doc-src/IsarAdvanced/Classes/Thy/code_examples/Classes.hs

 module Classes where {
 
-import qualified Integer;
-import qualified Nat;
+
+data Nat = Zero_nat | Suc Nat;
+
+data Bit = B0 | B1;
+
+nat_aux :: Integer -> Nat -> Nat;
+nat_aux i n = (if i <= 0 then n else nat_aux (i - 1) (Suc n));
+
+nat :: Integer -> Nat;
+nat i = nat_aux i Zero_nat;
 
 class Semigroup a where {
   mult :: a -> a -> a;
 };
 
-class (Classes.Semigroup a) => Monoidl a where {
+class (Semigroup a) => Monoidl a where {
   neutral :: a;
 };
 
-class (Classes.Monoidl a) => Group a where {
+class (Monoidl a) => Group a where {
   inverse :: a -> a;
 };
 
-inverse_int :: Integer.Inta -> Integer.Inta;
-inverse_int i = Integer.uminus_int i;
+inverse_int :: Integer -> Integer;
+inverse_int i = negate i;
 
-neutral_int :: Integer.Inta;
-neutral_int = Integer.Number_of_int Integer.Pls;
+neutral_int :: Integer;
+neutral_int = 0;
 
-mult_int :: Integer.Inta -> Integer.Inta -> Integer.Inta;
-mult_int i j = Integer.plus_int i j;
+mult_int :: Integer -> Integer -> Integer;
+mult_int i j = i + j;
 
-instance Classes.Semigroup Integer.Inta where {
-  mult = Classes.mult_int;
+instance Semigroup Integer where {
+  mult = mult_int;
 };
 
-instance Classes.Monoidl Integer.Inta where {
-  neutral = Classes.neutral_int;
+instance Monoidl Integer where {
+  neutral = neutral_int;
 };
 
-instance Classes.Group Integer.Inta where {
-  inverse = Classes.inverse_int;
+instance Group Integer where {
+  inverse = inverse_int;
 };
 
-pow_nat :: (Classes.Monoidl b) => Nat.Nat -> b -> b;
-pow_nat (Nat.Suc n) x = Classes.mult x (Classes.pow_nat n x);
-pow_nat Nat.Zero_nat x = Classes.neutral;
+pow_nat :: (Group b) => Nat -> b -> b;
+pow_nat (Suc n) x = mult x (pow_nat n x);
+pow_nat Zero_nat x = neutral;
 
-pow_int :: (Classes.Group a) => Integer.Inta -> a -> a;
+pow_int :: (Group a) => Integer -> a -> a;
 pow_int k x =
-  (if Integer.less_eq_int (Integer.Number_of_int Integer.Pls) k
-    then Classes.pow_nat (Integer.nat k) x
-    else Classes.inverse
-           (Classes.pow_nat (Integer.nat (Integer.uminus_int k)) x));
+  (if 0 <= k then pow_nat (nat k) x
+    else inverse (pow_nat (nat (negate k)) x));
 
-example :: Integer.Inta;
-example =
-  Classes.pow_int
-    (Integer.Number_of_int
-      (Integer.Bit
-        (Integer.Bit
-          (Integer.Bit (Integer.Bit Integer.Pls Integer.B1) Integer.B0)
-          Integer.B1)
-        Integer.B0))
-    (Integer.Number_of_int (Integer.Bit Integer.Min Integer.B0));
+example :: Integer;
+example = pow_int 10 (-2);
 
 }