notation "\b H" non associative with precedence 90
for @{(proj2 … (eqb_true ???) $H)}.
+notation < "𝐃" non associative with precedence 90
+ for @{'bigD}.
+interpretation "DeqSet" 'bigD = (mk_DeqSet ???).
+
lemma eqb_false: ∀S:DeqSet.∀a,b:S.
(eqb ? a b) = false ↔ a ≠ b.
#S #a #b % #H
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
(* ---------------------------------------- *) ⊢
example hint2: ∀b1,b2.
〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.
-#b1 #b2 #H @(\P H).
\ No newline at end of file
+#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.
+