ExtLib.Tactics.Parametric

Require Import Setoid.
Require Import RelationClasses.
Require Import Morphisms.

Set Implicit Arguments.
Set Strict Implicit.

The purpose of this tactic is to try to automatically derive morphisms for functions

Theorem Proper_red : forall T U (rT : relation T) (rU : relation U) (f : T -> U),
  (forall x x', rT x x' -> rU (f x) (f x')) ->
  Proper (rT ==> rU) f.
intuition.
Qed.

Theorem respectful_red : forall T U (rT : relation T) (rU : relation U) (f g : T -> U),
  (forall x x', rT x x' -> rU (f x) (g x')) ->
  respectful rT rU f g.
intuition.
Qed.
Theorem respectful_if_bool T : forall (x x' : bool) (t t' f f' : T) eqT,
  x = x' ->
  eqT t t' -> eqT f f' ->
  eqT (if x then t else f) (if x' then t' else f') .
intros; subst; auto; destruct x'; auto.
Qed.

Ltac derive_morph :=
repeat
  first [ lazymatch goal with
          | |- Proper _ _ => red; intros
          | |- (_ ==> _)%signature _ _ => red; intros
          end
        | apply respectful_red; intros
        | apply respectful_if_bool; intros
        | match goal with
          | [ H : (_ ==> ?EQ)%signature ?F ?F' |- ?EQ (?F _) (?F' _) ] =>
            apply H
          | [ |- ?EQ (?F _) (?F _) ] =>
            let inst := constr:(_ : Proper (_ ==> EQ) F) in
            apply inst
          | [ H : (_ ==> _ ==> ?EQ)%signature ?F ?F' |- ?EQ (?F _ _) (?F' _ _) ] =>
            apply H
          | [ |- ?EQ (?F _ _) (?F' _ _) ] =>
            let inst := constr:(_ : Proper (_ ==> _ ==> EQ) F) in
            apply inst
          | [ |- ?EQ (?F _ _ _) (?F _ _ _) ] =>
            let inst := constr:(_ : Proper (_ ==> _ ==> _ ==> EQ) F) in
            apply inst
          | [ |- ?EQ (?F _) (?F _) ] => unfold F
          | [ |- ?EQ (?F _ _) (?F _ _) ] => unfold F
          | [ |- ?EQ (?F _ _ _) (?F _ _ _) ] => unfold F
          end ].

Global Instance Proper_andb : Proper (@eq bool ==> @eq bool ==> @eq bool) andb.
derive_morph; auto.
Qed.

Section K.
  Variable F : bool -> bool -> bool.
  Hypothesis Fproper : Proper (@eq bool ==> @eq bool ==> @eq bool) F.
  Existing Instance Fproper.

  Definition food (x y z : bool) : bool :=
    F x (F y z).

  Global Instance Proper_food : Proper (@eq bool ==> @eq bool ==> @eq bool ==> @eq bool) food.
  Proof.
    derive_morph; auto.
  Qed.

  Global Instance Proper_S : Proper (@eq nat ==> @eq nat) S.
  Proof.
    derive_morph; auto.
  Qed.
End K.

Require Import List.

Section Map.
  Variable T : Type.
  Variable eqT : relation T.
  Inductive listEq {T} (eqT : relation T) : relation (list T) :=
  | listEq_nil : listEq eqT nil nil
  | listEq_cons : forall x x' y y', eqT x x' -> listEq eqT y y' ->listEq eqT (x :: y) (x' :: y').

  Theorem listEq_match V U (eqV : relation V) (eqU : relation U) : forall x x' : list V,
    forall X X' Y Y',
    eqU X X' ->
    (eqV ==> listEq eqV ==> eqU)%signature Y Y' ->
    listEq eqV x x' ->
    eqU (match x with
           | nil => X
           | x :: xs => Y x xs
         end)
        (match x' with
           | nil => X'
           | x :: xs => Y' x xs
         end).
  Proof.
    intros. induction H1; auto. derive_morph; auto.
  Qed.

  Variable U : Type.
  Variable eqU : relation U.
  Variable f : T -> U.
  Variable fproper : Proper (eqT ==> eqU) f.

  Definition hd (l : list T) : option T :=
    match l with
      | nil => None
      | l :: _ => Some l
    end.

(*
  Global Instance Proper_hd : Proper (listEq eqT ==> optionEq eqT) hd.
  Proof.
    foo. (** This has binders in the match... **)

  Abort.
*)


  Fixpoint map' (l : list T) : list U :=
    match l with
      | nil => nil
      | l :: ls => f l :: map' ls
    end.

  Global Instance Proper_map' : Proper (listEq eqT ==> listEq eqU) map'.
  Proof.
    derive_morph. induction H; econstructor; derive_morph; auto.
  Qed.
End Map.