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