Nat.thy

Deleted extra space in clos_mk.

(*  Title:      HOL/Nat.thy
    ID:         $Id$
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

Definition of types ind and nat.

Type nat is defined as a set Nat over type ind.
*)

Nat = WF +

(** type ind **)

types
  ind

arities
  ind :: term

consts
  Zero_Rep      :: "ind"
  Suc_Rep       :: "ind => ind"

rules
  (*the axiom of infinity in 2 parts*)
  inj_Suc_Rep           "inj(Suc_Rep)"
  Suc_Rep_not_Zero_Rep  "Suc_Rep(x) ~= Zero_Rep"



(** type nat **)

(* type definition *)

subtype (Nat)
  nat = "lfp(%X. {Zero_Rep} Un (Suc_Rep``X))"   (lfp_def)

instance
  nat :: ord


(* abstract constants and syntax *)

consts
  "0"           :: "nat"                ("0")
  Suc           :: "nat => nat"
  nat_case      :: "['a, nat => 'a, nat] => 'a"
  pred_nat      :: "(nat * nat) set"
  nat_rec       :: "[nat, 'a, [nat, 'a] => 'a] => 'a"

translations
  "case p of 0 => a | Suc(y) => b" == "nat_case(a, %y.b, p)"

defs
  Zero_def      "0 == Abs_Nat(Zero_Rep)"
  Suc_def       "Suc == (%n. Abs_Nat(Suc_Rep(Rep_Nat(n))))"

  (*nat operations and recursion*)
  nat_case_def  "nat_case(a, f, n) == @z.  (n=0 --> z=a)  \
\                                        & (!x. n=Suc(x) --> z=f(x))"
  pred_nat_def  "pred_nat == {p. ? n. p = <n, Suc(n)>}"

  less_def "m<n == <m,n>:trancl(pred_nat)"

  le_def   "m<=(n::nat) == ~(n<m)"

  nat_rec_def   "nat_rec(n, c, d) == wfrec(pred_nat, n,   \
\                        nat_case(%g.c, %m g. d(m, g(m))))"
end