src/HOL/Nat.thy
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Wed, 29 Nov 1995 16:44:59 +0100
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(*  Title:      HOL/Nat.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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Definition of types ind and nat.
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Type nat is defined as a set Nat over type ind.
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*)
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Nat = WF +
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(** type ind **)
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types
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  ind
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arities
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  ind :: term
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consts
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  Zero_Rep      :: ind
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  Suc_Rep       :: ind => ind
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rules
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  (*the axiom of infinity in 2 parts*)
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  inj_Suc_Rep           "inj(Suc_Rep)"
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  Suc_Rep_not_Zero_Rep  "Suc_Rep(x) ~= Zero_Rep"
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(** type nat **)
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(* type definition *)
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subtype (Nat)
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  nat = "lfp(%X. {Zero_Rep} Un (Suc_Rep``X))"   (lfp_def)
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instance
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  nat :: ord
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(* abstract constants and syntax *)
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consts
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  "0"           :: nat                ("0")
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  Suc           :: nat => nat
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  nat_case      :: ['a, nat => 'a, nat] => 'a
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  pred_nat      :: "(nat * nat) set"
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  nat_rec       :: [nat, 'a, [nat, 'a] => 'a] => 'a
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translations
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  "case p of 0 => a | Suc(y) => b" == "nat_case a (%y.b) p"
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defs
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  Zero_def      "0 == Abs_Nat(Zero_Rep)"
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  Suc_def       "Suc == (%n. Abs_Nat(Suc_Rep(Rep_Nat(n))))"
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  (*nat operations and recursion*)
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  nat_case_def  "nat_case a f n == @z.  (n=0 --> z=a)  
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                                        & (!x. n=Suc(x) --> z=f(x))"
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  pred_nat_def  "pred_nat == {p. ? n. p = (n, Suc(n))}"
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  less_def "m<n == (m,n):trancl(pred_nat)"
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  le_def   "m<=(n::nat) == ~(n<m)"
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  nat_rec_def   "nat_rec n c d == wfrec pred_nat n  
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                        (nat_case (%g.c) (%m g.(d m (g m))))"
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end