Sequence c (seqself c n')
+  end.
+
+(* Now consider a more abstract way of describing optimizations concisely. *)
+
+Record rule := {
+  OnCommand : cmd -> cmd;
+  OnExpression : arith -> arith
+}.
+
+Fixpoint bottomUp (r : rule) (c : cmd) : cmd :=
+  match c with
+  | Skip => r.(OnCommand) Skip
+  | Assign x e => r.(OnCommand) (Assign x (r.(OnExpression) e))
+  | Sequence c1 c2 => r.(OnCommand) (Sequence (bottomUp r c1) (bottomUp r c2))
+  | Repeat e body => r.(OnCommand) (Repeat (r.(OnExpression) e) (bottomUp r body))
+  end.
+
+Definition crule (f : cmd -> cmd) : rule :=
+  {| OnCommand := f; OnExpression := fun e => e |}.
+Definition erule (f : arith -> arith) : rule :=
+  {| OnCommand := fun c => c; OnExpression := f |}.
+Definition andThen (r1 r2 : rule) : rule.
+Admitted.
+
+Definition plus0 := erule (fun e =>
+                             match e with
+                             | Plus e' (Const 0) => e'
+                             | _ => e
+                             end).
+Definition unrollLoops := crule (fun c =>
+                                   match c with
+                                   | Repeat (Const n) body => seqself body n
+                                   | _ => c
+                                   end).
+
+Compute bottomUp plus0
+        (Sequence (Assign "x" (Plus (Var "x") (Const 0)))
+                  (Assign "y" (Var "x"))).
+
+Compute bottomUp unrollLoops
+        (Repeat (Plus (Const 2) (Const 0))
+                (Sequence (Assign "x" (Plus (Var "x") (Const 0)))
+                          (Assign "y" (Var "x")))).
+
+Compute bottomUp (andThen plus0 unrollLoops)
+        (Repeat (Plus (Const 2) (Const 0))
+                (Sequence (Assign "x" (Plus (Var "x") (Const 0)))
+                          (Assign "y" (Var "x")))).
+
+Definition correct (r : rule) : Prop.
+Admitted.
+
+Theorem crule_correct : forall f,
+    (forall c v, exec (f c) v = exec c v)
+    -> correct (crule f).
+Proof.
+Admitted.
+
+Theorem erule_correct : forall f,
+    (forall e v, interp (f e) v = interp e v)
+    -> correct (erule f).
+Proof.
+Admitted.
+
+Theorem andThen_correct : forall r1 r2,
+    correct r1
+    -> correct r2
+    -> correct (andThen r1 r2).
+Proof.
+Admitted.
+
+Theorem bottomUp_correct : forall r,
+    correct r
+    -> forall c v, exec (bottomUp r c) v = exec c v.
+Proof.
+Admitted.
+
+Definition rBottomUp (r : rule) : rule.
+Admitted.
+
+Theorem rBottomUp_correct : forall r,
+    correct r
+    -> correct (rBottomUp r).
+Proof.
+Admitted.
+
+Definition zzz := Assign "x" (Plus (Plus (Plus (Var "x") (Const 0)) (Const 0)) (Const 0)).
+
+Compute bottomUp plus0 zzz.
+Compute bottomUp (rBottomUp plus0) zzz.
+Compute bottomUp (rBottomUp (rBottomUp plus0)) zzz.
+Compute bottomUp (rBottomUp (rBottomUp (rBottomUp plus0))) zzz.
+       
+
+
 (** * Motivating continuations with search problems *)
 
 Fixpoint allSublists {A} (ls : list A) : list (list A) :=
@@ -146,8 +431,6 @@ Fixpoint allSublists {A} (ls : list A) : list (list A) :=
 
 Compute allSublists [1; 2; 3].
 
-Definition sum ls := fold_left plus ls 0.
-
 Fixpoint sublistSummingTo (ns : list nat) (target : nat) : option (list nat) :=
   match filter (fun ns' => if sum ns' ==n target then true else false) (allSublists ns) with
   | ns' :: _ => Some ns'