| #a #H #p normalize @p @refl
] qed.
+(* dependent pair *)
+record DPair (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
+ dpi1:> A
+ ; dpi2: f dpi1
+ }.
+
+interpretation "DPair" 'dpair x = (DPair ? x).
+
+interpretation "mk_DPair" 'mk_DPair x y = (mk_DPair ?? x y).
+
(* sigma *)
record Sig (A:Type[0]) (f:A→Prop) : Type[0] ≝ {
pi1: A
interpretation "Sigma" 'sigma x = (Sig ? x).
-notation "hvbox(« term 19 a, break term 19 b»)"
-with precedence 90 for @{ 'dp $a $b }.
-
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] ≝ {
match ${fresh xy} return λx. ? = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
λ${ident E}.$s ] ] (refl ? $t) }.
-notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y,ident z〉 \nbsp'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y,ident z〉 \nbsp 'as' \nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
with precedence 10
-for @{ match $t return λ${ident x}.$eq $T $x $t → $U with [ mk_Prod (${fresh xy}:$V) (${ident z}:$Z) ⇒
- match ${fresh xy} return λ${ident y}. $eq $R $r $t → ? with [ mk_Prod (${ident x}:$L) (${ident y}:$I) ⇒
+for @{ match $t return λ${ident k}:$X.$eq $T $k $t → $U with [ mk_Prod (${ident xy}:$V) (${ident z}:$Z) ⇒
+ match $xy return λ${ident a}. $eq $R $r $t → ? with [ mk_Prod (${ident x}:$L) (${ident y}:$I) ⇒
λ${ident E}:$J.$s ] ] ($refl $A $t) }.
notation > "hvbox('let' 〈ident w,ident x,ident y,ident z〉 ≝ t 'in' s)"
∀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 ??)).