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+Generalizable All Variables.
+
+Open Scope type_scope.
+
+Section type_reification.
+
+Inductive term :Type :=
+   Fun : term -> term -> term
+ | Prod : term -> term -> term
+ | Bool : term
+ | SET :term
+ | PROP :term
+ | TYPE :term
+ | Var : Type -> term.
+
+Fixpoint interp (t:term) :=
+  match t with
+    Bool => bool
+  | SET => Set
+  | PROP => Prop
+  | TYPE => Type
+  | Fun a b => interp a -> interp b
+  | Prod a b => interp a * interp b
+  | Var x => x
+end.
+
+Class interp_pair (abs : Type) :=
+ { repr : term;
+  link: abs = interp repr }.
+
+Implicit Arguments repr [[interp_pair]].
+Implicit Arguments link [[interp_pair]].
+
+Lemma prod_interp `{interp_pair a, interp_pair b} : a * b = interp (Prod (repr a) (repr b)).
+  simpl. intros. rewrite <- link. rewrite <- (link b). reflexivity.
+Qed.
+
+Lemma fun_interp :forall `{interp_pair a, interp_pair b}, (a -> b) = interp (Fun (repr a) (repr b)).
+  simpl. intros. rewrite <- link. rewrite <- (link b). reflexivity.
+Qed.
+
+Coercion repr : interp_pair >-> term.
+
+Definition abs `{interp_pair a} : Type := a.
+Coercion abs : interp_pair >-> Sortclass.
+
+Lemma fun_interp' :forall `{ia : interp_pair, ib : interp_pair}, (ia -> ib) = interp (Fun ia ib).
+  simpl. intros a ia b ib. rewrite <- link. rewrite <- (link b). reflexivity.
+Qed.
+
+Instance ProdCan `(interp_pair a, interp_pair b) : interp_pair (a * b) :=
+  { repr := Prod (repr a) (repr b) ; link := prod_interp }.
+
+Instance FunCan `(interp_pair a, interp_pair b) : interp_pair (a -> b) :=
+  { link := fun_interp }.
+
+Instance BoolCan : interp_pair bool :=
+  { repr := Bool ; link := refl_equal _ }.
+
+Instance VarCan x : interp_pair x | 10 := { repr := Var x ; link := refl_equal _ }.
+Instance SetCan : interp_pair Set := { repr := SET ; link := refl_equal _ }.
+Instance PropCan : interp_pair Prop := { repr := PROP ; link := refl_equal _ }.
+Instance TypeCan : interp_pair Type := { repr := TYPE ; link := refl_equal _ }.
+
+(* Print Canonical Projections. *)
+
+Variable A:Type.
+
+Variable Inhabited: term -> Prop.
+
+Variable Inhabited_correct: forall `{interp_pair p}, Inhabited (repr p) -> p.
+
+Lemma L : Prop * A -> bool * (Type -> Set) .
+apply (Inhabited_correct _ _).
+change (Inhabited (Fun (Prod PROP (Var A)) (Prod Bool (Fun TYPE SET)))).
+Admitted.
+
+End type_reification.