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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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352
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types nat
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arities nat :: term
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1322
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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
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