definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
unification hint 0 ≔ ;
X ≟ mk_DeqSet bool beqb beqb_true
(* ---------------------------------------- *) ⊢
#b #H @(\P H).
qed.
+(* option *)
+
+definition eq_option ≝
+ λA:DeqSet.λa1,a2:option A.
+ match a1 with
+ [ None ⇒ match a2 with [None ⇒ true | _ ⇒ false]
+ | Some a1' ⇒ match a2 with [None ⇒ false | Some a2' ⇒ a1'==a2']].
+
+lemma eq_option_true: ∀A:DeqSet.∀a1,a2:option A.
+ eq_option A a1 a2 = true ↔ a1 = a2.
+#A *
+ [*
+ [% //
+ |#a1 % normalize #H destruct
+ ]
+ |#a1 *
+ [normalize % #H destruct
+ |#a2 normalize %
+ [#Heq >(\P Heq) //
+ |#H destruct @(\b ?) //
+ ]
+ ]
+qed.
+
+definition DeqOption ≝ λA:DeqSet.
+ mk_DeqSet (option A) (eq_option A) (eq_option_true A).
+
+unification hint 0 ≔ C;
+ T ≟ carr C,
+ X ≟ DeqOption C
+(* ---------------------------------------- *) ⊢
+ option T ≡ carr X.
+
+unification hint 0 ≔ T,a1,a2;
+ X ≟ DeqOption T
+(* ---------------------------------------- *) ⊢
+ eq_option T a1 a2 ≡ eqb X a1 a2.
+
+
(* pairs *)
definition eq_pairs ≝
λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
(* ---------------------------------------- *) ⊢
eq_sum T1 T2 p1 p2 ≡ eqb X p1 p2.
+(* sigma *)
+definition eq_sigma ≝
+ λA:DeqSet.λP:A→Prop.λp1,p2:Σx:A.P x.
+ match p1 with
+ [mk_Sig a1 h1 ⇒
+ match p2 with
+ [mk_Sig a2 h2 ⇒ a1==a2]].
+
+(* uses proof irrelevance *)
+lemma eq_sigma_true: ∀A:DeqSet.∀P.∀p1,p2:Σx.P x.
+ eq_sigma A P p1 p2 = true ↔ p1 = p2.
+#A #P * #a1 #Ha1 * #a2 #Ha2 %
+ [normalize #eqa generalize in match Ha1; >(\P eqa) //
+ |#H >H @(\b ?) //
+ ]
+qed.
+
+definition DeqSig ≝ λA:DeqSet.λP:A→Prop.
+ mk_DeqSet (Σx:A.P x) (eq_sigma A P) (eq_sigma_true A P).
+
+(*
+unification hint 0 ≔ C,P;
+ T ≟ carr C,
+ X ≟ DeqSig C P
+(* ---------------------------------------- *) ⊢
+ Σx:T.P x ≡ carr X.
+
+unification hint 0 ≔ T,P,p1,p2;
+ X ≟ DeqSig T P
+(* ---------------------------------------- *) ⊢
+ eq_sigma T P p1 p2 ≡ eqb X p1 p2.
+*)