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(* Title: FOL/ex/Nat.thy

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ID: $Id$

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory

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Copyright 1992 University of Cambridge


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Examples for the manuals.

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Theory of the natural numbers: Peano's axioms, primitive recursion


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*)


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Nat = FOL +

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types nat


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arities nat :: term

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consts "0" :: nat ("0")


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Suc :: nat=>nat


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rec :: [nat, 'a, [nat,'a]=>'a] => 'a


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"+" :: [nat, nat] => nat (infixl 60)

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rules induct "[ P(0); !!x. P(x) ==> P(Suc(x)) ] ==> P(n)"


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Suc_inject "Suc(m)=Suc(n) ==> m=n"


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Suc_neq_0 "Suc(m)=0 ==> R"


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rec_0 "rec(0,a,f) = a"


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rec_Suc "rec(Suc(m), a, f) = f(m, rec(m,a,f))"


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add_def "m+n == rec(m, n, %x y. Suc(y))"


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end
