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more in ArithmEng

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aarne
2006-11-27 21:03:15 +00:00
parent 8cd9a329fa
commit 0c5f2c1288
5 changed files with 39 additions and 26 deletions
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16
examples/logic/Arithm.gf
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@@ -2,13 +2,11 @@ abstract Arithm = Logic ** {
-- arithmetic
fun
Nat, Real : Dom ;
Nat : 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 ;
@@ -24,8 +22,10 @@ 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)) ;
eqax2 : (m,n : Elem Nat) -> Proof (EqNat m n) ->
Proof (EqNat (Succ m) (Succ n)) ;
fun
IndNat : (C : Elem Nat -> Prop) ->
Proof (C Zero) ->
@@ -41,9 +41,9 @@ def
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))) ;
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 =
@@ -52,7 +52,7 @@ def ex1 =
(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)))
(\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)))