inductive empty : Set \def .
-let rec pos2nat (x:pos) : nat \def
+let rec pos2nat x \def
match x with
[ one \Rightarrow (S O)
| (next z) \Rightarrow S (pos2nat z)].
-let rec nat2int (x:nat) :int \def
- match x with
- [ O \Rightarrow positive O
- | (S z) \Rightarrow positive (S z)].
+definition nat2int \def \lambda x. positive x.
coercion pos2nat.
coercion nat2int.
-let rec plus x y : int \def
- match x with
- [ (positive n) \Rightarrow x
- | (negative z) \Rightarrow y].
-
-theorem a: plus O one.
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+definition fst \def \lambda x,y:int.x.
+alias symbol "eq" (instance 0) = "leibnitz's equality".
+theorem a: fst O one = fst (positive O) (next one).
+reflexivity.
+qed.