FirstClassFunctions: CPS versions of classics

FirstClassFunctions.v

@@ -5,8 +5,6 @@
 
 Require Import Frap.
 
-Set Implicit Arguments.
-
 
 (** * Classic list functions *)
 
@@ -28,7 +26,7 @@ Fixpoint foldl {A B : Set} (f : A -> B -> B) (acc : B) (ls : list A) : B :=
   | x :: ls' => foldl f (f x acc) ls'
   end.
 
-Record programming_languages := {
+Record programming_language := {
   Name : string;
   PurelyFunctional : bool;
   AppearedInYear : nat
@@ -71,3 +69,111 @@ Compute map Name (filter PurelyFunctional languages).
 Compute foldl max 0 (map AppearedInYear languages).
 Compute foldl max 0 (map AppearedInYear (filter PurelyFunctional languages)).
 
+
+(** * The classics in continuation-passing style *)
+
+Fixpoint mapK {A B R : Set} (f : A -> (B -> R) -> R) (ls : list A) (k : list B -> R) : R :=
+  match ls with
+  | nil => k nil
+  | x :: ls' => f x (fun x' => mapK f ls' (fun ls'' => k (x' :: ls'')))
+  end.
+
+Fixpoint filterK {A R : Set} (f : A -> (bool -> R) -> R) (ls : list A) (k : list A -> R) : R :=
+  match ls with
+  | nil => k nil
+  | x :: ls' => f x (fun b => filterK f ls' (fun ls'' => k (if b then x :: ls'' else ls'')))
+  end.
+
+Fixpoint foldlK {A B R : Set} (f : A -> B -> (B -> R) -> R) (acc : B) (ls : list A) (k : B -> R) : R :=
+  match ls with
+  | nil => k acc
+  | x :: ls' => f x acc (fun x' => foldlK f x' ls' k)
+  end.
+
+Definition NameK {R : Set} (l : programming_language) (k : string -> R) : R :=
+  k (Name l).
+Definition PurelyFunctionalK {R : Set} (l : programming_language) (k : bool -> R) : R :=
+  k (PurelyFunctional l).
+Definition AppearedInYearK {R : Set} (l : programming_language) (k : nat -> R) : R :=
+  k (AppearedInYear l).
+Definition maxK {R : Set} (n1 n2 : nat) (k : nat -> R) : R :=
+  k (max n1 n2).
+
+Compute mapK NameK languages (fun ls => ls).
+Compute filterK PurelyFunctionalK languages (fun ls => mapK NameK ls (fun x => x)).
+Compute mapK AppearedInYearK languages (fun ls => foldlK maxK 0 ls (fun x => x)).
+Compute filterK PurelyFunctionalK languages
+        (fun ls1 => mapK AppearedInYearK ls1
+                         (fun ls2 => foldlK maxK 0 ls2 (fun x => x))).
+
+Theorem mapK_ok : forall {A B R : Set} (f : A -> (B -> R) -> R) (f_base : A -> B),
+    (forall x k, f x k = k (f_base x))
+    -> forall (ls : list A) (k : list B -> R),
+      mapK f ls k = k (map f_base ls).
+Proof.
+  induct ls; simplify; try equality.
+
+  rewrite H.
+  apply IHls.
+Qed.
+
+Theorem names_ok : forall langs,
+    mapK NameK langs (fun ls => ls) = map Name langs.
+Proof.
+  simplify.
+  apply mapK_ok with (f_base := Name).
+  unfold NameK.
+  trivial.
+Qed.
+
+Theorem filterK_ok : forall {A R : Set} (f : A -> (bool -> R) -> R) (f_base : A -> bool),
+    (forall x k, f x k = k (f_base x))
+    -> forall (ls : list A) (k : list A -> R),
+      filterK f ls k = k (filter f_base ls).
+Proof.
+  induct ls; simplify; try equality.
+
+  rewrite H.
+  apply IHls.
+Qed.
+
+Theorem purenames_ok : forall langs,
+    filterK PurelyFunctionalK langs (fun ls => mapK NameK ls (fun x => x))
+    = map Name (filter PurelyFunctional langs).
+Proof.
+  simplify.
+  rewrite filterK_ok with (f_base := PurelyFunctional); trivial.
+  apply mapK_ok with (f_base := Name); trivial.
+Qed.
+
+Theorem foldlK_ok : forall {A B R : Set} (f : A -> B -> (B -> R) -> R) (f_base : A -> B -> B),
+    (forall x acc k, f x acc k = k (f_base x acc))
+    -> forall (ls : list A) (acc : B) (k : B -> R),
+      foldlK f acc ls k = k (foldl f_base acc ls).
+Proof.
+  induct ls; simplify; try equality.
+
+  rewrite H.
+  apply IHls.
+Qed.
+
+Theorem latest_ok : forall langs,
+    mapK AppearedInYearK langs (fun ls => foldlK maxK 0 ls (fun x => x))
+    = foldl max 0 (map AppearedInYear langs).
+Proof.
+  simplify.
+  rewrite mapK_ok with (f_base := AppearedInYear); trivial.
+  apply foldlK_ok with (f_base := max); trivial.
+Qed.
+
+Theorem latestpure_ok : forall langs,
+    filterK PurelyFunctionalK langs
+            (fun ls1 => mapK AppearedInYearK ls1
+                             (fun ls2 => foldlK maxK 0 ls2 (fun x => x)))
+    = foldl max 0 (map AppearedInYear (filter PurelyFunctional langs)).
+Proof.
+  simplify.
+  rewrite filterK_ok with (f_base := PurelyFunctional); trivial.
+  rewrite mapK_ok with (f_base := AppearedInYear); trivial.
+  apply foldlK_ok with (f_base := max); trivial.
+Qed.

_CoqProject

@@ -17,6 +17,7 @@ DataAbstraction.v
 DataAbstraction_template.v
 Interpreters_template.v
 Interpreters.v
+FirstClassFunctions.v
 TransitionSystems_template.v
 TransitionSystems.v
 IntroToProofScripting.v