(* dependent pair *)
record DPair (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
- dpi1: A
+ dpi1:> A
; dpi2: f dpi1
}.
-interpretation "DPair" 'sigma x = (DPair ? x).
+interpretation "DPair" 'dpair x = (DPair ? x).
-notation "hvbox(« term 19 a, break term 19 b»)"
-with precedence 90 for @{ 'dp $a $b }.
-
-interpretation "mk_DPair" 'dp x y = (mk_DPair ?? x y).
+interpretation "mk_DPair" 'mk_DPair x y = (mk_DPair ?? x y).
(* sigma *)
record Sig (A:Type[0]) (f:A→Prop) : Type[0] ≝ {
interpretation "mk_Sig" 'dp x y = (mk_Sig ?? x y).
+lemma sub_pi2 : ∀A.∀P,P':A → Prop. (∀x.P x → P' x) → ∀x:Σx:A.P x. P' (pi1 … x).
+#A #P #P' #H1 * #x #H2 @H1 @H2
+qed.
+
(* Prod *)
record Prod (A,B:Type[0]) : Type[0] ≝ {
∀A,B.∀a:A.∀b:B.∀c. 〈a,b〉 = c → b = \snd c.
#A #B #a #b *; /2/
qed.
+
+lemma coerc_pair_sigma:
+ ∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x).
+#A #B #P * #a #b #p % [@a | /2/]
+qed.
+coercion coerc_pair_sigma:∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x)
+≝ coerc_pair_sigma on p: (? × ?) to (? × (Sig ??)).