+#b1 #b2 #H @(\P H)
+qed.
+
+(* sum *)
+definition eq_sum ≝
+ λA,B:DeqSet.λp1,p2:A+B.
+ match p1 with
+ [ inl a1 ⇒ match p2 with
+ [ inl a2 ⇒ a1 == a2 | inr b2 ⇒ false ]
+ | inr b1 ⇒ match p2 with
+ [ inl a2 ⇒ false | inr b2 ⇒ b1 == b2 ]
+ ].
+
+lemma eq_sum_true: ∀A,B:DeqSet.∀p1,p2:A+B.
+ eq_sum A B p1 p2 = true ↔ p1 = p2.
+#A #B *
+ [#a1 *
+ [#a2 normalize %
+ [#eqa >(\P eqa) // | #H destruct @(\b ?) //]
+ |#b2 normalize % #H destruct
+ ]
+ |#b1 *
+ [#a2 normalize % #H destruct
+ |#b2 normalize %
+ [#eqb >(\P eqb) // | #H destruct @(\b ?) //]
+ ]
+ ]
+qed.
+
+definition DeqSum ≝ λA,B:DeqSet.
+ mk_DeqSet (A+B) (eq_sum A B) (eq_sum_true A B).
+
+unification hint 0 ≔ C1,C2;
+ T1 ≟ carr C1,
+ T2 ≟ carr C2,
+ X ≟ DeqSum C1 C2
+(* ---------------------------------------- *) ⊢
+ T1+T2 ≡ carr X.
+
+unification hint 0 ≔ T1,T2,p1,p2;
+ X ≟ DeqSum T1 T2
+(* ---------------------------------------- *) ⊢
+ 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.
+*)