arithm example

examples/logic/Arithm.gf

@@ -0,0 +1,64 @@
+abstract Arithm = Logic ** {
+
+-- arithmetic
+fun 
+  Nat, Real  : Dom ;
+data
+  Zero : Elem Nat ;
+  Succ : Elem Nat -> Elem Nat ;
+fun
+  trunc : Elem Real -> Elem Nat ;
+
+  EqNat : (m,n : Elem Nat) -> Prop ;
+  LtNat : (m,n : Elem Nat) -> Prop ;
+  Div   : (m,n : Elem Nat) -> Prop ;
+  Even  : Elem Nat -> Prop ;
+  Odd   : Elem Nat -> Prop ;
+  Prime : Elem Nat -> Prop ;
+
+  one  : Elem Nat ;
+  two  : Elem Nat ;
+  sum  : (m,n : Elem Nat) -> Elem Nat ;
+  prod : (m,n : Elem Nat) -> Elem Nat ;
+data
+  evax1 : Proof (Even Zero) ;
+  evax2 : (n : Elem Nat) -> Proof (Even n) -> Proof (Odd (Succ n)) ;
+  evax3 : (n : Elem Nat) -> Proof (Odd n)  -> Proof (Even (Succ n)) ;
+  eqax1 : Proof (EqNat Zero Zero) ;
+  eqax2 : (m,n : Elem Nat) -> Proof (EqNat m n) -> Proof (EqNat (Succ m) (Succ n)) ;
+fun
+  IndNat : (C : Elem Nat -> Prop) -> 
+             Proof (C Zero) -> 
+             ((x : Elem Nat) -> Hypo (C x) -> Proof (C (Succ x))) -> 
+                Proof (Univ Nat C) ;
+
+def 
+  one = Succ Zero ;
+  two = Succ one ;
+  sum m (Succ n) = Succ (sum m n) ;
+  sum m Zero = m ;
+  prod m (Succ n) = sum (prod m n) m ;
+  prod m Zero = Zero ;
+  LtNat m n = Exist Nat (\x -> EqNat n (sum m (Succ x))) ;
+  Div m n = Exist Nat (\x -> EqNat m (prod x n)) ;
+  Prime n = Conj 
+              (LtNat one n) 
+              (Univ Nat (\x -> Impl (Conj (LtNat one x) (Div n x)) (EqNat x n))) ;
+
+fun ex1 : Text ;
+def ex1 = 
+  ThmWithProof 
+    (Univ Nat (\x -> Disj (Even x) (Odd x))) 
+    (IndNat 
+       (\x -> Disj (Even x) (Odd x)) 
+       (DisjIl (Even Zero) (Odd Zero) evax1) 
+       (\x -> \h -> DisjE (Even x) (Odd x) (Disj (Even (Succ x)) (Odd (Succ x))) 
+         (Hypoth (Disj (Even x) (Odd x)) h) 
+         (\a -> DisjIr (Even (Succ x)) (Odd (Succ x)) 
+            (evax2 x (Hypoth (Even x) a))) 
+         (\b -> DisjIl (Even (Succ x)) (Odd (Succ x)) 
+            (evax3 x (Hypoth (Odd x) b))
+         )
+       )
+     ) ;
+} ;