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(* Title: FOLP/ex/nat.thy
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ID: $Id$
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1477
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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 manual "Introduction to Isabelle"
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Theory of the natural numbers: Peano's axioms, primitive recursion
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*)
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Nat = IFOLP +
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353
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types nat
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0
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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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(*Proof terms*)
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nrec :: "[nat,p,[nat,p]=>p]=>p"
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ninj,nneq :: "p=>p"
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rec0, recSuc :: "p"
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rules
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1149
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induct "[| b:P(0); !!x u. u:P(x) ==> c(x,u):P(Suc(x))
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|] ==> nrec(n,b,c):P(n)"
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0
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Suc_inject "p:Suc(m)=Suc(n) ==> ninj(p) : m=n"
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Suc_neq_0 "p:Suc(m)=0 ==> nneq(p) : R"
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rec_0 "rec0 : rec(0,a,f) = a"
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rec_Suc "recSuc : 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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nrecB0 "b: A ==> nrec(0,b,c) = b : A"
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nrecBSuc "c(n,nrec(n,b,c)) : A ==> nrec(Suc(n),b,c) = c(n,nrec(n,b,c)) : A"
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end
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