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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "pts_dummy/rc_sat.ma".
17 (* HIGHER ORDER REDUCIBILITY CANDIDATES ***************************************)
19 (* An arity is a type of λ→ to be used as carrier for a h.o. r.c. *)
21 (* The type of the higher order r.c.'s having a given carrier.
22 * a h.o. r.c is implemented as an inductively defined metalinguistic function
23 * [ a CIC function in the present case ].
25 let rec HRC P ≝ match P with
27 | ABST Q P ⇒ HRC Q → HRC P
30 (* The default h.o r.c.
31 * This is needed to complete the partial interpretation of types.
33 let rec defHRC P ≝ match P return λP. HRC P with
35 | ABST Q P ⇒ λ_. defHRC P
38 (* extensional equality *******************************************************)
40 (* This is the extensional equalty of functions
41 * modulo the extensional equality on the domain.
42 * The functions may not respect extensional equality so reflexivity fails.
44 let rec hrceq P ≝ match P return λP. HRC P → HRC P → Prop with
45 [ SORT ⇒ λC1,C2. C1 ≅ C2
46 | ABST Q P ⇒ λC1,C2. ∀B1,B2. hrceq Q B1 B2 → hrceq P (C1 B1) (C2 B2)
50 "extensional equality (h.o. reducibility candidate)"
51 'Eq1 P C1 C2 = (hrceq P C1 C2).
53 lemma symmetric_hrceq: ∀P. symmetric ? (hrceq P).
57 lemma transitive_hrceq: ∀P. transitive ? (hrceq P).
61 lemma reflexive_defHRC: ∀P. defHRC P ≅^P defHRC P.
62 #P (elim P) -P (normalize) /2/