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+(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
+  * Chapter 16: Deriving Programs from Specifications
+  * Author: Adam Chlipala
+  * License: https://creativecommons.org/licenses/by-nc-nd/4.0/
+  * Some material borrowed from Fiat <http://plv.csail.mit.edu/fiat/> *)
+
+Require Import FrapWithoutSets.
+Require Import Program Setoids.Setoid Classes.RelationClasses Classes.Morphisms Morphisms_Prop.
+
+
+(** * The computation monad *)
+
+Definition comp (A : Type) := A -> Prop.
+
+Definition ret {A} (x : A) : comp A := eq x.
+Definition bind {A B} (c1 : comp A) (c2 : A -> comp B) : comp B :=
+  fun b => exists a, c1 a /\ c2 a b.
+Definition pick_ {A} (P : A -> Prop) : comp A := P.
+
+Definition refine {A} (c1 c2 : comp A) :=
+  forall a, c2 a -> c1 a.
+
+Ltac morphisms := unfold refine, impl, pointwise_relation, bind; hnf; first_order.
+
+Global Instance refine_PreOrder A : PreOrder (@refine A).
+Proof.
+  constructor; morphisms.
+Qed.
+
+Add Parametric Morphism A : (@refine A)
+    with signature (@refine A) --> (@refine A) ++> impl
+      as refine_refine.
+Proof.
+  morphisms.
+Qed.
+
+Add Parametric Morphism A B : (@bind A B)
+    with signature (@refine A) 
+                     ==> (pointwise_relation _ (@refine B))
+                     ==> (@refine B)
+    as refine_bind.
+Proof.
+  morphisms.
+Qed.
+
+Add Parametric Morphism A B : (@bind A B)
+    with signature (flip (@refine A))
+                     ==> (pointwise_relation _ (flip (@refine B)))
+                     ==> (flip (@refine B))
+      as refine_bind_flip.
+Proof.
+  morphisms.
+Qed.
+
+Theorem bind_ret : forall A B (v : A) (c2 : A -> comp B),
+    refine (bind (ret v) c2) (c2 v).
+Proof.
+  morphisms.
+Qed.
+
+Notation "'pick' x 'where' P" := (pick_ (fun x => P)) (at level 80, x at level 0).
+Notation "x <- c1 ; c2" := (bind c1 (fun x => c2)) (at level 81, right associativity).
+
+
+(** * Picking a number not in a list *)
+
+(* A specification of what it means to choose a number that is not in a
+ * particular list *)
+Definition notInList (ls : list nat) :=
+  pick n where ~In n ls.
+
+(* We can use a simple property to justify a decomposition of the original
+ * spec. *)
+Theorem notInList_decompose : forall ls,
+  refine (notInList ls) (upper <- pick upper where forall n, In n ls -> upper >= n;
+                         pick beyond where beyond > upper).
+Proof.
+  simplify.
+  unfold notInList, refine, bind, pick_, not.
+  first_order.
+  apply H in H0.
+  linear_arithmetic.
+Qed.
+
+(* A simple traversal will find the maximum list element, which is a good upper
+ * bound. *)
+Definition listMax := fold_right max 0.
+
+(* ...and we can prove it! *)
+Theorem listMax_upperBound : forall init ls,
+  forall n, In n ls -> fold_right max init ls >= n.
+Proof.
+  induct ls; simplify; propositional.
+  linear_arithmetic.
+  apply IHls in H0.
+  linear_arithmetic.
+Qed.
+
+(* Now we restate that result as a computation refinement. *)
+Theorem listMax_refines : forall ls,
+  refine (pick upper where forall n, In n ls -> upper >= n) (ret (listMax ls)).
+Proof.
+  unfold refine, pick_, ret; simplify; subst.
+  apply listMax_upperBound; assumption.
+Qed.
+
+(* An easy way to find a number higher than another: add 1! *)
+Theorem increment_refines : forall n,
+  refine (pick higher where higher > n) (ret (n + 1)).
+Proof.
+  unfold refine, pick_, ret; simplify; subst.
+  linear_arithmetic.
+Qed.
+
+Ltac begin :=
+  eexists; simplify;
+    (* We run this next step to hide an evar, so that rewriting isn't too eager to
+     * make up values for it. *)
+    match goal with
+    | [ |- refine _ (?f _) ] => set f
+    end.
+
+Ltac finish :=
+  match goal with
+  | [ |- refine ?e (?f ?arg) ] =>
+    let g := eval pattern arg in e in
+    match g with
+    | ?g' _ =>
+      let f' := eval unfold f in f in
+      unify f' g'; reflexivity
+    end
+  end.
+
+(* Let's derive an efficient implementation. *)
+Theorem implementation : { f : list nat -> comp nat | forall ls, refine (notInList ls) (f ls) }.
+Proof.
+  begin.
+  rewrite notInList_decompose.
+  rewrite listMax_refines.
+  setoid_rewrite increment_refines.
+  (* ^-- Different tactic here to let us rewrite under a binder! *)
+  rewrite bind_ret.
+  finish.
+Defined.
+
+(* We can extract the program that we found as a standlone, executable Gallina
+ * term. *)
+Definition impl := Eval simpl in proj1_sig implementation.
+Print impl.
+
+(* We'll temporarily expose the definition of [max], so we can compute neatly
+ * here. *)
+Transparent max.
+Eval compute in impl (1 :: 7 :: 8 :: 2 :: 13 :: 6 :: nil).