(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/datatypes/constructors/".
include "logic/equality.ma".
inductive void : Set \def.
-inductive Prod (A,B:Set) : Set \def
+inductive unit : Set ≝ something: unit.
+
+inductive Prod (A,B:Type) : Type \def
pair : A \to B \to Prod A B.
-definition fst \def \lambda A,B:Set.\lambda p: Prod A B.
+interpretation "Pair construction" 'pair x y =
+ (cic:/matita/datatypes/constructors/Prod.ind#xpointer(1/1/1) _ _ x y).
+
+interpretation "Product" 'product x y =
+ (cic:/matita/datatypes/constructors/Prod.ind#xpointer(1/1) x y).
+
+definition fst \def \lambda A,B:Type.\lambda p: Prod A B.
match p with
[(pair a b) \Rightarrow a].
-definition snd \def \lambda A,B:Set.\lambda p: Prod A B.
+definition snd \def \lambda A,B:Type.\lambda p: Prod A B.
match p with
[(pair a b) \Rightarrow b].
-theorem eq_pair_fst_snd: \forall A,B:Set.\forall p: Prod A B.
-p = pair A B (fst A B p) (snd A B p).
+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 〉.
intros.elim p.simplify.reflexivity.
qed.
-inductive Sum (A,B:Set) : Set \def
+inductive Sum (A,B:Type) : Type \def
inl : A \to Sum A B
| inr : B \to Sum A B.
-inductive ProdT (A,B:Type) : Type \def
-pairT : A \to B \to ProdT A B.
+interpretation "Disjoint union" 'plus A B =
+ (cic:/matita/datatypes/constructors/Sum.ind#xpointer(1/1) A B).
-definition fstT \def \lambda A,B:Type.\lambda p: ProdT A B.
-match p with
-[(pairT a b) \Rightarrow a].
-
-definition sndT \def \lambda A,B:Type.\lambda p: ProdT A B.
-match p with
-[(pairT a b) \Rightarrow b].
\ No newline at end of file
+inductive option (A:Type) : Type ≝
+ None : option A
+ | Some : A → option A.