propagating the arithmetics library, partial commit

[helm.git] / matita / matita / lib / basics / types.ma

diff --git a/matita/matita/lib/basics/types.ma b/matita/matita/lib/basics/types.ma
index 1dcd79b7a49892702ba1290f86f3cb0099d3e335..efc213f95da759cb305ef3d1522ee13691de78e0 100644 (file)
--- a/matita/matita/lib/basics/types.ma
+++ b/matita/matita/lib/basics/types.ma
@@ -9,6 +9,7 @@
      \ /   GNU General Public License Version 2   
       V_______________________________________________________________ *)
 
+include "basics/core_notation/pair_2.ma".
 include "basics/logic.ma".
 
 (* void *)
@@ -67,7 +68,7 @@ interpretation "mk_DPair" 'mk_DPair x y = (mk_DPair ?? x y).
 
 (* sigma *)
 record Sig (A:Type[0]) (f:A→Prop) : Type[0] ≝ {
-    pi1: A
+    pi1: A  (* not a coercion due to problems with Cerco *)
   ; pi2: f pi1
   }.
   
@@ -79,6 +80,9 @@ lemma sub_pi2 : ∀A.∀P,P':A → Prop. (∀x.P x → P' x) → ∀x:Σx:A.P x.
 #A #P #P' #H1 * #x #H2 @H1 @H2
 qed.
 
+lemma inj_mk_Sig: ∀A,P.∀x. x = mk_Sig A P (pi1 A P x) (pi2 A P x).
+#A #P #x cases x //
+qed-. 
 (* Prod *)
 
 record Prod (A,B:Type[0]) : Type[0] ≝ {
@@ -145,10 +149,10 @@ for @{ match $t return λx.x = $t → ? with [ mk_Prod ${fresh xy} ${ident z} 
        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)"