+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "lambda/rc_sat.ma".
+
+(* HIGHER ORDER REDUCIBILITY CANDIDATES ***************************************)
+
+(* An arity is a type of λ→ to be used as carrier for a h.o. r.c. *)
+inductive ARITY: Type[0] ≝
+ | SORT: ARITY
+ | IMPL: ARITY → ARITY → ARITY
+.
+
+(* The type of the higher order r.c.'s having a given carrier.
+ * a h.o. r.c is implemented as an inductively defined metalinguistic function
+ * [ a CIC function in the present case ].
+ *)
+let rec HRC P ≝ match P with
+ [ SORT ⇒ RC
+ | IMPL Q P ⇒ HRC Q → HRC P
+ ].
+
+(* The default h.o r.c.
+ * This is needed to complete the partial interpretation of types.
+ *)
+let rec defHRC P ≝ match P return λP. HRC P with
+ [ SORT ⇒ snRC
+ | IMPL Q P ⇒ λ_. defHRC P
+ ].
+
+(* extensional equality *******************************************************)
+
+(* This is the extensional equalty of functions
+ * modulo the extensional equality on the domain.
+ * The functions may not respect extensional equality so reflexivity fails.
+ *)
+let rec hrceq P ≝ match P return λP. HRC P → HRC P → Prop with
+ [ SORT ⇒ λC1,C2. C1 ≅ C2
+ | IMPL Q P ⇒ λC1,C2. ∀B1,B2. hrceq Q B1 B2 → hrceq P (C1 B1) (C2 B2)
+ ].
+
+interpretation
+ "extensional equality (h.o. reducibility candidate)"
+ 'Eq1 P C1 C2 = (hrceq P C1 C2).
+
+lemma symmetric_hrceq: ∀P. symmetric ? (hrceq P).
+#P (elim P) -P /4/
+qed.
+
+lemma transitive_hrceq: ∀P. transitive ? (hrceq P).
+#P (elim P) -P /5/
+qed.
+
+lemma reflexive_defHRC: ∀P. defHRC P ≅^P defHRC P.
+#P (elim P) -P (normalize) /2/
+qed.