interpretation "Pair construction" 'pair x y =
(cic:/matita/datatypes/constructors/Prod.ind#xpointer(1/1/1) _ _ x y).
-notation "hvbox(\langle x break , y \rangle )" with precedence 89
-for @{ 'pair $x $y}.
-
interpretation "Product" 'product x y =
(cic:/matita/datatypes/constructors/Prod.ind#xpointer(1/1) x y).
-notation "hvbox(x break \times y)" with precedence 89
-for @{ 'product $x $y}.
-
definition fst \def \lambda A,B:Type.\lambda p: Prod A B.
match p with
[(pair a b) \Rightarrow a].
match p with
[(pair a b) \Rightarrow b].
-interpretation "First projection" 'fst x =
- (cic:/matita/datatypes/constructors/fst.con _ _ x).
-
-notation "\fst x" with precedence 89
-for @{ 'fst $x}.
-
-interpretation "Second projection" 'snd x =
- (cic:/matita/datatypes/constructors/snd.con _ _ x).
-
-notation "\snd x" with precedence 89
-for @{ 'snd $x}.
+interpretation "pair pi1" 'pi1 = (fst _ _).
+interpretation "pair pi2" 'pi2 = (snd _ _).
+interpretation "pair pi1" 'pi1a x = (fst _ _ x).
+interpretation "pair pi2" 'pi2a x = (snd _ _ x).
+interpretation "pair pi1" 'pi1b x y = (fst _ _ x y).
+interpretation "pair pi2" 'pi2b x y = (snd _ _ x y).
theorem eq_pair_fst_snd: \forall A,B:Type.\forall p:Prod A B.
-p = 〈 (\fst p), (\snd p) 〉.
+p = 〈 \fst p, \snd p 〉.
intros.elim p.simplify.reflexivity.
qed.